From 48c607603526df27749930ed286f8ad48617ba66 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Mon, 8 Sep 2025 07:55:06 +0200 Subject: [PATCH] update of book --- .../_build/.doctrees/environment.pickle | Bin 305380 -> 308882 bytes .../_build/.doctrees/week37.doctree | Bin 260633 -> 305906 bytes .../_build/html/_sources/week37.ipynb | 726 +++++++++++++----- doc/LectureNotes/_build/html/searchindex.js | 2 +- doc/LectureNotes/_build/html/week37.html | 243 ++++++ .../_build/jupyter_execute/week37.ipynb | 726 +++++++++++++----- doc/LectureNotes/week37.ipynb | 726 +++++++++++++----- 7 files changed, 1888 insertions(+), 535 deletions(-) diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle index 47056f27ef274f2c18436810a0d2ecfd84cf28fa..c6c70ce1d8ee744a699671998b3086fa9aad308c 100644 GIT binary patch delta 13327 zcmZ`=d3=q>*EjP_awCzD#S$dKrB#B6+PB0`D)v2=E4d_BB$0(!6Z=gBsV5Azq_lRb 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"a24010ae", + "id": "a5769b3d", "metadata": { "editable": true }, @@ -52,7 +52,7 @@ }, { "cell_type": "markdown", - "id": "4a291d59", + "id": "21f2937f", "metadata": { "editable": true }, @@ -69,7 +69,7 @@ }, { "cell_type": "markdown", - "id": "85c747e2", + "id": "da32a24e", "metadata": { "editable": true }, @@ -79,7 +79,7 @@ }, { "cell_type": "markdown", - "id": "6580dfe2", + "id": "2b7f8433", "metadata": { "editable": true }, @@ -103,7 +103,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "c2ddcfe5", + "id": "2a8c5baa", "metadata": { "collapsed": false, "editable": true @@ -117,7 +117,7 @@ }, { "cell_type": "markdown", - "id": "e1e8a5b2", + "id": "47d8423d", "metadata": { "editable": true }, @@ -128,7 +128,7 @@ }, { "cell_type": "markdown", - "id": "c8a5100b", + "id": "08d37b91", "metadata": { "editable": true }, @@ -140,7 +140,7 @@ }, { "cell_type": "markdown", - "id": "b026883e", + "id": "a3c412ef", "metadata": { "editable": true }, @@ -150,7 +150,7 @@ }, { "cell_type": "markdown", - "id": "3a2f7b75", + "id": "3f1b2071", "metadata": { "editable": true }, @@ -162,7 +162,7 @@ }, { "cell_type": "markdown", - "id": "6380eed5", + "id": "07536cbd", "metadata": { "editable": true }, @@ -176,7 +176,7 @@ }, { "cell_type": "markdown", - "id": "c5d3766d", + "id": "34287135", "metadata": { "editable": true }, @@ -192,7 +192,7 @@ }, { "cell_type": "markdown", - "id": "1d313807", + "id": "c1063bfb", "metadata": { "editable": true }, @@ -202,7 +202,7 @@ }, { "cell_type": "markdown", - "id": "bee64882", + "id": "d327d2e3", "metadata": { "editable": true }, @@ -214,7 +214,7 @@ }, { "cell_type": "markdown", - "id": "7ffe8d02", + "id": "b2c9a9cd", "metadata": { "editable": true }, @@ -224,7 +224,7 @@ }, { "cell_type": "markdown", - "id": "97225362", + "id": "062a7534", "metadata": { "editable": true }, @@ -236,7 +236,7 @@ }, { "cell_type": "markdown", - "id": "9fe2a0b3", + "id": "b1b15536", "metadata": { "editable": true }, @@ -250,7 +250,7 @@ }, { "cell_type": "markdown", - "id": "2e678439", + "id": "a259e250", "metadata": { "editable": true }, @@ -260,7 +260,7 @@ }, { "cell_type": "markdown", - "id": "5f45e358", + "id": "002197c1", "metadata": { "editable": true }, @@ -271,7 +271,7 @@ }, { "cell_type": "markdown", - "id": "1713ee43", + "id": "55e8dca9", "metadata": { "editable": true }, @@ -286,7 +286,7 @@ }, { "cell_type": "markdown", - "id": "671ea0fc", + "id": "a97ffaec", "metadata": { "editable": true }, @@ -296,7 +296,7 @@ }, { "cell_type": "markdown", - "id": "7df56d17", + "id": "b59a4220", "metadata": { "editable": true }, @@ -308,7 +308,7 @@ }, { "cell_type": "markdown", - "id": "5887c657", + "id": "ba5dcc08", "metadata": { "editable": true }, @@ -320,7 +320,7 @@ }, { "cell_type": "markdown", - "id": "5a012ac0", + "id": "d8907fed", "metadata": { "editable": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "markdown", - "id": "cf1fd4f4", + "id": "728d5b78", "metadata": { "editable": true }, @@ -348,7 +348,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "4417d3aa", + "id": "02d7e401", "metadata": { "collapsed": false, "editable": true @@ -407,7 +407,7 @@ }, { "cell_type": "markdown", - "id": "7d39d005", + "id": "4dd147c2", "metadata": { "editable": true }, @@ -419,7 +419,7 @@ }, { "cell_type": "markdown", - "id": "45a85d32", + "id": "75ca7f80", "metadata": { "editable": true }, @@ -431,7 +431,7 @@ }, { "cell_type": "markdown", - "id": "31d267ea", + "id": "5b897c75", "metadata": { "editable": true }, @@ -441,7 +441,7 @@ }, { "cell_type": "markdown", - "id": "f8f50b02", + "id": "46aa12f6", "metadata": { "editable": true }, @@ -455,7 +455,7 @@ }, { "cell_type": "markdown", - "id": "ac21d44c", + "id": "8ac05816", "metadata": { "editable": true }, @@ -465,7 +465,7 @@ }, { "cell_type": "markdown", - "id": "aae5aaa1", + "id": "cee76d94", "metadata": { "editable": true }, @@ -477,7 +477,7 @@ }, { "cell_type": "markdown", - "id": "319922a5", + "id": "88cf9577", "metadata": { "editable": true }, @@ -488,7 +488,7 @@ }, { "cell_type": "markdown", - "id": "724078a1", + "id": "0108d67e", "metadata": { "editable": true }, @@ -503,7 +503,7 @@ }, { "cell_type": "markdown", - "id": "dbc443e3", + "id": "1e307469", "metadata": { "editable": true }, @@ -517,7 +517,7 @@ }, { "cell_type": "markdown", - "id": "2ea2bf50", + "id": "c8dc7485", "metadata": { "editable": true }, @@ -528,7 +528,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "9f431da1", + "id": "2909407a", "metadata": { "collapsed": false, "editable": true @@ -589,7 +589,7 @@ }, { "cell_type": "markdown", - "id": "8aa155a9", + "id": "d25693ff", "metadata": { "editable": true }, @@ -611,7 +611,7 @@ }, { "cell_type": "markdown", - "id": "03bd2e44", + "id": "78b0bf65", "metadata": { "editable": true }, @@ -626,7 +626,7 @@ }, { "cell_type": "markdown", - "id": "0e101e2d", + "id": "9ee803d8", "metadata": { "editable": true }, @@ -637,7 +637,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "09ecede4", + "id": "ac420f7a", "metadata": { "collapsed": false, "editable": true @@ -703,7 +703,7 @@ }, { "cell_type": "markdown", - "id": "3489dbbc", + "id": "c548d574", "metadata": { "editable": true }, @@ -714,7 +714,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "426eaa39", + "id": "687e9d89", "metadata": { "collapsed": false, "editable": true @@ -788,7 +788,7 @@ }, { "cell_type": "markdown", - "id": "6220214d", + "id": "27a27a67", "metadata": { "editable": true }, @@ -807,7 +807,7 @@ }, { "cell_type": "markdown", - "id": "bf86ac65", + "id": "a12c19b2", "metadata": { "editable": true }, @@ -828,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "4ac61edb", + "id": "b8436434", "metadata": { "editable": true }, @@ -844,7 +844,7 @@ }, { "cell_type": "markdown", - "id": "0058008d", + "id": "f00e1864", "metadata": { "editable": true }, @@ -858,7 +858,7 @@ }, { "cell_type": "markdown", - "id": "f994e1e2", + "id": "ea91af30", "metadata": { "editable": true }, @@ -886,7 +886,7 @@ }, { "cell_type": "markdown", - "id": "842a8611", + "id": "bc9502a0", "metadata": { "editable": true }, @@ -918,7 +918,7 @@ }, { "cell_type": "markdown", - "id": "90bd121a", + "id": "6a236a2a", "metadata": { "editable": true }, @@ -935,7 +935,7 @@ }, { "cell_type": "markdown", - "id": "5cd81303", + "id": "29dc562b", "metadata": { "editable": true }, @@ -948,7 +948,7 @@ }, { "cell_type": "markdown", - "id": "60e085a9", + "id": "6a34f155", "metadata": { "editable": true }, @@ -961,7 +961,7 @@ }, { "cell_type": "markdown", - "id": "fef0100e", + "id": "0afd8cd8", "metadata": { "editable": true }, @@ -974,7 +974,7 @@ }, { "cell_type": "markdown", - "id": "aaba7f05", + "id": "f0b27e71", "metadata": { "editable": true }, @@ -988,7 +988,7 @@ }, { "cell_type": "markdown", - "id": "038b47ae", + "id": "3b04b9c6", "metadata": { "editable": true }, @@ -1010,7 +1010,7 @@ }, { "cell_type": "markdown", - "id": "0ad42833", + "id": "05eca708", "metadata": { "editable": true }, @@ -1025,7 +1025,7 @@ }, { "cell_type": "markdown", - "id": "64b15ba2", + "id": "473025f4", "metadata": { "editable": true }, @@ -1037,7 +1037,7 @@ }, { "cell_type": "markdown", - "id": "49c6adb0", + "id": "26e0b288", "metadata": { "editable": true }, @@ -1050,7 +1050,7 @@ }, { "cell_type": "markdown", - "id": "82873545", + "id": "091efee5", "metadata": { "editable": true }, @@ -1064,7 +1064,7 @@ }, { "cell_type": "markdown", - "id": "35a8e70d", + "id": "22c5f80e", "metadata": { "editable": true }, @@ -1075,7 +1075,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "6aa32b90", + "id": "102b1658", "metadata": { "collapsed": false, "editable": true @@ -1100,7 +1100,7 @@ }, { "cell_type": "markdown", - "id": "6e20f534", + "id": "79448e46", "metadata": { "editable": true }, @@ -1116,7 +1116,7 @@ }, { "cell_type": "markdown", - "id": "71745d3e", + "id": "dbc8b940", "metadata": { "editable": true }, @@ -1137,7 +1137,7 @@ }, { "cell_type": "markdown", - "id": "bad95be2", + "id": "b63ae18d", "metadata": { "editable": true }, @@ -1157,7 +1157,7 @@ }, { "cell_type": "markdown", - "id": "40b4d87e", + "id": "c5ee074e", "metadata": { "editable": true }, @@ -1176,7 +1176,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "1208bbec", + "id": "cfc48413", "metadata": { "collapsed": false, "editable": true @@ -1211,7 +1211,7 @@ }, { "cell_type": "markdown", - "id": "b83b5ed1", + "id": "fbb8c0eb", "metadata": { "editable": true }, @@ -1224,7 +1224,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "1f669db6", + "id": "cc1e51cd", "metadata": { "collapsed": false, "editable": true @@ -1301,7 +1301,7 @@ }, { "cell_type": "markdown", - "id": "3e9ed564", + "id": "22a23ea0", "metadata": { "editable": true }, @@ -1316,7 +1316,377 @@ }, { "cell_type": "markdown", - "id": "9c0ac318", + "id": "f0258497", + "metadata": { + "editable": true + }, + "source": [ + "## SGD vs Full-Batch GD: Convergence Speed and Memory Comparison" + ] + }, + { + "cell_type": "markdown", + "id": "69f1d941", + "metadata": { + "editable": true + }, + "source": [ + "### Theoretical Convergence Speed and convex optimization\n", + "\n", + "Consider minimizing an empirical cost function" + ] + }, + { + "cell_type": "markdown", + "id": "876e1d2b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\theta) =\\frac{1}{N}\\sum_{i=1}^N l_i(\\theta),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "67381fd3", + "metadata": { + "editable": true + }, + "source": [ + "where each $l_i(\\theta)$ is a\n", + "differentiable loss term. Gradient Descent (GD) updates parameters\n", + "using the full gradient $\\nabla C(\\theta)$, while Stochastic Gradient\n", + "Descent (SGD) uses a single sample (or mini-batch) gradient $\\nabla\n", + "l_i(\\theta)$ selected at random. In equation form, one GD step is:" + ] + }, + { + "cell_type": "markdown", + "id": "f08ec530", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_{t+1} = \\theta_t-\\eta \\nabla C(\\theta_t) =\\theta_t -\\eta \\frac{1}{N}\\sum_{i=1}^N \\nabla l_i(\\theta_t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3d99c5ee", + "metadata": { + "editable": true + }, + "source": [ + "whereas one SGD step is:" + ] + }, + { + "cell_type": "markdown", + "id": "03ecb7f9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_{t+1} = \\theta_t -\\eta \\nabla l_{i_t}(\\theta_t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a6db0c6c", + "metadata": { + "editable": true + }, + "source": [ + "with $i_t$ randomly chosen. On smooth convex problems, GD and SGD both\n", + "converge to the global minimum, but their rates differ. GD can take\n", + "larger, more stable steps since it uses the exact gradient, achieving\n", + "an error that decreases on the order of $O(1/t)$ per iteration for\n", + "convex objectives (and even exponentially fast for strongly convex\n", + "cases). In contrast, plain SGD has more variance in each step, leading\n", + "to sublinear convergence in expectation – typically $O(1/\\sqrt{t})$\n", + "for general convex objectives (\\thetaith appropriate diminishing step\n", + "sizes) . Intuitively, GD’s trajectory is smoother and more\n", + "predictable, while SGD’s path oscillates due to noise but costs far\n", + "less per iteration, enabling many more updates in the same time." + ] + }, + { + "cell_type": "markdown", + "id": "167f76aa", + "metadata": { + "editable": true + }, + "source": [ + "### Strongly Convex Case\n", + "\n", + "If $C(\\theta)$ is strongly convex and $L$-smooth (so GD enjoys linear\n", + "convergence), the gap $C(\\theta_t)-C(\\theta^*)$ for GD shrinks as" + ] + }, + { + "cell_type": "markdown", + "id": "d8d5cb23", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\theta_t) - C(\\theta^* ) \\le \\Big(1 - \\frac{\\mu}{L}\\Big)^t [C(\\theta_0)-C(\\theta^*)],\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e127c141", + "metadata": { + "editable": true + }, + "source": [ + "a geometric (linear) convergence per iteration . Achieving an\n", + "$\\epsilon$-accurate solution thus takes on the order of\n", + "$\\log(1/\\epsilon)$ iterations for GD. However, each GD iteration costs\n", + "$O(N)$ gradient evaluations. SGD cannot exploit strong convexity to\n", + "obtain a linear rate – instead, with a properly decaying step size\n", + "(e.g. $\\eta_t = \\frac{1}{\\mu t}$) or iterate averaging, SGD attains an\n", + "$O(1/t)$ convergence rate in expectation . For example, one result\n", + "of Moulines and Bach 2011, see shows that with $\\eta_t = \\Theta(1/t)$," + ] + }, + { + "cell_type": "markdown", + "id": "7ea2c03c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathbb{E}[C(\\theta_t) - C(\\theta^*)] = O(1/t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e2e33c54", + "metadata": { + "editable": true + }, + "source": [ + "for strongly convex, smooth $F$ . This $1/t$ rate is slower per\n", + "iteration than GD’s exponential decay, but each SGD iteration is $N$\n", + "times cheaper. In fact, to reach error $\\epsilon$, plain SGD needs on\n", + "the order of $T=O(1/\\epsilon)$ iterations (sub-linear convergence),\n", + "while GD needs $O(\\log(1/\\epsilon))$ iterations. When accounting for\n", + "cost-per-iteration, GD requires $O(N \\log(1/\\epsilon))$ total gradient\n", + "computations versus SGD’s $O(1/\\epsilon)$ single-sample\n", + "computations. In large-scale regimes (huge $N$), SGD can be\n", + "faster in wall-clock time because $N \\log(1/\\epsilon)$ may far exceed\n", + "$1/\\epsilon$ for reasonable accuracy levels. In other words,\n", + "with millions of data points, one epoch of GD (one full gradient) is\n", + "extremely costly, whereas SGD can make $N$ cheap updates in the time\n", + "GD makes one – often yielding a good solution faster in practice, even\n", + "though SGD’s asymptotic error decays more slowly. As one lecture\n", + "succinctly puts it: “SGD can be super effective in terms of iteration\n", + "cost and memory, but SGD is slow to converge and can’t adapt to strong\n", + "convexity” . Thus, the break-even point depends on $N$ and the desired\n", + "accuracy: for moderate accuracy on very large $N$, SGD’s cheaper\n", + "updates win; for extremely high precision (very small $\\epsilon$) on a\n", + "modest $N$, GD’s fast convergence per step can be advantageous." + ] + }, + { + "cell_type": "markdown", + "id": "88f943e6", + "metadata": { + "editable": true + }, + "source": [ + "### Non-Convex Problems\n", + "\n", + "In non-convex optimization (e.g. deep neural networks), neither GD nor\n", + "SGD guarantees global minima, but SGD often displays faster progress\n", + "in finding useful minima. Theoretical results here are weaker, usually\n", + "showing convergence to a stationary point $\\theta$ ($|\\nabla C|$ is\n", + "small) in expectation. For example, GD might require $O(1/\\epsilon^2)$\n", + "iterations to ensure $|\\nabla C(\\theta)| < \\epsilon$, and SGD typically has\n", + "similar polynomial complexity (often worse due to gradient\n", + "noise). However, a noteworthy difference is that SGD’s stochasticity\n", + "can help escape saddle points or poor local minima. Random gradient\n", + "fluctuations act like implicit noise, helping the iterate “jump” out\n", + "of flat saddle regions where full-batch GD could stagnate . In fact,\n", + "research has shown that adding noise to GD can guarantee escaping\n", + "saddle points in polynomial time, and the inherent noise in SGD often\n", + "serves this role. Empirically, this means SGD can sometimes find a\n", + "lower loss basin faster, whereas full-batch GD might get “stuck” near\n", + "saddle points or need a very small learning rate to navigate complex\n", + "error surfaces . Overall, in modern high-dimensional machine learning,\n", + "SGD (or mini-batch SGD) is the workhorse for large non-convex problems\n", + "because it converges to good solutions much faster in practice,\n", + "despite the lack of a linear convergence guarantee. Full-batch GD is\n", + "rarely used on large neural networks, as it would require tiny steps\n", + "to avoid divergence and is extremely slow per iteration ." + ] + }, + { + "cell_type": "markdown", + "id": "aa4a8927", + "metadata": { + "editable": true + }, + "source": [ + "## Memory Usage and Scalability\n", + "\n", + "A major advantage of SGD is its memory efficiency in handling large\n", + "datasets. Full-batch GD requires access to the entire training set for\n", + "each iteration, which often means the whole dataset (or a large\n", + "subset) must reside in memory to compute $\\nabla C(\\theta)$ . This results\n", + "in memory usage that scales linearly with the dataset size $N$. For\n", + "instance, if each training sample is large (e.g. high-dimensional\n", + "features), computing a full gradient may require storing a substantial\n", + "portion of the data or all intermediate gradients until they are\n", + "aggregated. In contrast, SGD needs only a single (or a small\n", + "mini-batch of) training example(s) in memory at any time . The\n", + "algorithm processes one sample (or mini-batch) at a time and\n", + "immediately updates the model, discarding that sample before moving to\n", + "the next. This streaming approach means that memory footprint is\n", + "essentially independent of $N$ (apart from storing the model\n", + "parameters themselves). As one source notes, gradient descent\n", + "“requires more memory than SGD” because it “must store the entire\n", + "dataset for each iteration,” whereas SGD “only needs to store the\n", + "current training example” . In practical terms, if you have a dataset\n", + "of size, say, 1 million examples, full-batch GD would need memory for\n", + "all million every step, while SGD could be implemented to load just\n", + "one example at a time – a crucial benefit if data are too large to fit\n", + "in RAM or GPU memory. This scalability makes SGD suitable for\n", + "large-scale learning: as long as you can stream data from disk, SGD\n", + "can handle arbitrarily large datasets with fixed memory. In fact, SGD\n", + "“does not need to remember which examples were visited” in the past,\n", + "allowing it to run in an online fashion on infinite data streams\n", + ". Full-batch GD, on the other hand, would require multiple passes\n", + "through a giant dataset per update (or a complex distributed memory\n", + "system), which is often infeasible.\n", + "\n", + "There is also a secondary memory effect: computing a full-batch\n", + "gradient in deep learning requires storing all intermediate\n", + "activations for backpropagation across the entire batch. A very large\n", + "batch (approaching the full dataset) might exhaust GPU memory due to\n", + "the need to hold activation gradients for thousands or millions of\n", + "examples simultaneously. SGD/minibatches mitigate this by splitting\n", + "the workload – e.g. with a mini-batch of size 32 or 256, memory use\n", + "stays bounded, whereas a full-batch (size = $N$) forward/backward pass\n", + "could not even be executed if $N$ is huge. Techniques like gradient\n", + "accumulation exist to simulate large-batch GD by summing many\n", + "small-batch gradients – but these still process data in manageable\n", + "chunks to avoid memory overflow. In summary, memory complexity for GD\n", + "grows with $N$, while for SGD it remains $O(1)$ w.r.t. dataset size\n", + "(only the model and perhaps a mini-batch reside in memory) . This is a\n", + "key reason why batch GD “does not scale” to very large data and why\n", + "virtually all large-scale machine learning algorithms rely on\n", + "stochastic or mini-batch methods." + ] + }, + { + "cell_type": "markdown", + "id": "72d0192b", + "metadata": { + "editable": true + }, + "source": [ + "## Empirical Evidence: Convergence Time and Memory in Practice\n", + "\n", + "Empirical studies strongly support the theoretical trade-offs\n", + "above. In large-scale machine learning tasks, SGD often converges to a\n", + "good solution much faster in wall-clock time than full-batch GD, and\n", + "it uses far less memory. For example, Bottou & Bousquet (2008)\n", + "analyzed learning time under a fixed computational budget and\n", + "concluded that when data is abundant, it’s better to use a faster\n", + "(even if less precise) optimization method to process more examples in\n", + "the same time . This analysis showed that for large-scale problems,\n", + "processing more data with SGD yields lower error than spending the\n", + "time to do exact (batch) optimization on fewer data . In other words,\n", + "if you have a time budget, it’s often optimal to accept slightly\n", + "slower convergence per step (as with SGD) in exchange for being able\n", + "to use many more training samples in that time. This phenomenon is\n", + "borne out by experiments:" + ] + }, + { + "cell_type": "markdown", + "id": "44fcd423", + "metadata": { + "editable": true + }, + "source": [ + "### Deep Neural Networks\n", + "\n", + "In modern deep learning, full-batch GD is so slow that it is rarely\n", + "attempted; instead, mini-batch SGD is standard. A recent study\n", + "demonstrated that it is possible to train a ResNet-50 on ImageNet\n", + "using full-batch gradient descent, but it required careful tuning\n", + "(e.g. gradient clipping, tiny learning rates) and vast computational\n", + "resources – and even then, each full-batch update was extremely\n", + "expensive.\n", + "\n", + "Using a huge batch\n", + "(closer to full GD) tends to slow down convergence if the learning\n", + "rate is not scaled up, and often encounters optimization difficulties\n", + "(plateaus) that small batches avoid.\n", + "Empirically, small or medium\n", + "batch SGD finds minima in fewer clock hours because it can rapidly\n", + "loop over the data with gradient noise aiding exploration." + ] + }, + { + "cell_type": "markdown", + "id": "8de0942f", + "metadata": { + "editable": true + }, + "source": [ + "### Memory constraints\n", + "\n", + "From a memory standpoint, practitioners note that batch GD becomes\n", + "infeasible on large data. For example, if one tried to do full-batch\n", + "training on a dataset that doesn’t fit in RAM or GPU memory, the\n", + "program would resort to heavy disk I/O or simply crash. SGD\n", + "circumvents this by processing mini-batches. Even in cases where data\n", + "does fit in memory, using a full batch can spike memory usage due to\n", + "storing all gradients. One empirical observation is that mini-batch\n", + "training has a “lower, fluctuating usage pattern” of memory, whereas\n", + "full-batch loading “quickly consumes memory (often exceeding limits)”\n", + ". This is especially relevant for graph neural networks or other\n", + "models where a “batch” may include a huge chunk of a graph: full-batch\n", + "gradient computation can exhaust GPU memory, whereas mini-batch\n", + "methods keep memory usage manageable .\n", + "\n", + "In summary, SGD converges faster than full-batch GD in terms of actual\n", + "training time for large-scale problems, provided we measure\n", + "convergence as reaching a good-enough solution. Theoretical bounds\n", + "show SGD needs more iterations, but because it performs many more\n", + "updates per unit time (and requires far less memory), it often\n", + "achieves lower loss in a given time frame than GD. Full-batch GD might\n", + "take slightly fewer iterations in theory, but each iteration is so\n", + "costly that it is “slower… especially for large datasets” . Meanwhile,\n", + "memory scaling strongly favors SGD: GD’s memory cost grows with\n", + "dataset size, making it impractical beyond a point, whereas SGD’s\n", + "memory use is modest and mostly constant w.r.t. $N$ . These\n", + "differences have made SGD (and mini-batch variants) the de facto\n", + "choice for training large machine learning models, from logistic\n", + "regression on millions of examples to deep neural networks with\n", + "billions of parameters. The consensus in both research and practice is\n", + "that for large-scale or high-dimensional tasks, SGD-type methods\n", + "converge quicker per unit of computation and handle memory constraints\n", + "better than standard full-batch gradient descent ." + ] + }, + { + "cell_type": "markdown", + "id": "f08a4bbe", "metadata": { "editable": true }, @@ -1347,7 +1717,7 @@ }, { "cell_type": "markdown", - "id": "d8f518c4", + "id": "dc1fa30f", "metadata": { "editable": true }, @@ -1369,7 +1739,7 @@ }, { "cell_type": "markdown", - "id": "3dcb89bd", + "id": "1fbfcb5e", "metadata": { "editable": true }, @@ -1389,7 +1759,7 @@ }, { "cell_type": "markdown", - "id": "8f258bc2", + "id": "83d5dfc2", "metadata": { "editable": true }, @@ -1405,7 +1775,7 @@ }, { "cell_type": "markdown", - "id": "2a3715f8", + "id": "4cf425f2", "metadata": { "editable": true }, @@ -1425,7 +1795,7 @@ }, { "cell_type": "markdown", - "id": "a1d9578a", + "id": "a8de083c", "metadata": { "editable": true }, @@ -1437,7 +1807,7 @@ }, { "cell_type": "markdown", - "id": "b6b5bc5e", + "id": "f8b98ecd", "metadata": { "editable": true }, @@ -1449,7 +1819,7 @@ }, { "cell_type": "markdown", - "id": "44b313c8", + "id": "c41121c9", "metadata": { "editable": true }, @@ -1461,7 +1831,7 @@ }, { "cell_type": "markdown", - "id": "b56c85b9", + "id": "0c9cde87", "metadata": { "editable": true }, @@ -1473,7 +1843,7 @@ }, { "cell_type": "markdown", - "id": "5bcc6bd2", + "id": "9079853e", "metadata": { "editable": true }, @@ -1484,7 +1854,7 @@ }, { "cell_type": "markdown", - "id": "41fc9f01", + "id": "1b2340aa", "metadata": { "editable": true }, @@ -1496,7 +1866,7 @@ }, { "cell_type": "markdown", - "id": "8151719b", + "id": "1c63eff7", "metadata": { "editable": true }, @@ -1506,7 +1876,7 @@ }, { "cell_type": "markdown", - "id": "bb75b0ad", + "id": "e05e89e4", "metadata": { "editable": true }, @@ -1518,7 +1888,7 @@ }, { "cell_type": "markdown", - "id": "3c71fd46", + "id": "b3cbe567", "metadata": { "editable": true }, @@ -1528,7 +1898,7 @@ }, { "cell_type": "markdown", - "id": "1d835a18", + "id": "5d2f1096", "metadata": { "editable": true }, @@ -1549,7 +1919,7 @@ }, { "cell_type": "markdown", - "id": "77dcc8c3", + "id": "4c4f3846", "metadata": { "editable": true }, @@ -1562,7 +1932,7 @@ }, { "cell_type": "markdown", - "id": "21161d57", + "id": "57d24251", "metadata": { "editable": true }, @@ -1574,7 +1944,7 @@ }, { "cell_type": "markdown", - "id": "e87e09a9", + "id": "caff3ad3", "metadata": { "editable": true }, @@ -1591,7 +1961,7 @@ }, { "cell_type": "markdown", - "id": "1a98c681", + "id": "67133da8", "metadata": { "editable": true }, @@ -1607,7 +1977,7 @@ }, { "cell_type": "markdown", - "id": "8b337277", + "id": "d2d2d644", "metadata": { "editable": true }, @@ -1629,7 +1999,7 @@ }, { "cell_type": "markdown", - "id": "af77b83f", + "id": "897d1ca3", "metadata": { "editable": true }, @@ -1647,7 +2017,7 @@ }, { "cell_type": "markdown", - "id": "bc924f77", + "id": "549532b3", "metadata": { "editable": true }, @@ -1667,7 +2037,7 @@ }, { "cell_type": "markdown", - "id": "86e5ab5e", + "id": "f014a3a2", "metadata": { "editable": true }, @@ -1681,7 +2051,7 @@ }, { "cell_type": "markdown", - "id": "949f359d", + "id": "67bed63f", "metadata": { "editable": true }, @@ -1693,7 +2063,7 @@ }, { "cell_type": "markdown", - "id": "0ba26be3", + "id": "3014fe59", "metadata": { "editable": true }, @@ -1705,7 +2075,7 @@ }, { "cell_type": "markdown", - "id": "4fb9b2a2", + "id": "a99d9c1c", "metadata": { "editable": true }, @@ -1717,7 +2087,7 @@ }, { "cell_type": "markdown", - "id": "8711e597", + "id": "907f9915", "metadata": { "editable": true }, @@ -1729,7 +2099,7 @@ }, { "cell_type": "markdown", - "id": "49e6e73d", + "id": "551eb7db", "metadata": { "editable": true }, @@ -1740,7 +2110,7 @@ }, { "cell_type": "markdown", - "id": "ca5bb491", + "id": "ea8ae470", "metadata": { "editable": true }, @@ -1752,7 +2122,7 @@ }, { "cell_type": "markdown", - "id": "5e19d7bf", + "id": "9f5d78fd", "metadata": { "editable": true }, @@ -1766,7 +2136,7 @@ }, { "cell_type": "markdown", - "id": "f79d952e", + "id": "8291642f", "metadata": { "editable": true }, @@ -1777,7 +2147,7 @@ }, { "cell_type": "markdown", - "id": "13e9862f", + "id": "ee0e74ec", "metadata": { "editable": true }, @@ -1789,7 +2159,7 @@ }, { "cell_type": "markdown", - "id": "5693500e", + "id": "3699f7e5", "metadata": { "editable": true }, @@ -1811,7 +2181,7 @@ }, { "cell_type": "markdown", - "id": "65a5e1e7", + "id": "16cdd781", "metadata": { "editable": true }, @@ -1835,7 +2205,7 @@ }, { "cell_type": "markdown", - "id": "27686255", + "id": "779881a9", "metadata": { "editable": true }, @@ -1851,7 +2221,7 @@ }, { "cell_type": "markdown", - "id": "f3dfc1e2", + "id": "0724c747", "metadata": { "editable": true }, @@ -1867,7 +2237,7 @@ }, { "cell_type": "markdown", - "id": "045d399c", + "id": "02a113cb", "metadata": { "editable": true }, @@ -1881,7 +2251,7 @@ }, { "cell_type": "markdown", - "id": "4e75ee41", + "id": "e013ce1a", "metadata": { "editable": true }, @@ -1899,7 +2269,7 @@ }, { "cell_type": "markdown", - "id": "ddbb28ab", + "id": "1baa5b4e", "metadata": { "editable": true }, @@ -1920,7 +2290,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "dae38b6c", + "id": "8aab54bd", "metadata": { "collapsed": false, "editable": true @@ -1980,7 +2350,7 @@ }, { "cell_type": "markdown", - "id": "ca5a343a", + "id": "26c6e4f1", "metadata": { "editable": true }, @@ -1991,7 +2361,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "08d97c1e", + "id": "226c13ec", "metadata": { "collapsed": false, "editable": true @@ -2055,7 +2425,7 @@ }, { "cell_type": "markdown", - "id": "727d8fc3", + "id": "6895dbfe", "metadata": { "editable": true }, @@ -2070,7 +2440,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "4e41c003", + "id": "f8d01982", "metadata": { "collapsed": false, "editable": true @@ -2154,7 +2524,7 @@ }, { "cell_type": "markdown", - "id": "fe00db52", + "id": "cffe8367", "metadata": { "editable": true }, @@ -2165,7 +2535,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "8f22105b", + "id": "b57871a9", "metadata": { "collapsed": false, "editable": true @@ -2243,7 +2613,7 @@ }, { "cell_type": "markdown", - "id": "8956bf7a", + "id": "37b273c1", "metadata": { "editable": true }, @@ -2256,7 +2626,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "044275ef", + "id": "9ff0eb69", "metadata": { "collapsed": false, "editable": true @@ -2300,7 +2670,7 @@ }, { "cell_type": "markdown", - "id": "353b50b3", + "id": "e7d143b6", "metadata": { "editable": true }, @@ -2311,7 +2681,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "fdc8debd", + "id": "9b5e2d1d", "metadata": { "collapsed": false, "editable": true @@ -2370,7 +2740,7 @@ }, { "cell_type": "markdown", - "id": "b738f1b8", + "id": "8b6fb13f", "metadata": { "editable": true }, @@ -2380,7 +2750,7 @@ }, { "cell_type": "markdown", - "id": "65ce93ba", + "id": "06c3f4bb", "metadata": { "editable": true }, @@ -2391,7 +2761,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "604d7286", + "id": "4abf9ccd", "metadata": { "collapsed": false, "editable": true @@ -2456,7 +2826,7 @@ }, { "cell_type": "markdown", - "id": "e663a714", + "id": "18d42e29", "metadata": { "editable": true }, @@ -2467,7 +2837,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "749fa687", + "id": "03415114", "metadata": { "collapsed": false, "editable": true @@ -2537,7 +2907,7 @@ }, { "cell_type": "markdown", - "id": "8801fcd5", + "id": "41120d8f", "metadata": { "editable": true }, @@ -2554,7 +2924,7 @@ }, { "cell_type": "markdown", - "id": "8ea68725", + "id": "16eb2a88", "metadata": { "editable": true }, @@ -2582,7 +2952,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "04811786", + "id": "a6df3a5c", "metadata": { "collapsed": false, "editable": true @@ -2602,7 +2972,7 @@ }, { "cell_type": "markdown", - "id": "b0e7cc2c", + "id": "fc15d89b", "metadata": { "editable": true }, @@ -2618,7 +2988,7 @@ }, { "cell_type": "markdown", - "id": "f8a8132d", + "id": "4e4b5ee0", "metadata": { "editable": true }, @@ -2638,7 +3008,7 @@ }, { "cell_type": "markdown", - "id": "03eca41f", + "id": "4455b9a0", "metadata": { "editable": true }, @@ -2665,7 +3035,7 @@ }, { "cell_type": "markdown", - "id": "710e8f88", + "id": "9592eb20", "metadata": { "editable": true }, @@ -2678,7 +3048,7 @@ }, { "cell_type": "markdown", - "id": "5d3df9bf", + "id": "e3022ea8", "metadata": { "editable": true }, @@ -2690,7 +3060,7 @@ }, { "cell_type": "markdown", - "id": "be0fd5f1", + "id": "246a28bf", "metadata": { "editable": true }, @@ -2705,7 +3075,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "2a0924bb", + "id": "d132a060", "metadata": { "collapsed": false, "editable": true @@ -2732,7 +3102,7 @@ }, { "cell_type": "markdown", - "id": "d116f448", + "id": "9e70a01f", "metadata": { "editable": true }, @@ -2746,7 +3116,7 @@ }, { "cell_type": "markdown", - "id": "41caea07", + "id": "2c0ad6b4", "metadata": { "editable": true }, @@ -2758,7 +3128,7 @@ }, { "cell_type": "markdown", - "id": "1fa96f7c", + "id": "3c6081b8", "metadata": { "editable": true }, @@ -2775,7 +3145,7 @@ }, { "cell_type": "markdown", - "id": "70038d6a", + "id": "845af933", "metadata": { "editable": true }, @@ -2787,7 +3157,7 @@ }, { "cell_type": "markdown", - "id": "852a77d0", + "id": "564afbbd", "metadata": { "editable": true }, @@ -2797,7 +3167,7 @@ }, { "cell_type": "markdown", - "id": "fc4afaaf", + "id": "c4088263", "metadata": { "editable": true }, @@ -2809,7 +3179,7 @@ }, { "cell_type": "markdown", - "id": "94b18ced", + "id": "96983e3d", "metadata": { "editable": true }, @@ -2819,7 +3189,7 @@ }, { "cell_type": "markdown", - "id": "d7a95314", + "id": "91d029d7", "metadata": { "editable": true }, @@ -2831,7 +3201,7 @@ }, { "cell_type": "markdown", - "id": "eaf6a485", + "id": "20d351f6", "metadata": { "editable": true }, @@ -2842,7 +3212,7 @@ }, { "cell_type": "markdown", - "id": "3d9442a2", + "id": "7a8e79fd", 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Reminder on Lagrangian Multipliers": [[8, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[4, "a-simple-example"]], "A soft classifier": [[8, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[1, "a-top-down-perspective-on-neural-networks"]], "ADAM algorithm, taken from Goodfellow et al": [[30, "adam-algorithm-taken-from-goodfellow-et-al"]], "ADAM optimizer": [[13, "adam-optimizer"], [30, "id2"]], "Accuracy": [[30, "accuracy"]], "Activation functions": [[12, "activation-functions"]], "AdaGrad Properties": [[30, "adagrad-properties"]], "AdaGrad Update Rule Derivation": [[30, "adagrad-update-rule-derivation"]], "AdaGrad algorithm, taken from Goodfellow et al": [[30, "adagrad-algorithm-taken-from-goodfellow-et-al"]], "Adam Optimizer": [[30, "adam-optimizer"]], "Adam vs. AdaGrad and RMSProp": [[30, "adam-vs-adagrad-and-rmsprop"]], "Adam: Bias Correction": [[30, "adam-bias-correction"]], "Adam: Exponential Moving Averages (Moments)": [[30, 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[[24, "expectation-values"]], "Extending to more than one variable": [[29, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[27, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[28, "fixing-the-singularity"], [29, "fixing-the-singularity"]], "Format for electronic delivery of report and programs": [[22, "format-for-electronic-delivery-of-report-and-programs"]], "Frequently used scaling functions": [[28, "frequently-used-scaling-functions"], [30, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[29, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[12, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[28, "functionality-in-scikit-learn"], [30, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[3, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[5, "further-properties-important-for-our-analyses-later"], [28, "further-properties-important-for-our-analyses-later"], [29, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[21, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[27, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[27, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [27, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[27, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[10, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[10, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[1, "gradient-clipping"]], "Gradient Descent Example": [[29, "id1"], [30, "id1"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "Gradient descent and Ridge": [[29, "gradient-descent-and-ridge"], [30, "gradient-descent-and-ridge"]], "Gradient descent and revisiting Ordinary Least Squares from last week": [[30, "gradient-descent-and-revisiting-ordinary-least-squares-from-last-week"]], "Gradient descent example": [[29, "gradient-descent-example"], [30, "gradient-descent-example"]], "Grading": [[25, "grading"], [25, "id2"], [27, "grading"]], "How to take derivatives of Matrix-Vector expressions": [[16, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[8, "hyperplanes-and-all-that"]], "Important Matrix and vector handling packages": [[21, "important-matrix-and-vector-handling-packages"]], "Important technicalities: More on Rescaling data": [[28, "important-technicalities-more-on-rescaling-data"]], "Improving gradient descent with momentum": [[30, "improving-gradient-descent-with-momentum"]], "Improving performance": [[1, "improving-performance"]], "In summary": [[25, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[13, "including-stochastic-gradient-descent-with-autograd"], [30, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[11, "incremental-pca"]], "Installing R, C++, cython or Julia": [[27, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[27, "installing-r-c-cython-numba-etc"]], "Instructor information": [[25, "instructor-information"]], "Interpretations and optimizing our parameters": [[27, "interpretations-and-optimizing-our-parameters"], [27, "id2"], [27, "id3"], [28, "interpretations-and-optimizing-our-parameters"], [28, "id1"], [28, "id2"]], "Interpreting the Ridge results": [[28, "interpreting-the-ridge-results"], [29, "interpreting-the-ridge-results"], [29, "id4"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [28, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [20, "introduction"], [21, "introduction"]], "Introduction to numerical projects": [[22, "introduction-to-numerical-projects"]], "Iterative Fitting, Classification and AdaBoost": [[10, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[10, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[11, "kernel-pca"]], "Kernels and non-linearity": [[8, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[21, "lu-decomposition-the-inverse-of-a-matrix"]], "Lasso Regression": [[29, "lasso-regression"]], "Lasso case": [[29, "lasso-case"]], "Layers": [[1, "layers"]], "Layers used to build CNNs": [[3, "layers-used-to-build-cnns"]], "Learning goals": [[15, "learning-goals"], [16, "learning-goals"], [17, "learning-goals"], [18, "learning-goals"], [19, "learning-goals"]], "Learning outcomes": [[20, "learning-outcomes"], [27, "learning-outcomes"]], "Lectures and ComputerLab": [[27, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[1, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[21, null]], "Linear Regression": [[0, null]], "Linear Regression Problems": [[28, "linear-regression-problems"], [29, "linear-regression-problems"]], "Linear Regression and the SVD": [[29, "linear-regression-and-the-svd"]], "Linear Regression, basic elements": [[0, "linear-regression-basic-elements"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[5, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[5, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[5, "linking-with-the-svd"], [28, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[26, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[7, null], [7, "id1"]], "MNIST and GANs": [[4, "mnist-and-gans"]], "Machine Learning": [[27, "machine-learning"]], "Machine learning": [[20, "machine-learning"]], "Main textbooks": [[27, "main-textbooks"]], "Making a tree": [[9, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[10, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[28, "making-your-own-test-train-splitting"]], "Material for exercises week 35": [[28, "material-for-exercises-week-35"]], "Material for lab sessions sessions Tuesday and Wednesday": [[29, "material-for-lab-sessions-sessions-tuesday-and-wednesday"]], "Material for lecture Monday September 2": [[29, "material-for-lecture-monday-september-2"]], "Material for lecture Monday September 8": [[30, "material-for-lecture-monday-september-8"]], "Material for the lab sessions": [[30, "material-for-the-lab-sessions"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [28, "mathematical-interpretation-of-ordinary-least-squares"], [29, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[8, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[3, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[5, "mathematics-of-the-svd-and-implications"], [28, "mathematics-of-the-svd-and-implications"], [29, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[27, "matrices-in-python"]], "Matrix multiplication": [[1, "matrix-multiplication"]], "Matrix-vector notation and activation": [[12, "matrix-vector-notation-and-activation"]], "Meet the covariance!": [[24, "meet-the-covariance"]], "Meet the Covariance Matrix": [[5, "meet-the-covariance-matrix"], [28, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[28, "meet-the-hessian-matrix"]], "Meet the Pandas": [[27, "meet-the-pandas"]], "Min-Max Scaling": [[28, "min-max-scaling"]], "Momentum based GD": [[13, "momentum-based-gd"], [30, "momentum-based-gd"]], "More complicated Example: The Ising model": [[6, "more-complicated-example-the-ising-model"]], "More interpretations": [[28, "more-interpretations"], [29, "more-interpretations"], [29, "id5"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "More on Steepest descent": [[29, "more-on-steepest-descent"]], "More on convex functions": [[29, "more-on-convex-functions"]], "More preprocessing": [[28, "more-preprocessing"], [30, "more-preprocessing"]], "Motivation for Adaptive Step Sizes": [[30, "motivation-for-adaptive-step-sizes"]], "Multilayer perceptrons": [[12, "multilayer-perceptrons"]], "Network requirements": [[2, "network-requirements"]], "Neural Networks vs CNNs": [[3, "neural-networks-vs-cnns"]], "Neural networks": [[12, null]], "Note about SVD Calculations": [[28, "note-about-svd-calculations"], [29, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[29, "note-on-scikit-learn"]], "Numerical experiments and the covariance, central limit theorem": [[24, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[21, "numpy-and-arrays"], [27, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[27, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization and gradient descent, the central part of any Machine Learning algortithm": [[29, "optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm"]], "Optimization, the central part of any Machine Learning algortithm": [[13, null]], "Optimizing our parameters": [[27, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[27, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[1, "optimizing-the-cost-function"]], "Organizing our data": [[0, "organizing-our-data"], [27, "organizing-our-data"]], "Other Matrix and Vector Operations": [[21, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[4, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning at UiO": [[27, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[27, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other popular texts": [[27, "other-popular-texts"]], "Other techniques": [[11, "other-techniques"]], "Other types of networks": [[12, "other-types-of-networks"]], "Other ways of visualizing the trees": [[9, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[27, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[27, "overview-of-first-week"]], "Overview video on Stochastic Gradient Descent (SGD)": [[30, "overview-video-on-stochastic-gradient-descent-sgd"]], "Own code for Ordinary Least Squares": [[27, "own-code-for-ordinary-least-squares"], [28, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[27, "pandas-ai"]], "Part a : Ordinary Least Square (OLS) for the Runge function": [[22, "part-a-ordinary-least-square-ols-for-the-runge-function"]], "Part b: Adding Ridge regression for the Runge function": [[22, "part-b-adding-ridge-regression-for-the-runge-function"]], "Part c: Writing your own gradient descent code": [[22, "part-c-writing-your-own-gradient-descent-code"]], "Part d: Including momentum and more advanced ways to update the learning the rate": [[22, "part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate"]], "Part e: Writing our own code for Lasso regression": [[22, "part-e-writing-our-own-code-for-lasso-regression"]], "Part f: Stochastic gradient descent": [[22, "part-f-stochastic-gradient-descent"]], "Part g: Bias-variance trade-off and resampling techniques": [[22, "part-g-bias-variance-trade-off-and-resampling-techniques"]], "Part h): Cross-validation as resampling techniques, adding more complexity": [[22, "part-h-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Partial Differential Equations": [[2, "partial-differential-equations"]], "Plans for week 35": [[28, "plans-for-week-35"]], "Plans for week 36": [[29, "plans-for-week-36"]], "Plans for week 37, lecture Monday": [[30, "plans-for-week-37-lecture-monday"]], "Practical tips": [[13, "practical-tips"], [30, "practical-tips"]], "Practicalities": [[25, "practicalities"], [25, "id1"]], "Preamble: Note on writing reports, using reference material, AI and other tools": [[22, "preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools"]], "Predicting New Points With A Trained Recurrent Neural Network": [[4, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our data": [[28, "preprocessing-our-data"]], "Prerequisites": [[27, "prerequisites"]], "Prerequisites and background": [[20, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[3, "prerequisites-collect-and-pre-process-data"]], "Probability Distribution Functions": [[24, "probability-distribution-functions"]], "Program example for gradient descent with Ridge Regression": [[29, "program-example-for-gradient-descent-with-ridge-regression"], [30, "program-example-for-gradient-descent-with-ridge-regression"]], "Program for stochastic gradient": [[13, "program-for-stochastic-gradient"]], "Project 1 on Machine Learning, deadline October 6 (midnight), 2025": [[22, null]], "Properties of PDFs": [[24, "properties-of-pdfs"]], "Pros and cons": [[30, "pros-and-cons"]], "Pros and cons of trees, pros": [[9, "pros-and-cons-of-trees-pros"]], "Python installers": [[20, "python-installers"], [27, "python-installers"]], "RMS prop": [[13, "rms-prop"]], "RMSProp algorithm, taken from Goodfellow et al": [[30, "rmsprop-algorithm-taken-from-goodfellow-et-al"]], "RMSProp: Adaptive Learning Rates": [[30, "rmsprop-adaptive-learning-rates"]], "RMSprop for adaptive learning rate with Stochastic Gradient Descent": [[30, "rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"]], "Random Numbers": [[24, "random-numbers"]], "Random forests": [[10, "random-forests"]], "Randomized PCA": [[11, "randomized-pca"]], "Reading material": [[27, "reading-material"]], "Reading recommendations:": [[28, "reading-recommendations"]], "Reading suggestions week 34": [[27, "reading-suggestions-week-34"]], "Readings and Videos:": [[30, "readings-and-videos"]], "Recurrent neural networks": [[12, "recurrent-neural-networks"]], "Recurrent neural networks: Overarching view": [[4, null]], "Reducing the number of degrees of freedom, overarching view": [[0, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [28, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reformulating the problem": [[2, "reformulating-the-problem"]], "Regression Case": [[10, "regression-case"]], "Regression analysis and resampling methods": [[22, "regression-analysis-and-resampling-methods"]], "Regression analysis, overarching aims": [[27, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[27, "regression-analysis-overarching-aims-ii"]], "Regularization": [[1, "regularization"]], "Reminder from last week": [[28, "reminder-from-last-week"]], "Reminder on Newton-Raphson\u2019s method": [[29, "reminder-on-newton-raphson-s-method"]], "Reminder on Statistics": [[6, "reminder-on-statistics"]], "Reminder on different scaling methods": [[30, "reminder-on-different-scaling-methods"]], "Replace or not": [[13, "replace-or-not"], [30, "replace-or-not"]], "Required Technologies": [[20, "required-technologies"]], "Resampling Methods": [[6, null]], "Resampling and the Bias-Variance Trade-off": [[19, "resampling-and-the-bias-variance-trade-off"]], "Resampling methods": [[6, "id1"]], "Residual Error": [[28, "residual-error"], [29, "residual-error"]], "Resources on differential equations and deep learning": [[2, "resources-on-differential-equations-and-deep-learning"]], "Revisiting Ordinary Least Squares": [[29, "revisiting-ordinary-least-squares"]], "Revisiting our Linear Regression Solvers": [[13, "revisiting-our-linear-regression-solvers"]], "Rewriting the Covariance and/or Correlation Matrix": [[28, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the fitting procedure as a linear algebra problem": [[27, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[27, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge Regression": [[29, "ridge-regression"]], "Ridge and LASSO Regression": [[28, "ridge-and-lasso-regression"], [29, "ridge-and-lasso-regression"], [29, "id2"]], "Ridge and Lasso Regression": [[5, null], [5, "id1"]], "SGD example": [[30, "sgd-example"]], "SVD analysis": [[29, "svd-analysis"]], "Same code but now with momentum gradient descent": [[13, "same-code-but-now-with-momentum-gradient-descent"], [30, "same-code-but-now-with-momentum-gradient-descent"], [30, "id3"], [30, "id4"]], "Schedule first week": [[27, "schedule-first-week"]], "Schematic Regression Procedure": [[9, "schematic-regression-procedure"]], "Second moment of the gradient": [[30, "second-moment-of-the-gradient"]], "September 15-19": [[19, "september-15-19"]], "Setting up the Back propagation algorithm": [[12, "setting-up-the-back-propagation-algorithm"]], "Setting up the Matrix to be inverted": [[28, "setting-up-the-matrix-to-be-inverted"], [29, "setting-up-the-matrix-to-be-inverted"]], "Setting up the network using Autograd; The full program": [[2, "setting-up-the-network-using-autograd-the-full-program"]], "Similar (second order function now) problem but now with AdaGrad": [[13, "similar-second-order-function-now-problem-but-now-with-adagrad"], [30, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[9, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple case": [[28, "simple-case"], [29, "simple-case"]], "Simple code for solving the above problem": [[29, "simple-code-for-solving-the-above-problem"]], "Simple example code": [[30, "simple-example-code"]], "Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression": [[29, "simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression"]], "Simple geometric interpretation": [[29, "simple-geometric-interpretation"]], "Simple linear regression model using scikit-learn": [[0, "simple-linear-regression-model-using-scikit-learn"], [27, "simple-linear-regression-model-using-scikit-learn"]], "Simple one-dimensional second-order polynomial": [[18, "simple-one-dimensional-second-order-polynomial"]], "Simple program": [[29, "simple-program"], [30, "simple-program"]], "Slightly different approach": [[30, "slightly-different-approach"]], "Sneaking in automatic differentiation using Autograd": [[30, "sneaking-in-automatic-differentiation-using-autograd"]], "Software and needed installations": [[22, "software-and-needed-installations"], [27, "software-and-needed-installations"]], "Solving Differential Equations with Deep Learning": [[2, null]], "Solving the one dimensional Poisson equation": [[2, "solving-the-one-dimensional-poisson-equation"]], "Solving the wave equation with Neural Networks": [[2, "solving-the-wave-equation-with-neural-networks"]], "Some famous Matrices": [[21, "some-famous-matrices"]], "Some simple problems": [[13, "some-simple-problems"], [29, "some-simple-problems"]], "Some useful matrix and vector expressions": [[28, "some-useful-matrix-and-vector-expressions"]], "Splitting our Data in Training and Test data": [[0, "splitting-our-data-in-training-and-test-data"], [28, "splitting-our-data-in-training-and-test-data"]], "Standard steepest descent": [[13, "standard-steepest-descent"]], "Statistical analysis and optimization of data": [[20, "statistical-analysis-and-optimization-of-data"], [27, "statistical-analysis-and-optimization-of-data"]], "Steepest descent": [[13, "steepest-descent"], [29, "steepest-descent"]], "Stochastic Gradient Descent": [[30, "stochastic-gradient-descent"]], "Stochastic Gradient Descent (SGD)": [[13, "stochastic-gradient-descent-sgd"], [30, "stochastic-gradient-descent-sgd"]], "Stochastic variables and the main concepts, the discrete case": [[24, "stochastic-variables-and-the-main-concepts-the-discrete-case"]], "Support Vector Machines, overarching aims": [[8, null]], "Systematic reduction": [[3, "systematic-reduction"]], "Teachers": [[27, "teachers"]], "Teachers and Grading": [[25, null]], "Teaching Assistants Fall semester 2023": [[25, "teaching-assistants-fall-semester-2023"]], "Tentative deadllines for projects": [[25, "tentative-deadllines-for-projects"]], "Testing the Means Squared Error as function of Complexity": [[0, "testing-the-means-squared-error-as-function-of-complexity"], [28, "testing-the-means-squared-error-as-function-of-complexity"]], "Textbooks": [[26, null]], "The Algorithm before theorem": [[11, "the-algorithm-before-theorem"]], "The Breast Cancer Data, now with Keras": [[1, "the-breast-cancer-data-now-with-keras"]], "The CART algorithm for Classification": [[9, "the-cart-algorithm-for-classification"]], "The CART algorithm for Regression": [[9, "the-cart-algorithm-for-regression"]], "The CIFAR01 data set": [[3, "the-cifar01-data-set"]], "The Hessian matrix": [[29, "the-hessian-matrix"], [30, "the-hessian-matrix"]], "The Hessian matrix for Ridge Regression": [[29, "the-hessian-matrix-for-ridge-regression"], [30, "the-hessian-matrix-for-ridge-regression"]], "The Jacobian": [[28, "the-jacobian"]], "The MNIST dataset again": [[3, "the-mnist-dataset-again"]], "The OLS case": [[29, "the-ols-case"]], "The RELU function family": [[1, "the-relu-function-family"]], "The Ridge case": [[29, "the-ridge-case"]], "The SVD, a Fantastic Algorithm": [[28, "the-svd-a-fantastic-algorithm"], [29, "the-svd-a-fantastic-algorithm"]], "The Softmax function": [[1, "the-softmax-function"]], "The \\chi^2 function": [[0, "the-chi-2-function"], [27, "the-chi-2-function"], [27, "id4"], [27, "id5"], [27, "id6"], [27, "id7"], [27, "id8"]], "The bias-variance tradeoff": [[6, "the-bias-variance-tradeoff"]], "The code for solving the ODE": [[2, "the-code-for-solving-the-ode"]], "The complete code with a simple data set": [[28, "the-complete-code-with-a-simple-data-set"]], "The cost/loss function": [[28, "the-cost-loss-function"]], "The course has two central parts": [[20, "the-course-has-two-central-parts"]], "The derivative of the cost/loss function": [[29, "the-derivative-of-the-cost-loss-function"], [30, "the-derivative-of-the-cost-loss-function"]], "The equations": [[29, "the-equations"]], "The equations for ordinary least squares": [[28, "the-equations-for-ordinary-least-squares"]], "The first Case": [[29, "the-first-case"]], "The gradient step": [[30, "the-gradient-step"]], "The ideal": [[29, "the-ideal"]], "The logistic function": [[7, "the-logistic-function"]], "The mean squared error and its derivative": [[28, "the-mean-squared-error-and-its-derivative"]], "The moons example": [[8, "the-moons-example"]], "The multilayer perceptron (MLP)": [[12, "the-multilayer-perceptron-mlp"]], "The network with one input layer, specified number of hidden layers, and one output layer": [[2, "the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer"]], "The plethora of machine learning algorithms/methods": [[27, "the-plethora-of-machine-learning-algorithms-methods"]], "The sensitiveness of the gradient descent": [[29, "the-sensitiveness-of-the-gradient-descent"]], "The singular value decomposition": [[5, "the-singular-value-decomposition"], [28, "the-singular-value-decomposition"], [29, "the-singular-value-decomposition"]], "The two-dimensional case": [[8, "the-two-dimensional-case"]], "Time decay rate": [[30, "time-decay-rate"]], "To our real data: nuclear binding energies. 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"a-frequentist-approach-to-data-analysis"]], "A better approach": [[8, "a-better-approach"]], "A first summary": [[27, "a-first-summary"]], "A quick Reminder on Lagrangian Multipliers": [[8, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[4, "a-simple-example"]], "A soft classifier": [[8, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[1, "a-top-down-perspective-on-neural-networks"]], "ADAM algorithm, taken from Goodfellow et al": [[30, "adam-algorithm-taken-from-goodfellow-et-al"]], "ADAM optimizer": [[13, "adam-optimizer"], [30, "id2"]], "Accuracy": [[30, "accuracy"]], "Activation functions": [[12, "activation-functions"]], "AdaGrad Properties": [[30, "adagrad-properties"]], "AdaGrad Update Rule Derivation": [[30, "adagrad-update-rule-derivation"]], "AdaGrad algorithm, taken from Goodfellow et al": [[30, "adagrad-algorithm-taken-from-goodfellow-et-al"]], "Adam Optimizer": [[30, "adam-optimizer"]], "Adam vs. AdaGrad and RMSProp": [[30, "adam-vs-adagrad-and-rmsprop"]], "Adam: Bias Correction": [[30, "adam-bias-correction"]], "Adam: Exponential Moving Averages (Moments)": [[30, "adam-exponential-moving-averages-moments"]], "Adam: Update Rule Derivation": [[30, "adam-update-rule-derivation"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[10, "adaptive-boosting-adaboost-basic-algorithm"]], "Adaptivity Across Dimensions": [[30, "adaptivity-across-dimensions"]], "Adding error analysis and training set up": [[27, "adding-error-analysis-and-training-set-up"], [28, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[1, "adjust-hyperparameters"]], "Algorithms and codes for Adagrad, RMSprop and Adam": [[30, "algorithms-and-codes-for-adagrad-rmsprop-and-adam"]], "Algorithms for Setting up Decision Trees": [[9, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[10, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[4, "an-extrapolation-example"]], "An optimization/minimization problem": [[27, "an-optimization-minimization-problem"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[28, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And finally ADAM": [[30, "and-finally-adam"]], "And what about using neural networks?": [[27, "and-what-about-using-neural-networks"]], "Another Example, now with a polynomial fit": [[29, "another-example-now-with-a-polynomial-fit"]], "Another example, the moons again": [[9, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[20, null]], "Autocorrelation function": [[24, "autocorrelation-function"]], "Automatic differentiation": [[13, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[28, "back-to-ridge-and-lasso-regression"], [29, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[11, "back-to-the-cancer-data"]], "Background literature": [[22, "background-literature"]], "Bagging": [[10, "bagging"]], "Bagging Examples": [[10, "bagging-examples"]], "Basic Matrix Features": [[21, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[11, null]], "Basic math of the SVD": [[5, "basic-math-of-the-svd"], [28, "basic-math-of-the-svd"], [29, "basic-math-of-the-svd"]], "Basics": [[7, "basics"]], "Basics of a tree": [[9, "basics-of-a-tree"]], "Batch Normalization": [[1, "batch-normalization"]], "Batches and mini-batches": [[30, "batches-and-mini-batches"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[5, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[10, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[6, "bootstrap"]], "Bringing it together, first back propagation equation": [[12, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[1, null]], "Building a tree, regression": [[9, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[1, "building-neural-networks-in-tensorflow-and-keras"]], "But none of these can compete with Newton\u2019s method": [[30, "but-none-of-these-can-compete-with-newton-s-method"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[3, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[9, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Challenge: Choosing a Fixed Learning Rate": [[30, "challenge-choosing-a-fixed-learning-rate"]], "Choose cost function and optimizer": [[1, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[11, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[14, null]], "Code for SVD and Inversion of Matrices": [[5, "code-for-svd-and-inversion-of-matrices"]], "Code with a Number of Minibatches which varies": [[30, "code-with-a-number-of-minibatches-which-varies"]], "Codes and Approaches": [[14, "codes-and-approaches"]], "Codes for the SVD": [[5, "codes-for-the-svd"], [28, "codes-for-the-svd"], [29, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[15, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[1, "collect-and-pre-process-data"]], "Communication channels": [[27, "communication-channels"]], "Compare Bagging on Trees with Random Forests": [[10, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[2, "comparing-with-a-numerical-scheme"]], "Comparison with OLS": [[29, "comparison-with-ols"]], "Computation of gradients": [[30, "computation-of-gradients"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conditions on convex functions": [[29, "conditions-on-convex-functions"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convergence rates": [[30, "convergence-rates"]], "Convex function": [[29, "convex-function"]], "Convex functions": [[13, "convex-functions"], [29, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[3, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[3, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[12, "convolutional-neural-network"]], "Convolutional Neural Networks": [[3, null]], "Correlation Function and Design/Feature Matrix": [[28, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[11, "correlation-matrix"], [28, "correlation-matrix"]], "Correlation Matrix with Pandas": [[28, "correlation-matrix-with-pandas"]], "Course Format": [[27, "course-format"]], "Course setting": [[23, null]], "Covariance Matrix Examples": [[28, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[28, "covariance-and-correlation-matrix"]], "Cross-validation": [[6, "cross-validation"]], "Deadlines for projects (tentative)": [[27, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[9, null]], "Deep Neural Networks": [[30, "deep-neural-networks"]], "Deep learning methods": [[27, "deep-learning-methods"]], "Define model and architecture": [[1, "define-model-and-architecture"]], "Defining the cost function": [[1, "defining-the-cost-function"]], "Deliverables": [[15, "deliverables"], [16, "deliverables"]], "Derivation of the AdaGrad Algorithm": [[30, "derivation-of-the-adagrad-algorithm"]], "Derivatives and the chain rule": [[12, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[28, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[5, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[16, "deriving-and-implementing-ordinary-least-squares"]], "Deriving and Implementing Ridge Regression": [[17, "deriving-and-implementing-ridge-regression"]], "Deriving the Lasso Regression Equations": [[28, "deriving-the-lasso-regression-equations"], [29, "deriving-the-lasso-regression-equations"], [29, "id6"]], "Deriving the Ridge Regression Equations": [[28, "deriving-the-ridge-regression-equations"], [29, "deriving-the-ridge-regression-equations"], [29, "id3"]], "Deriving the back propagation code for a multilayer perceptron model": [[12, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[1, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[11, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[8, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[9, "disadvantages"]], "Discriminative Modeling": [[27, "discriminative-modeling"]], "Domains and probabilities": [[24, "domains-and-probabilities"]], "Dropout": [[1, "dropout"]], "Economy-size SVD": [[28, "economy-size-svd"], [29, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[24, null]], "Empirical Evidence: Convergence Time and Memory in Practice": [[30, "empirical-evidence-convergence-time-and-memory-in-practice"]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[10, null]], "Entropy and the ID3 algorithm": [[9, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[27, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[1, "evaluate-model-performance-on-test-data"]], "Example 2": [[28, "example-2"]], "Example 3": [[28, "example-3"]], "Example 4": [[28, "example-4"]], "Example Matrix": [[28, "example-matrix"], [29, "example-matrix"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[27, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[27, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[28, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[28, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[2, "example-exponential-decay"]], "Example: Population growth": [[2, "example-population-growth"]], "Example: The diffusion equation": [[2, "example-the-diffusion-equation"]], "Example: binary classification problem": [[1, "example-binary-classification-problem"]], "Examples": [[27, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[7, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Exercise 1 - Choice of model and degrees of freedom": [[17, "exercise-1-choice-of-model-and-degrees-of-freedom"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[16, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[15, "exercise-1-github-setup"]], "Exercise 1, scale your data": [[18, "exercise-1-scale-your-data"]], "Exercise 1: Expectation values for ordinary least squares expressions": [[19, "exercise-1-expectation-values-for-ordinary-least-squares-expressions"]], "Exercise 1: Setting up various Python environments": [[0, "exercise-1-setting-up-various-python-environments"]], "Exercise 2 - Deriving the expression for OLS": [[16, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[17, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[15, "exercise-2-setting-up-a-github-repository"]], "Exercise 2, calculate the gradients": [[18, "exercise-2-calculate-the-gradients"]], "Exercise 2: Expectation values for Ridge regression": [[19, "exercise-2-expectation-values-for-ridge-regression"]], "Exercise 2: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[16, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[15, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[17, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[15, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters \\boldsymbol{\\theta}": [[18, "exercise-3-using-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta"]], "Exercise 3: Deriving the expression for the Bias-Variance Trade-off": [[19, "exercise-3-deriving-the-expression-for-the-bias-variance-trade-off"]], "Exercise 3: Normalizing our data": [[0, "exercise-3-normalizing-our-data"]], "Exercise 4 - Fitting a polynomial": [[16, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - Implementing Ridge Regression": [[17, "exercise-4-implementing-ridge-regression"]], "Exercise 4 - Testing multiple hyperparameters": [[17, "exercise-4-testing-multiple-hyperparameters"]], "Exercise 4 - The train-test split": [[15, "exercise-4-the-train-test-split"]], "Exercise 4, Implementing the simplest form for gradient descent": [[18, "exercise-4-implementing-the-simplest-form-for-gradient-descent"]], "Exercise 4: Adding Ridge Regression": [[0, "exercise-4-adding-ridge-regression"]], "Exercise 4: Computing the Bias and Variance": [[19, "exercise-4-computing-the-bias-and-variance"]], "Exercise 5 - Comparing your code with sklearn": [[16, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5, Ridge regression and a new Synthetic Dataset": [[18, "exercise-5-ridge-regression-and-a-new-synthetic-dataset"]], "Exercise 5: Analytical exercises": [[0, "exercise-5-analytical-exercises"]], "Exercise: Cross-validation as resampling techniques, adding more complexity": [[6, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[6, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[6, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function with resampling": [[6, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[6, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function with resampling": [[6, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[0, "exercises"]], "Exercises and Projects": [[6, "exercises-and-projects"]], "Exercises week 34": [[15, null]], "Exercises week 35": [[16, null]], "Exercises week 36": [[17, null]], "Exercises week 37": [[18, null]], "Exercises week 38": [[19, null]], "Expectation values": [[24, "expectation-values"]], "Extending to more than one variable": [[29, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[27, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[28, "fixing-the-singularity"], [29, "fixing-the-singularity"]], "Format for electronic delivery of report and programs": [[22, "format-for-electronic-delivery-of-report-and-programs"]], "Frequently used scaling functions": [[28, "frequently-used-scaling-functions"], [30, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[29, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[12, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[28, "functionality-in-scikit-learn"], [30, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[3, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[5, "further-properties-important-for-our-analyses-later"], [28, "further-properties-important-for-our-analyses-later"], [29, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[21, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[27, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[27, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [27, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[27, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[10, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[10, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[1, "gradient-clipping"]], "Gradient Descent Example": [[29, "id1"], [30, "id1"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "Gradient descent and Ridge": [[29, "gradient-descent-and-ridge"], [30, "gradient-descent-and-ridge"]], "Gradient descent and revisiting Ordinary Least Squares from last week": [[30, "gradient-descent-and-revisiting-ordinary-least-squares-from-last-week"]], "Gradient descent example": [[29, "gradient-descent-example"], [30, "gradient-descent-example"]], "Grading": [[25, "grading"], [25, "id2"], [27, "grading"]], "How to take derivatives of Matrix-Vector expressions": [[16, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[8, "hyperplanes-and-all-that"]], "Important Matrix and vector handling packages": [[21, "important-matrix-and-vector-handling-packages"]], "Important technicalities: More on Rescaling data": [[28, "important-technicalities-more-on-rescaling-data"]], "Improving gradient descent with momentum": [[30, "improving-gradient-descent-with-momentum"]], "Improving performance": [[1, "improving-performance"]], "In summary": [[25, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[13, "including-stochastic-gradient-descent-with-autograd"], [30, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[11, "incremental-pca"]], "Installing R, C++, cython or Julia": [[27, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[27, "installing-r-c-cython-numba-etc"]], "Instructor information": [[25, "instructor-information"]], "Interpretations and optimizing our parameters": [[27, "interpretations-and-optimizing-our-parameters"], [27, "id2"], [27, "id3"], [28, "interpretations-and-optimizing-our-parameters"], [28, "id1"], [28, "id2"]], "Interpreting the Ridge results": [[28, "interpreting-the-ridge-results"], [29, "interpreting-the-ridge-results"], [29, "id4"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [28, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [20, "introduction"], [21, "introduction"]], "Introduction to numerical projects": [[22, "introduction-to-numerical-projects"]], "Iterative Fitting, Classification and AdaBoost": [[10, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[10, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[11, "kernel-pca"]], "Kernels and non-linearity": [[8, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[21, "lu-decomposition-the-inverse-of-a-matrix"]], "Lasso Regression": [[29, "lasso-regression"]], "Lasso case": [[29, "lasso-case"]], "Layers": [[1, "layers"]], "Layers used to build CNNs": [[3, "layers-used-to-build-cnns"]], "Learning goals": [[15, "learning-goals"], [16, "learning-goals"], [17, "learning-goals"], [18, "learning-goals"], [19, "learning-goals"]], "Learning outcomes": [[20, "learning-outcomes"], [27, "learning-outcomes"]], "Lectures and ComputerLab": [[27, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[1, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[21, null]], "Linear Regression": [[0, null]], "Linear Regression Problems": [[28, "linear-regression-problems"], [29, "linear-regression-problems"]], "Linear Regression and the SVD": [[29, "linear-regression-and-the-svd"]], "Linear Regression, basic elements": [[0, "linear-regression-basic-elements"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[5, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[5, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[5, "linking-with-the-svd"], [28, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[26, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[7, null], [7, "id1"]], "MNIST and GANs": [[4, "mnist-and-gans"]], "Machine Learning": [[27, "machine-learning"]], "Machine learning": [[20, "machine-learning"]], "Main textbooks": [[27, "main-textbooks"]], "Making a tree": [[9, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[10, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[28, "making-your-own-test-train-splitting"]], "Material for exercises week 35": [[28, "material-for-exercises-week-35"]], "Material for lab sessions sessions Tuesday and Wednesday": [[29, "material-for-lab-sessions-sessions-tuesday-and-wednesday"]], "Material for lecture Monday September 2": [[29, "material-for-lecture-monday-september-2"]], "Material for lecture Monday September 8": [[30, "material-for-lecture-monday-september-8"]], "Material for the lab sessions": [[30, "material-for-the-lab-sessions"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [28, "mathematical-interpretation-of-ordinary-least-squares"], [29, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[8, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[3, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[5, "mathematics-of-the-svd-and-implications"], [28, "mathematics-of-the-svd-and-implications"], [29, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[27, "matrices-in-python"]], "Matrix multiplication": [[1, "matrix-multiplication"]], "Matrix-vector notation and activation": [[12, "matrix-vector-notation-and-activation"]], "Meet the covariance!": [[24, "meet-the-covariance"]], "Meet the Covariance Matrix": [[5, "meet-the-covariance-matrix"], [28, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[28, "meet-the-hessian-matrix"]], "Meet the Pandas": [[27, "meet-the-pandas"]], "Memory Usage and Scalability": [[30, "memory-usage-and-scalability"]], "Memory constraints": [[30, "memory-constraints"]], "Min-Max Scaling": [[28, "min-max-scaling"]], "Momentum based GD": [[13, "momentum-based-gd"], [30, "momentum-based-gd"]], "More complicated Example: The Ising model": [[6, "more-complicated-example-the-ising-model"]], "More interpretations": [[28, "more-interpretations"], [29, "more-interpretations"], [29, "id5"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "More on Steepest descent": [[29, "more-on-steepest-descent"]], "More on convex functions": [[29, "more-on-convex-functions"]], "More preprocessing": [[28, "more-preprocessing"], [30, "more-preprocessing"]], "Motivation for Adaptive Step Sizes": [[30, "motivation-for-adaptive-step-sizes"]], "Multilayer perceptrons": [[12, "multilayer-perceptrons"]], "Network requirements": [[2, "network-requirements"]], "Neural Networks vs CNNs": [[3, "neural-networks-vs-cnns"]], "Neural networks": [[12, null]], "Non-Convex Problems": [[30, "non-convex-problems"]], "Note about SVD Calculations": [[28, "note-about-svd-calculations"], [29, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[29, "note-on-scikit-learn"]], "Numerical experiments and the covariance, central limit theorem": [[24, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[21, "numpy-and-arrays"], [27, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[27, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization and gradient descent, the central part of any Machine Learning algortithm": [[29, "optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm"]], "Optimization, the central part of any Machine Learning algortithm": [[13, null]], "Optimizing our parameters": [[27, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[27, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[1, "optimizing-the-cost-function"]], "Organizing our data": [[0, "organizing-our-data"], [27, "organizing-our-data"]], "Other Matrix and Vector Operations": [[21, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[4, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning at UiO": [[27, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[27, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other popular texts": [[27, "other-popular-texts"]], "Other techniques": [[11, "other-techniques"]], "Other types of networks": [[12, "other-types-of-networks"]], "Other ways of visualizing the trees": [[9, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[27, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[27, "overview-of-first-week"]], "Overview video on Stochastic Gradient Descent (SGD)": [[30, "overview-video-on-stochastic-gradient-descent-sgd"]], "Own code for Ordinary Least Squares": [[27, "own-code-for-ordinary-least-squares"], [28, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[27, "pandas-ai"]], "Part a : Ordinary Least Square (OLS) for the Runge function": [[22, "part-a-ordinary-least-square-ols-for-the-runge-function"]], "Part b: Adding Ridge regression for the Runge function": [[22, "part-b-adding-ridge-regression-for-the-runge-function"]], "Part c: Writing your own gradient descent code": [[22, "part-c-writing-your-own-gradient-descent-code"]], "Part d: Including momentum and more advanced ways to update the learning the rate": [[22, "part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate"]], "Part e: Writing our own code for Lasso regression": [[22, "part-e-writing-our-own-code-for-lasso-regression"]], "Part f: Stochastic gradient descent": [[22, "part-f-stochastic-gradient-descent"]], "Part g: Bias-variance trade-off and resampling techniques": [[22, "part-g-bias-variance-trade-off-and-resampling-techniques"]], "Part h): Cross-validation as resampling techniques, adding more complexity": [[22, "part-h-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Partial Differential Equations": [[2, "partial-differential-equations"]], "Plans for week 35": [[28, "plans-for-week-35"]], "Plans for week 36": [[29, "plans-for-week-36"]], "Plans for week 37, lecture Monday": [[30, "plans-for-week-37-lecture-monday"]], "Practical tips": [[13, "practical-tips"], [30, "practical-tips"]], "Practicalities": [[25, "practicalities"], [25, "id1"]], "Preamble: Note on writing reports, using reference material, AI and other tools": [[22, "preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools"]], "Predicting New Points With A Trained Recurrent Neural Network": [[4, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our data": [[28, "preprocessing-our-data"]], "Prerequisites": [[27, "prerequisites"]], "Prerequisites and background": [[20, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[3, "prerequisites-collect-and-pre-process-data"]], "Probability Distribution Functions": [[24, "probability-distribution-functions"]], "Program example for gradient descent with Ridge Regression": [[29, "program-example-for-gradient-descent-with-ridge-regression"], [30, "program-example-for-gradient-descent-with-ridge-regression"]], "Program for stochastic gradient": [[13, "program-for-stochastic-gradient"]], "Project 1 on Machine Learning, deadline October 6 (midnight), 2025": [[22, null]], "Properties of PDFs": [[24, "properties-of-pdfs"]], "Pros and cons": [[30, "pros-and-cons"]], "Pros and cons of trees, pros": [[9, "pros-and-cons-of-trees-pros"]], "Python installers": [[20, "python-installers"], [27, "python-installers"]], "RMS prop": [[13, "rms-prop"]], "RMSProp algorithm, taken from Goodfellow et al": [[30, "rmsprop-algorithm-taken-from-goodfellow-et-al"]], "RMSProp: Adaptive Learning Rates": [[30, "rmsprop-adaptive-learning-rates"]], "RMSprop for adaptive learning rate with Stochastic Gradient Descent": [[30, "rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"]], "Random Numbers": [[24, "random-numbers"]], "Random forests": [[10, "random-forests"]], "Randomized PCA": [[11, "randomized-pca"]], "Reading material": [[27, "reading-material"]], "Reading recommendations:": [[28, "reading-recommendations"]], "Reading suggestions week 34": [[27, "reading-suggestions-week-34"]], "Readings and Videos:": [[30, "readings-and-videos"]], "Recurrent neural networks": [[12, "recurrent-neural-networks"]], "Recurrent neural networks: Overarching view": [[4, null]], "Reducing the number of degrees of freedom, overarching view": [[0, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [28, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reformulating the problem": [[2, "reformulating-the-problem"]], "Regression Case": [[10, "regression-case"]], "Regression analysis and resampling methods": [[22, "regression-analysis-and-resampling-methods"]], "Regression analysis, overarching aims": [[27, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[27, "regression-analysis-overarching-aims-ii"]], "Regularization": [[1, "regularization"]], "Reminder from last week": [[28, "reminder-from-last-week"]], "Reminder on Newton-Raphson\u2019s method": [[29, "reminder-on-newton-raphson-s-method"]], "Reminder on Statistics": [[6, "reminder-on-statistics"]], "Reminder on different scaling methods": [[30, "reminder-on-different-scaling-methods"]], "Replace or not": [[13, "replace-or-not"], [30, "replace-or-not"]], "Required Technologies": [[20, "required-technologies"]], "Resampling Methods": [[6, null]], "Resampling and the Bias-Variance Trade-off": [[19, "resampling-and-the-bias-variance-trade-off"]], "Resampling methods": [[6, "id1"]], "Residual Error": [[28, "residual-error"], [29, "residual-error"]], "Resources on differential equations and deep learning": [[2, "resources-on-differential-equations-and-deep-learning"]], "Revisiting Ordinary Least Squares": [[29, "revisiting-ordinary-least-squares"]], "Revisiting our Linear Regression Solvers": [[13, "revisiting-our-linear-regression-solvers"]], "Rewriting the Covariance and/or Correlation Matrix": [[28, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the fitting procedure as a linear algebra problem": [[27, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[27, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge Regression": [[29, "ridge-regression"]], "Ridge and LASSO Regression": [[28, "ridge-and-lasso-regression"], [29, "ridge-and-lasso-regression"], [29, "id2"]], "Ridge and Lasso Regression": [[5, null], [5, "id1"]], "SGD example": [[30, "sgd-example"]], "SGD vs Full-Batch GD: Convergence Speed and Memory Comparison": [[30, "sgd-vs-full-batch-gd-convergence-speed-and-memory-comparison"]], "SVD analysis": [[29, "svd-analysis"]], "Same code but now with momentum gradient descent": [[13, "same-code-but-now-with-momentum-gradient-descent"], [30, "same-code-but-now-with-momentum-gradient-descent"], [30, "id3"], [30, "id4"]], "Schedule first week": [[27, "schedule-first-week"]], "Schematic Regression Procedure": [[9, "schematic-regression-procedure"]], "Second moment of the gradient": [[30, "second-moment-of-the-gradient"]], "September 15-19": [[19, "september-15-19"]], "Setting up the Back propagation algorithm": [[12, "setting-up-the-back-propagation-algorithm"]], "Setting up the Matrix to be inverted": [[28, "setting-up-the-matrix-to-be-inverted"], [29, "setting-up-the-matrix-to-be-inverted"]], "Setting up the network using Autograd; The full program": [[2, "setting-up-the-network-using-autograd-the-full-program"]], "Similar (second order function now) problem but now with AdaGrad": [[13, "similar-second-order-function-now-problem-but-now-with-adagrad"], [30, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[9, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple case": [[28, "simple-case"], [29, "simple-case"]], "Simple code for solving the above problem": [[29, "simple-code-for-solving-the-above-problem"]], "Simple example code": [[30, "simple-example-code"]], "Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression": [[29, "simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression"]], "Simple geometric interpretation": [[29, "simple-geometric-interpretation"]], "Simple linear regression model using scikit-learn": [[0, "simple-linear-regression-model-using-scikit-learn"], [27, "simple-linear-regression-model-using-scikit-learn"]], "Simple one-dimensional second-order polynomial": [[18, "simple-one-dimensional-second-order-polynomial"]], "Simple program": [[29, "simple-program"], [30, "simple-program"]], "Slightly different approach": [[30, "slightly-different-approach"]], "Sneaking in automatic differentiation using Autograd": [[30, "sneaking-in-automatic-differentiation-using-autograd"]], "Software and needed installations": [[22, "software-and-needed-installations"], [27, "software-and-needed-installations"]], "Solving Differential Equations with Deep Learning": [[2, null]], "Solving the one dimensional Poisson equation": [[2, "solving-the-one-dimensional-poisson-equation"]], "Solving the wave equation with Neural Networks": [[2, "solving-the-wave-equation-with-neural-networks"]], "Some famous Matrices": [[21, "some-famous-matrices"]], "Some simple problems": [[13, "some-simple-problems"], [29, "some-simple-problems"]], "Some useful matrix and vector expressions": [[28, "some-useful-matrix-and-vector-expressions"]], "Splitting our Data in Training and Test data": [[0, "splitting-our-data-in-training-and-test-data"], [28, "splitting-our-data-in-training-and-test-data"]], "Standard steepest descent": [[13, "standard-steepest-descent"]], "Statistical analysis and optimization of data": [[20, "statistical-analysis-and-optimization-of-data"], [27, "statistical-analysis-and-optimization-of-data"]], "Steepest descent": [[13, "steepest-descent"], [29, "steepest-descent"]], "Stochastic Gradient Descent": [[30, "stochastic-gradient-descent"]], "Stochastic Gradient Descent (SGD)": [[13, "stochastic-gradient-descent-sgd"], [30, "stochastic-gradient-descent-sgd"]], "Stochastic variables and the main concepts, the discrete case": [[24, "stochastic-variables-and-the-main-concepts-the-discrete-case"]], "Strongly Convex Case": [[30, "strongly-convex-case"]], "Support Vector Machines, overarching aims": [[8, null]], "Systematic reduction": [[3, "systematic-reduction"]], "Teachers": [[27, "teachers"]], "Teachers and Grading": [[25, null]], "Teaching Assistants Fall semester 2023": [[25, "teaching-assistants-fall-semester-2023"]], "Tentative deadllines for projects": [[25, "tentative-deadllines-for-projects"]], "Testing the Means Squared Error as function of Complexity": [[0, "testing-the-means-squared-error-as-function-of-complexity"], [28, "testing-the-means-squared-error-as-function-of-complexity"]], "Textbooks": [[26, null]], "The Algorithm before theorem": [[11, "the-algorithm-before-theorem"]], "The Breast Cancer Data, now with Keras": [[1, "the-breast-cancer-data-now-with-keras"]], "The CART algorithm for Classification": [[9, "the-cart-algorithm-for-classification"]], "The CART algorithm for Regression": [[9, "the-cart-algorithm-for-regression"]], "The CIFAR01 data set": [[3, "the-cifar01-data-set"]], "The Hessian matrix": [[29, "the-hessian-matrix"], [30, "the-hessian-matrix"]], "The Hessian matrix for Ridge Regression": [[29, "the-hessian-matrix-for-ridge-regression"], [30, "the-hessian-matrix-for-ridge-regression"]], "The Jacobian": [[28, "the-jacobian"]], "The MNIST dataset again": [[3, "the-mnist-dataset-again"]], "The OLS case": [[29, "the-ols-case"]], "The RELU function family": [[1, "the-relu-function-family"]], "The Ridge case": [[29, "the-ridge-case"]], "The SVD, a Fantastic Algorithm": [[28, "the-svd-a-fantastic-algorithm"], [29, "the-svd-a-fantastic-algorithm"]], "The Softmax function": [[1, "the-softmax-function"]], "The \\chi^2 function": [[0, "the-chi-2-function"], [27, "the-chi-2-function"], [27, "id4"], [27, "id5"], [27, "id6"], [27, "id7"], [27, "id8"]], "The bias-variance tradeoff": [[6, "the-bias-variance-tradeoff"]], "The code for solving the ODE": [[2, "the-code-for-solving-the-ode"]], "The complete code with a simple data set": [[28, "the-complete-code-with-a-simple-data-set"]], "The cost/loss function": [[28, "the-cost-loss-function"]], "The course has two central parts": [[20, "the-course-has-two-central-parts"]], "The derivative of the cost/loss function": [[29, "the-derivative-of-the-cost-loss-function"], [30, "the-derivative-of-the-cost-loss-function"]], "The equations": [[29, "the-equations"]], "The equations for ordinary least squares": [[28, "the-equations-for-ordinary-least-squares"]], "The first Case": [[29, "the-first-case"]], "The gradient step": [[30, "the-gradient-step"]], "The ideal": [[29, "the-ideal"]], "The logistic function": [[7, "the-logistic-function"]], "The mean squared error and its derivative": [[28, "the-mean-squared-error-and-its-derivative"]], "The moons example": [[8, "the-moons-example"]], "The multilayer perceptron (MLP)": [[12, "the-multilayer-perceptron-mlp"]], "The network with one input layer, specified number of hidden layers, and one output layer": [[2, "the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer"]], "The plethora of machine learning algorithms/methods": [[27, "the-plethora-of-machine-learning-algorithms-methods"]], "The sensitiveness of the gradient descent": [[29, "the-sensitiveness-of-the-gradient-descent"]], "The singular value decomposition": [[5, "the-singular-value-decomposition"], [28, "the-singular-value-decomposition"], [29, "the-singular-value-decomposition"]], "The two-dimensional case": [[8, "the-two-dimensional-case"]], "Theoretical Convergence Speed and convex optimization": [[30, "theoretical-convergence-speed-and-convex-optimization"]], "Time decay rate": [[30, "time-decay-rate"]], "To our real data: nuclear binding energies. 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a/doc/LectureNotes/_build/html/week37.html +++ b/doc/LectureNotes/_build/html/week37.html @@ -416,6 +416,18 @@ document.write(`
  • Time decay rate
  • Code with a Number of Minibatches which varies
  • Replace or not
    • +
  • SGD vs Full-Batch GD: Convergence Speed and Memory Comparison +
    • +
  • Memory Usage and Scalability
    • +
  • Empirical Evidence: Convergence Time and Memory in Practice +
  • Second moment of the gradient
  • Challenge: Choosing a Fixed Learning Rate
  • Motivation for Adaptive Step Sizes
    • @@ -1202,6 +1214,225 @@ mini-batches. The discussion here may be useful.

    +
    +

    SGD vs Full-Batch GD: Convergence Speed and Memory Comparison#

    +
    +

    Theoretical Convergence Speed and convex optimization#

    +

    Consider minimizing an empirical cost function

    +
    +\[ +C(\theta) =\frac{1}{N}\sum_{i=1}^N l_i(\theta), +\]
    +

    where each \(l_i(\theta)\) is a +differentiable loss term. Gradient Descent (GD) updates parameters +using the full gradient \(\nabla C(\theta)\), while Stochastic Gradient +Descent (SGD) uses a single sample (or mini-batch) gradient \(\nabla +l_i(\theta)\) selected at random. In equation form, one GD step is:

    +
    +\[ +\theta_{t+1} = \theta_t-\eta \nabla C(\theta_t) =\theta_t -\eta \frac{1}{N}\sum_{i=1}^N \nabla l_i(\theta_t), +\]
    +

    whereas one SGD step is:

    +
    +\[ +\theta_{t+1} = \theta_t -\eta \nabla l_{i_t}(\theta_t), +\]
    +

    with \(i_t\) randomly chosen. On smooth convex problems, GD and SGD both +converge to the global minimum, but their rates differ. GD can take +larger, more stable steps since it uses the exact gradient, achieving +an error that decreases on the order of \(O(1/t)\) per iteration for +convex objectives (and even exponentially fast for strongly convex +cases). In contrast, plain SGD has more variance in each step, leading +to sublinear convergence in expectation – typically \(O(1/\sqrt{t})\) +for general convex objectives (\thetaith appropriate diminishing step +sizes) . Intuitively, GD’s trajectory is smoother and more +predictable, while SGD’s path oscillates due to noise but costs far +less per iteration, enabling many more updates in the same time.

    +
    +
    +

    Strongly Convex Case#

    +

    If \(C(\theta)\) is strongly convex and \(L\)-smooth (so GD enjoys linear +convergence), the gap \(C(\theta_t)-C(\theta^*)\) for GD shrinks as

    +
    +\[ +C(\theta_t) - C(\theta^* ) \le \Big(1 - \frac{\mu}{L}\Big)^t [C(\theta_0)-C(\theta^*)], +\]
    +

    a geometric (linear) convergence per iteration . Achieving an +\(\epsilon\)-accurate solution thus takes on the order of +\(\log(1/\epsilon)\) iterations for GD. However, each GD iteration costs +\(O(N)\) gradient evaluations. SGD cannot exploit strong convexity to +obtain a linear rate – instead, with a properly decaying step size +(e.g. \(\eta_t = \frac{1}{\mu t}\)) or iterate averaging, SGD attains an +\(O(1/t)\) convergence rate in expectation . For example, one result +of Moulines and Bach 2011, see https://papers.nips.cc/paper_files/paper/2011/hash/40008b9a5380fcacce3976bf7c08af5b-Abstract.html shows that with \(\eta_t = \Theta(1/t)\),

    +
    +\[ +\mathbb{E}[C(\theta_t) - C(\theta^*)] = O(1/t), +\]
    +

    for strongly convex, smooth \(F\) . This \(1/t\) rate is slower per +iteration than GD’s exponential decay, but each SGD iteration is \(N\) +times cheaper. In fact, to reach error \(\epsilon\), plain SGD needs on +the order of \(T=O(1/\epsilon)\) iterations (sub-linear convergence), +while GD needs \(O(\log(1/\epsilon))\) iterations. When accounting for +cost-per-iteration, GD requires \(O(N \log(1/\epsilon))\) total gradient +computations versus SGD’s \(O(1/\epsilon)\) single-sample +computations. In large-scale regimes (huge \(N\)), SGD can be +faster in wall-clock time because \(N \log(1/\epsilon)\) may far exceed +\(1/\epsilon\) for reasonable accuracy levels. In other words, +with millions of data points, one epoch of GD (one full gradient) is +extremely costly, whereas SGD can make \(N\) cheap updates in the time +GD makes one – often yielding a good solution faster in practice, even +though SGD’s asymptotic error decays more slowly. As one lecture +succinctly puts it: “SGD can be super effective in terms of iteration +cost and memory, but SGD is slow to converge and can’t adapt to strong +convexity” . Thus, the break-even point depends on \(N\) and the desired +accuracy: for moderate accuracy on very large \(N\), SGD’s cheaper +updates win; for extremely high precision (very small \(\epsilon\)) on a +modest \(N\), GD’s fast convergence per step can be advantageous.

    +
    +
    +

    Non-Convex Problems#

    +

    In non-convex optimization (e.g. deep neural networks), neither GD nor +SGD guarantees global minima, but SGD often displays faster progress +in finding useful minima. Theoretical results here are weaker, usually +showing convergence to a stationary point \(\theta\) (\(|\nabla C|\) is +small) in expectation. For example, GD might require \(O(1/\epsilon^2)\) +iterations to ensure \(|\nabla C(\theta)| < \epsilon\), and SGD typically has +similar polynomial complexity (often worse due to gradient +noise). However, a noteworthy difference is that SGD’s stochasticity +can help escape saddle points or poor local minima. Random gradient +fluctuations act like implicit noise, helping the iterate “jump” out +of flat saddle regions where full-batch GD could stagnate . In fact, +research has shown that adding noise to GD can guarantee escaping +saddle points in polynomial time, and the inherent noise in SGD often +serves this role. Empirically, this means SGD can sometimes find a +lower loss basin faster, whereas full-batch GD might get “stuck” near +saddle points or need a very small learning rate to navigate complex +error surfaces . Overall, in modern high-dimensional machine learning, +SGD (or mini-batch SGD) is the workhorse for large non-convex problems +because it converges to good solutions much faster in practice, +despite the lack of a linear convergence guarantee. Full-batch GD is +rarely used on large neural networks, as it would require tiny steps +to avoid divergence and is extremely slow per iteration .

    +
    +
    +
    +

    Memory Usage and Scalability#

    +

    A major advantage of SGD is its memory efficiency in handling large +datasets. Full-batch GD requires access to the entire training set for +each iteration, which often means the whole dataset (or a large +subset) must reside in memory to compute \(\nabla C(\theta)\) . This results +in memory usage that scales linearly with the dataset size \(N\). For +instance, if each training sample is large (e.g. high-dimensional +features), computing a full gradient may require storing a substantial +portion of the data or all intermediate gradients until they are +aggregated. In contrast, SGD needs only a single (or a small +mini-batch of) training example(s) in memory at any time . The +algorithm processes one sample (or mini-batch) at a time and +immediately updates the model, discarding that sample before moving to +the next. This streaming approach means that memory footprint is +essentially independent of \(N\) (apart from storing the model +parameters themselves). As one source notes, gradient descent +“requires more memory than SGD” because it “must store the entire +dataset for each iteration,” whereas SGD “only needs to store the +current training example” . In practical terms, if you have a dataset +of size, say, 1 million examples, full-batch GD would need memory for +all million every step, while SGD could be implemented to load just +one example at a time – a crucial benefit if data are too large to fit +in RAM or GPU memory. This scalability makes SGD suitable for +large-scale learning: as long as you can stream data from disk, SGD +can handle arbitrarily large datasets with fixed memory. In fact, SGD +“does not need to remember which examples were visited” in the past, +allowing it to run in an online fashion on infinite data streams +. Full-batch GD, on the other hand, would require multiple passes +through a giant dataset per update (or a complex distributed memory +system), which is often infeasible.

    +

    There is also a secondary memory effect: computing a full-batch +gradient in deep learning requires storing all intermediate +activations for backpropagation across the entire batch. A very large +batch (approaching the full dataset) might exhaust GPU memory due to +the need to hold activation gradients for thousands or millions of +examples simultaneously. SGD/minibatches mitigate this by splitting +the workload – e.g. with a mini-batch of size 32 or 256, memory use +stays bounded, whereas a full-batch (size = \(N\)) forward/backward pass +could not even be executed if \(N\) is huge. Techniques like gradient +accumulation exist to simulate large-batch GD by summing many +small-batch gradients – but these still process data in manageable +chunks to avoid memory overflow. In summary, memory complexity for GD +grows with \(N\), while for SGD it remains \(O(1)\) w.r.t. dataset size +(only the model and perhaps a mini-batch reside in memory) . This is a +key reason why batch GD “does not scale” to very large data and why +virtually all large-scale machine learning algorithms rely on +stochastic or mini-batch methods.

    +
    +
    +

    Empirical Evidence: Convergence Time and Memory in Practice#

    +

    Empirical studies strongly support the theoretical trade-offs +above. In large-scale machine learning tasks, SGD often converges to a +good solution much faster in wall-clock time than full-batch GD, and +it uses far less memory. For example, Bottou & Bousquet (2008) +analyzed learning time under a fixed computational budget and +concluded that when data is abundant, it’s better to use a faster +(even if less precise) optimization method to process more examples in +the same time . This analysis showed that for large-scale problems, +processing more data with SGD yields lower error than spending the +time to do exact (batch) optimization on fewer data . In other words, +if you have a time budget, it’s often optimal to accept slightly +slower convergence per step (as with SGD) in exchange for being able +to use many more training samples in that time. This phenomenon is +borne out by experiments:

    +
    +

    Deep Neural Networks#

    +

    In modern deep learning, full-batch GD is so slow that it is rarely +attempted; instead, mini-batch SGD is standard. A recent study +demonstrated that it is possible to train a ResNet-50 on ImageNet +using full-batch gradient descent, but it required careful tuning +(e.g. gradient clipping, tiny learning rates) and vast computational +resources – and even then, each full-batch update was extremely +expensive.

    +

    Using a huge batch +(closer to full GD) tends to slow down convergence if the learning +rate is not scaled up, and often encounters optimization difficulties +(plateaus) that small batches avoid. +Empirically, small or medium +batch SGD finds minima in fewer clock hours because it can rapidly +loop over the data with gradient noise aiding exploration.

    +
    +
    +

    Memory constraints#

    +

    From a memory standpoint, practitioners note that batch GD becomes +infeasible on large data. For example, if one tried to do full-batch +training on a dataset that doesn’t fit in RAM or GPU memory, the +program would resort to heavy disk I/O or simply crash. SGD +circumvents this by processing mini-batches. Even in cases where data +does fit in memory, using a full batch can spike memory usage due to +storing all gradients. One empirical observation is that mini-batch +training has a “lower, fluctuating usage pattern” of memory, whereas +full-batch loading “quickly consumes memory (often exceeding limits)” +. This is especially relevant for graph neural networks or other +models where a “batch” may include a huge chunk of a graph: full-batch +gradient computation can exhaust GPU memory, whereas mini-batch +methods keep memory usage manageable .

    +

    In summary, SGD converges faster than full-batch GD in terms of actual +training time for large-scale problems, provided we measure +convergence as reaching a good-enough solution. Theoretical bounds +show SGD needs more iterations, but because it performs many more +updates per unit time (and requires far less memory), it often +achieves lower loss in a given time frame than GD. Full-batch GD might +take slightly fewer iterations in theory, but each iteration is so +costly that it is “slower… especially for large datasets” . Meanwhile, +memory scaling strongly favors SGD: GD’s memory cost grows with +dataset size, making it impractical beyond a point, whereas SGD’s +memory use is modest and mostly constant w.r.t. \(N\) . These +differences have made SGD (and mini-batch variants) the de facto +choice for training large machine learning models, from logistic +regression on millions of examples to deep neural networks with +billions of parameters. The consensus in both research and practice is +that for large-scale or high-dimensional tasks, SGD-type methods +converge quicker per unit of computation and handle memory constraints +better than standard full-batch gradient descent .

    +
    +

    Second moment of the gradient#

    In stochastic gradient descent, with and without momentum, we still @@ -2560,6 +2791,18 @@ centered matrix and/or vector that enter the fitting procedure.

  • Time decay rate
  • Code with a Number of Minibatches which varies
  • Replace or not
    • +
  • SGD vs Full-Batch GD: Convergence Speed and Memory Comparison +
    • +
  • Memory Usage and Scalability
    • +
  • Empirical Evidence: Convergence Time and Memory in Practice +
  • Second moment of the gradient
  • Challenge: Choosing a Fixed Learning Rate
  • Motivation for Adaptive Step Sizes
    • diff --git a/doc/LectureNotes/_build/jupyter_execute/week37.ipynb b/doc/LectureNotes/_build/jupyter_execute/week37.ipynb index 7bc622930..6bd1dea13 100644 --- a/doc/LectureNotes/_build/jupyter_execute/week37.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/week37.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "311a2385", + "id": "53d0b4e7", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "9e4484dc", + "id": "0c919844", "metadata": { "editable": true }, @@ -29,7 +29,7 @@ }, { "cell_type": "markdown", - "id": "a24010ae", + "id": "a5769b3d", "metadata": { "editable": true }, @@ -52,7 +52,7 @@ }, { "cell_type": "markdown", - "id": "4a291d59", + "id": "21f2937f", "metadata": { "editable": true }, @@ -69,7 +69,7 @@ }, { "cell_type": "markdown", - "id": "85c747e2", + "id": "da32a24e", "metadata": { "editable": true }, @@ -79,7 +79,7 @@ }, { "cell_type": "markdown", - "id": "6580dfe2", + "id": "2b7f8433", "metadata": { "editable": true }, @@ -103,7 +103,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "c2ddcfe5", + "id": "2a8c5baa", "metadata": { "collapsed": false, "editable": true @@ -117,7 +117,7 @@ }, { "cell_type": "markdown", - "id": "e1e8a5b2", + "id": "47d8423d", "metadata": { "editable": true }, @@ -128,7 +128,7 @@ }, { "cell_type": "markdown", - "id": "c8a5100b", + "id": "08d37b91", "metadata": { "editable": true }, @@ -140,7 +140,7 @@ }, { "cell_type": "markdown", - "id": "b026883e", + "id": "a3c412ef", "metadata": { "editable": true }, @@ -150,7 +150,7 @@ }, { "cell_type": "markdown", - "id": "3a2f7b75", + "id": "3f1b2071", "metadata": { "editable": true }, @@ -162,7 +162,7 @@ }, { "cell_type": "markdown", - "id": "6380eed5", + "id": "07536cbd", "metadata": { "editable": true }, @@ -176,7 +176,7 @@ }, { "cell_type": "markdown", - "id": "c5d3766d", + "id": "34287135", "metadata": { "editable": true }, @@ -192,7 +192,7 @@ }, { "cell_type": "markdown", - "id": "1d313807", + "id": "c1063bfb", "metadata": { "editable": true }, @@ -202,7 +202,7 @@ }, { "cell_type": "markdown", - "id": "bee64882", + "id": "d327d2e3", "metadata": { "editable": true }, @@ -214,7 +214,7 @@ }, { "cell_type": "markdown", - "id": "7ffe8d02", + "id": "b2c9a9cd", "metadata": { "editable": true }, @@ -224,7 +224,7 @@ }, { "cell_type": "markdown", - "id": "97225362", + "id": "062a7534", "metadata": { "editable": true }, @@ -236,7 +236,7 @@ }, { "cell_type": "markdown", - "id": "9fe2a0b3", + "id": "b1b15536", "metadata": { "editable": true }, @@ -250,7 +250,7 @@ }, { "cell_type": "markdown", - "id": "2e678439", + "id": "a259e250", "metadata": { "editable": true }, @@ -260,7 +260,7 @@ }, { "cell_type": "markdown", - "id": "5f45e358", + "id": "002197c1", "metadata": { "editable": true }, @@ -271,7 +271,7 @@ }, { "cell_type": "markdown", - "id": "1713ee43", + "id": "55e8dca9", "metadata": { "editable": true }, @@ -286,7 +286,7 @@ }, { "cell_type": "markdown", - "id": "671ea0fc", + "id": "a97ffaec", "metadata": { "editable": true }, @@ -296,7 +296,7 @@ }, { "cell_type": "markdown", - "id": "7df56d17", + "id": "b59a4220", "metadata": { "editable": true }, @@ -308,7 +308,7 @@ }, { "cell_type": "markdown", - "id": "5887c657", + "id": "ba5dcc08", "metadata": { "editable": true }, @@ -320,7 +320,7 @@ }, { "cell_type": "markdown", - "id": "5a012ac0", + "id": "d8907fed", "metadata": { "editable": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "markdown", - "id": "cf1fd4f4", + "id": "728d5b78", "metadata": { "editable": true }, @@ -348,7 +348,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "4417d3aa", + "id": "02d7e401", "metadata": { "collapsed": false, "editable": true @@ -407,7 +407,7 @@ }, { "cell_type": "markdown", - "id": "7d39d005", + "id": "4dd147c2", "metadata": { "editable": true }, @@ -419,7 +419,7 @@ }, { "cell_type": "markdown", - "id": "45a85d32", + "id": "75ca7f80", "metadata": { "editable": true }, @@ -431,7 +431,7 @@ }, { "cell_type": "markdown", - "id": "31d267ea", + "id": "5b897c75", "metadata": { "editable": true }, @@ -441,7 +441,7 @@ }, { "cell_type": "markdown", - "id": "f8f50b02", + "id": "46aa12f6", "metadata": { "editable": true }, @@ -455,7 +455,7 @@ }, { "cell_type": "markdown", - "id": "ac21d44c", + "id": "8ac05816", "metadata": { "editable": true }, @@ -465,7 +465,7 @@ }, { "cell_type": "markdown", - "id": "aae5aaa1", + "id": "cee76d94", "metadata": { "editable": true }, @@ -477,7 +477,7 @@ }, { "cell_type": "markdown", - "id": "319922a5", + "id": "88cf9577", "metadata": { "editable": true }, @@ -488,7 +488,7 @@ }, { "cell_type": "markdown", - "id": "724078a1", + "id": "0108d67e", "metadata": { "editable": true }, @@ -503,7 +503,7 @@ }, { "cell_type": "markdown", - "id": "dbc443e3", + "id": "1e307469", "metadata": { "editable": true }, @@ -517,7 +517,7 @@ }, { "cell_type": "markdown", - "id": "2ea2bf50", + "id": "c8dc7485", "metadata": { "editable": true }, @@ -528,7 +528,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "9f431da1", + "id": "2909407a", "metadata": { "collapsed": false, "editable": true @@ -589,7 +589,7 @@ }, { "cell_type": "markdown", - "id": "8aa155a9", + "id": "d25693ff", "metadata": { "editable": true }, @@ -611,7 +611,7 @@ }, { "cell_type": "markdown", - "id": "03bd2e44", + "id": "78b0bf65", "metadata": { "editable": true }, @@ -626,7 +626,7 @@ }, { "cell_type": "markdown", - "id": "0e101e2d", + "id": "9ee803d8", "metadata": { "editable": true }, @@ -637,7 +637,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "09ecede4", + "id": "ac420f7a", "metadata": { "collapsed": false, "editable": true @@ -703,7 +703,7 @@ }, { "cell_type": "markdown", - "id": "3489dbbc", + "id": "c548d574", "metadata": { "editable": true }, @@ -714,7 +714,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "426eaa39", + "id": "687e9d89", "metadata": { "collapsed": false, "editable": true @@ -788,7 +788,7 @@ }, { "cell_type": "markdown", - "id": "6220214d", + "id": "27a27a67", "metadata": { "editable": true }, @@ -807,7 +807,7 @@ }, { "cell_type": "markdown", - "id": "bf86ac65", + "id": "a12c19b2", "metadata": { "editable": true }, @@ -828,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "4ac61edb", + "id": "b8436434", "metadata": { "editable": true }, @@ -844,7 +844,7 @@ }, { "cell_type": "markdown", - "id": "0058008d", + "id": "f00e1864", "metadata": { "editable": true }, @@ -858,7 +858,7 @@ }, { "cell_type": "markdown", - "id": "f994e1e2", + "id": "ea91af30", "metadata": { "editable": true }, @@ -886,7 +886,7 @@ }, { "cell_type": "markdown", - "id": "842a8611", + "id": "bc9502a0", "metadata": { "editable": true }, @@ -918,7 +918,7 @@ }, { "cell_type": "markdown", - "id": "90bd121a", + "id": "6a236a2a", "metadata": { "editable": true }, @@ -935,7 +935,7 @@ }, { "cell_type": "markdown", - "id": "5cd81303", + "id": "29dc562b", "metadata": { "editable": true }, @@ -948,7 +948,7 @@ }, { "cell_type": "markdown", - "id": "60e085a9", + "id": "6a34f155", "metadata": { "editable": true }, @@ -961,7 +961,7 @@ }, { "cell_type": "markdown", - "id": "fef0100e", + "id": "0afd8cd8", "metadata": { "editable": true }, @@ -974,7 +974,7 @@ }, { "cell_type": "markdown", - "id": "aaba7f05", + "id": "f0b27e71", "metadata": { "editable": true }, @@ -988,7 +988,7 @@ }, { "cell_type": "markdown", - "id": "038b47ae", + "id": "3b04b9c6", "metadata": { "editable": true }, @@ -1010,7 +1010,7 @@ }, { "cell_type": "markdown", - "id": "0ad42833", + "id": "05eca708", "metadata": { "editable": true }, @@ -1025,7 +1025,7 @@ }, { "cell_type": "markdown", - "id": "64b15ba2", + "id": "473025f4", "metadata": { "editable": true }, @@ -1037,7 +1037,7 @@ }, { "cell_type": "markdown", - "id": "49c6adb0", + "id": "26e0b288", "metadata": { "editable": true }, @@ -1050,7 +1050,7 @@ }, { "cell_type": "markdown", - "id": "82873545", + "id": "091efee5", "metadata": { "editable": true }, @@ -1064,7 +1064,7 @@ }, { "cell_type": "markdown", - "id": "35a8e70d", + "id": "22c5f80e", "metadata": { "editable": true }, @@ -1075,7 +1075,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "6aa32b90", + "id": "102b1658", "metadata": { "collapsed": false, "editable": true @@ -1100,7 +1100,7 @@ }, { "cell_type": "markdown", - "id": "6e20f534", + "id": "79448e46", "metadata": { "editable": true }, @@ -1116,7 +1116,7 @@ }, { "cell_type": "markdown", - "id": "71745d3e", + "id": "dbc8b940", "metadata": { "editable": true }, @@ -1137,7 +1137,7 @@ }, { "cell_type": "markdown", - "id": "bad95be2", + "id": "b63ae18d", "metadata": { "editable": true }, @@ -1157,7 +1157,7 @@ }, { "cell_type": "markdown", - "id": "40b4d87e", + "id": "c5ee074e", "metadata": { "editable": true }, @@ -1176,7 +1176,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "1208bbec", + "id": "cfc48413", "metadata": { "collapsed": false, "editable": true @@ -1211,7 +1211,7 @@ }, { "cell_type": "markdown", - "id": "b83b5ed1", + "id": "fbb8c0eb", "metadata": { "editable": true }, @@ -1224,7 +1224,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "1f669db6", + "id": "cc1e51cd", "metadata": { "collapsed": false, "editable": true @@ -1301,7 +1301,7 @@ }, { "cell_type": "markdown", - "id": "3e9ed564", + "id": "22a23ea0", "metadata": { "editable": true }, @@ -1316,7 +1316,377 @@ }, { "cell_type": "markdown", - "id": "9c0ac318", + "id": "f0258497", + "metadata": { + "editable": true + }, + "source": [ + "## SGD vs Full-Batch GD: Convergence Speed and Memory Comparison" + ] + }, + { + "cell_type": "markdown", + "id": "69f1d941", + "metadata": { + "editable": true + }, + "source": [ + "### Theoretical Convergence Speed and convex optimization\n", + "\n", + "Consider minimizing an empirical cost function" + ] + }, + { + "cell_type": "markdown", + "id": "876e1d2b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\theta) =\\frac{1}{N}\\sum_{i=1}^N l_i(\\theta),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "67381fd3", + "metadata": { + "editable": true + }, + "source": [ + "where each $l_i(\\theta)$ is a\n", + "differentiable loss term. Gradient Descent (GD) updates parameters\n", + "using the full gradient $\\nabla C(\\theta)$, while Stochastic Gradient\n", + "Descent (SGD) uses a single sample (or mini-batch) gradient $\\nabla\n", + "l_i(\\theta)$ selected at random. In equation form, one GD step is:" + ] + }, + { + "cell_type": "markdown", + "id": "f08ec530", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_{t+1} = \\theta_t-\\eta \\nabla C(\\theta_t) =\\theta_t -\\eta \\frac{1}{N}\\sum_{i=1}^N \\nabla l_i(\\theta_t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3d99c5ee", + "metadata": { + "editable": true + }, + "source": [ + "whereas one SGD step is:" + ] + }, + { + "cell_type": "markdown", + "id": "03ecb7f9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_{t+1} = \\theta_t -\\eta \\nabla l_{i_t}(\\theta_t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a6db0c6c", + "metadata": { + "editable": true + }, + "source": [ + "with $i_t$ randomly chosen. On smooth convex problems, GD and SGD both\n", + "converge to the global minimum, but their rates differ. GD can take\n", + "larger, more stable steps since it uses the exact gradient, achieving\n", + "an error that decreases on the order of $O(1/t)$ per iteration for\n", + "convex objectives (and even exponentially fast for strongly convex\n", + "cases). In contrast, plain SGD has more variance in each step, leading\n", + "to sublinear convergence in expectation – typically $O(1/\\sqrt{t})$\n", + "for general convex objectives (\\thetaith appropriate diminishing step\n", + "sizes) . Intuitively, GD’s trajectory is smoother and more\n", + "predictable, while SGD’s path oscillates due to noise but costs far\n", + "less per iteration, enabling many more updates in the same time." + ] + }, + { + "cell_type": "markdown", + "id": "167f76aa", + "metadata": { + "editable": true + }, + "source": [ + "### Strongly Convex Case\n", + "\n", + "If $C(\\theta)$ is strongly convex and $L$-smooth (so GD enjoys linear\n", + "convergence), the gap $C(\\theta_t)-C(\\theta^*)$ for GD shrinks as" + ] + }, + { + "cell_type": "markdown", + "id": "d8d5cb23", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\theta_t) - C(\\theta^* ) \\le \\Big(1 - \\frac{\\mu}{L}\\Big)^t [C(\\theta_0)-C(\\theta^*)],\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e127c141", + "metadata": { + "editable": true + }, + "source": [ + "a geometric (linear) convergence per iteration . Achieving an\n", + "$\\epsilon$-accurate solution thus takes on the order of\n", + "$\\log(1/\\epsilon)$ iterations for GD. However, each GD iteration costs\n", + "$O(N)$ gradient evaluations. SGD cannot exploit strong convexity to\n", + "obtain a linear rate – instead, with a properly decaying step size\n", + "(e.g. $\\eta_t = \\frac{1}{\\mu t}$) or iterate averaging, SGD attains an\n", + "$O(1/t)$ convergence rate in expectation . For example, one result\n", + "of Moulines and Bach 2011, see shows that with $\\eta_t = \\Theta(1/t)$," + ] + }, + { + "cell_type": "markdown", + "id": "7ea2c03c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathbb{E}[C(\\theta_t) - C(\\theta^*)] = O(1/t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e2e33c54", + "metadata": { + "editable": true + }, + "source": [ + "for strongly convex, smooth $F$ . This $1/t$ rate is slower per\n", + "iteration than GD’s exponential decay, but each SGD iteration is $N$\n", + "times cheaper. In fact, to reach error $\\epsilon$, plain SGD needs on\n", + "the order of $T=O(1/\\epsilon)$ iterations (sub-linear convergence),\n", + "while GD needs $O(\\log(1/\\epsilon))$ iterations. When accounting for\n", + "cost-per-iteration, GD requires $O(N \\log(1/\\epsilon))$ total gradient\n", + "computations versus SGD’s $O(1/\\epsilon)$ single-sample\n", + "computations. In large-scale regimes (huge $N$), SGD can be\n", + "faster in wall-clock time because $N \\log(1/\\epsilon)$ may far exceed\n", + "$1/\\epsilon$ for reasonable accuracy levels. In other words,\n", + "with millions of data points, one epoch of GD (one full gradient) is\n", + "extremely costly, whereas SGD can make $N$ cheap updates in the time\n", + "GD makes one – often yielding a good solution faster in practice, even\n", + "though SGD’s asymptotic error decays more slowly. As one lecture\n", + "succinctly puts it: “SGD can be super effective in terms of iteration\n", + "cost and memory, but SGD is slow to converge and can’t adapt to strong\n", + "convexity” . Thus, the break-even point depends on $N$ and the desired\n", + "accuracy: for moderate accuracy on very large $N$, SGD’s cheaper\n", + "updates win; for extremely high precision (very small $\\epsilon$) on a\n", + "modest $N$, GD’s fast convergence per step can be advantageous." + ] + }, + { + "cell_type": "markdown", + "id": "88f943e6", + "metadata": { + "editable": true + }, + "source": [ + "### Non-Convex Problems\n", + "\n", + "In non-convex optimization (e.g. deep neural networks), neither GD nor\n", + "SGD guarantees global minima, but SGD often displays faster progress\n", + "in finding useful minima. Theoretical results here are weaker, usually\n", + "showing convergence to a stationary point $\\theta$ ($|\\nabla C|$ is\n", + "small) in expectation. For example, GD might require $O(1/\\epsilon^2)$\n", + "iterations to ensure $|\\nabla C(\\theta)| < \\epsilon$, and SGD typically has\n", + "similar polynomial complexity (often worse due to gradient\n", + "noise). However, a noteworthy difference is that SGD’s stochasticity\n", + "can help escape saddle points or poor local minima. Random gradient\n", + "fluctuations act like implicit noise, helping the iterate “jump” out\n", + "of flat saddle regions where full-batch GD could stagnate . In fact,\n", + "research has shown that adding noise to GD can guarantee escaping\n", + "saddle points in polynomial time, and the inherent noise in SGD often\n", + "serves this role. Empirically, this means SGD can sometimes find a\n", + "lower loss basin faster, whereas full-batch GD might get “stuck” near\n", + "saddle points or need a very small learning rate to navigate complex\n", + "error surfaces . Overall, in modern high-dimensional machine learning,\n", + "SGD (or mini-batch SGD) is the workhorse for large non-convex problems\n", + "because it converges to good solutions much faster in practice,\n", + "despite the lack of a linear convergence guarantee. Full-batch GD is\n", + "rarely used on large neural networks, as it would require tiny steps\n", + "to avoid divergence and is extremely slow per iteration ." + ] + }, + { + "cell_type": "markdown", + "id": "aa4a8927", + "metadata": { + "editable": true + }, + "source": [ + "## Memory Usage and Scalability\n", + "\n", + "A major advantage of SGD is its memory efficiency in handling large\n", + "datasets. Full-batch GD requires access to the entire training set for\n", + "each iteration, which often means the whole dataset (or a large\n", + "subset) must reside in memory to compute $\\nabla C(\\theta)$ . This results\n", + "in memory usage that scales linearly with the dataset size $N$. For\n", + "instance, if each training sample is large (e.g. high-dimensional\n", + "features), computing a full gradient may require storing a substantial\n", + "portion of the data or all intermediate gradients until they are\n", + "aggregated. In contrast, SGD needs only a single (or a small\n", + "mini-batch of) training example(s) in memory at any time . The\n", + "algorithm processes one sample (or mini-batch) at a time and\n", + "immediately updates the model, discarding that sample before moving to\n", + "the next. This streaming approach means that memory footprint is\n", + "essentially independent of $N$ (apart from storing the model\n", + "parameters themselves). As one source notes, gradient descent\n", + "“requires more memory than SGD” because it “must store the entire\n", + "dataset for each iteration,” whereas SGD “only needs to store the\n", + "current training example” . In practical terms, if you have a dataset\n", + "of size, say, 1 million examples, full-batch GD would need memory for\n", + "all million every step, while SGD could be implemented to load just\n", + "one example at a time – a crucial benefit if data are too large to fit\n", + "in RAM or GPU memory. This scalability makes SGD suitable for\n", + "large-scale learning: as long as you can stream data from disk, SGD\n", + "can handle arbitrarily large datasets with fixed memory. In fact, SGD\n", + "“does not need to remember which examples were visited” in the past,\n", + "allowing it to run in an online fashion on infinite data streams\n", + ". Full-batch GD, on the other hand, would require multiple passes\n", + "through a giant dataset per update (or a complex distributed memory\n", + "system), which is often infeasible.\n", + "\n", + "There is also a secondary memory effect: computing a full-batch\n", + "gradient in deep learning requires storing all intermediate\n", + "activations for backpropagation across the entire batch. A very large\n", + "batch (approaching the full dataset) might exhaust GPU memory due to\n", + "the need to hold activation gradients for thousands or millions of\n", + "examples simultaneously. SGD/minibatches mitigate this by splitting\n", + "the workload – e.g. with a mini-batch of size 32 or 256, memory use\n", + "stays bounded, whereas a full-batch (size = $N$) forward/backward pass\n", + "could not even be executed if $N$ is huge. Techniques like gradient\n", + "accumulation exist to simulate large-batch GD by summing many\n", + "small-batch gradients – but these still process data in manageable\n", + "chunks to avoid memory overflow. In summary, memory complexity for GD\n", + "grows with $N$, while for SGD it remains $O(1)$ w.r.t. dataset size\n", + "(only the model and perhaps a mini-batch reside in memory) . This is a\n", + "key reason why batch GD “does not scale” to very large data and why\n", + "virtually all large-scale machine learning algorithms rely on\n", + "stochastic or mini-batch methods." + ] + }, + { + "cell_type": "markdown", + "id": "72d0192b", + "metadata": { + "editable": true + }, + "source": [ + "## Empirical Evidence: Convergence Time and Memory in Practice\n", + "\n", + "Empirical studies strongly support the theoretical trade-offs\n", + "above. In large-scale machine learning tasks, SGD often converges to a\n", + "good solution much faster in wall-clock time than full-batch GD, and\n", + "it uses far less memory. For example, Bottou & Bousquet (2008)\n", + "analyzed learning time under a fixed computational budget and\n", + "concluded that when data is abundant, it’s better to use a faster\n", + "(even if less precise) optimization method to process more examples in\n", + "the same time . This analysis showed that for large-scale problems,\n", + "processing more data with SGD yields lower error than spending the\n", + "time to do exact (batch) optimization on fewer data . In other words,\n", + "if you have a time budget, it’s often optimal to accept slightly\n", + "slower convergence per step (as with SGD) in exchange for being able\n", + "to use many more training samples in that time. This phenomenon is\n", + "borne out by experiments:" + ] + }, + { + "cell_type": "markdown", + "id": "44fcd423", + "metadata": { + "editable": true + }, + "source": [ + "### Deep Neural Networks\n", + "\n", + "In modern deep learning, full-batch GD is so slow that it is rarely\n", + "attempted; instead, mini-batch SGD is standard. A recent study\n", + "demonstrated that it is possible to train a ResNet-50 on ImageNet\n", + "using full-batch gradient descent, but it required careful tuning\n", + "(e.g. gradient clipping, tiny learning rates) and vast computational\n", + "resources – and even then, each full-batch update was extremely\n", + "expensive.\n", + "\n", + "Using a huge batch\n", + "(closer to full GD) tends to slow down convergence if the learning\n", + "rate is not scaled up, and often encounters optimization difficulties\n", + "(plateaus) that small batches avoid.\n", + "Empirically, small or medium\n", + "batch SGD finds minima in fewer clock hours because it can rapidly\n", + "loop over the data with gradient noise aiding exploration." + ] + }, + { + "cell_type": "markdown", + "id": "8de0942f", + "metadata": { + "editable": true + }, + "source": [ + "### Memory constraints\n", + "\n", + "From a memory standpoint, practitioners note that batch GD becomes\n", + "infeasible on large data. For example, if one tried to do full-batch\n", + "training on a dataset that doesn’t fit in RAM or GPU memory, the\n", + "program would resort to heavy disk I/O or simply crash. SGD\n", + "circumvents this by processing mini-batches. Even in cases where data\n", + "does fit in memory, using a full batch can spike memory usage due to\n", + "storing all gradients. One empirical observation is that mini-batch\n", + "training has a “lower, fluctuating usage pattern” of memory, whereas\n", + "full-batch loading “quickly consumes memory (often exceeding limits)”\n", + ". This is especially relevant for graph neural networks or other\n", + "models where a “batch” may include a huge chunk of a graph: full-batch\n", + "gradient computation can exhaust GPU memory, whereas mini-batch\n", + "methods keep memory usage manageable .\n", + "\n", + "In summary, SGD converges faster than full-batch GD in terms of actual\n", + "training time for large-scale problems, provided we measure\n", + "convergence as reaching a good-enough solution. Theoretical bounds\n", + "show SGD needs more iterations, but because it performs many more\n", + "updates per unit time (and requires far less memory), it often\n", + "achieves lower loss in a given time frame than GD. Full-batch GD might\n", + "take slightly fewer iterations in theory, but each iteration is so\n", + "costly that it is “slower… especially for large datasets” . Meanwhile,\n", + "memory scaling strongly favors SGD: GD’s memory cost grows with\n", + "dataset size, making it impractical beyond a point, whereas SGD’s\n", + "memory use is modest and mostly constant w.r.t. $N$ . These\n", + "differences have made SGD (and mini-batch variants) the de facto\n", + "choice for training large machine learning models, from logistic\n", + "regression on millions of examples to deep neural networks with\n", + "billions of parameters. The consensus in both research and practice is\n", + "that for large-scale or high-dimensional tasks, SGD-type methods\n", + "converge quicker per unit of computation and handle memory constraints\n", + "better than standard full-batch gradient descent ." + ] + }, + { + "cell_type": "markdown", + "id": "f08a4bbe", "metadata": { "editable": true }, @@ -1347,7 +1717,7 @@ }, { "cell_type": "markdown", - "id": "d8f518c4", + "id": "dc1fa30f", "metadata": { "editable": true }, @@ -1369,7 +1739,7 @@ }, { "cell_type": "markdown", - "id": "3dcb89bd", + "id": "1fbfcb5e", "metadata": { "editable": true }, @@ -1389,7 +1759,7 @@ }, { "cell_type": "markdown", - "id": "8f258bc2", + "id": "83d5dfc2", "metadata": { "editable": true }, @@ -1405,7 +1775,7 @@ }, { "cell_type": "markdown", - "id": "2a3715f8", + "id": "4cf425f2", "metadata": { "editable": true }, @@ -1425,7 +1795,7 @@ }, { "cell_type": "markdown", - "id": "a1d9578a", + "id": "a8de083c", "metadata": { "editable": true }, @@ -1437,7 +1807,7 @@ }, { "cell_type": "markdown", - "id": "b6b5bc5e", + "id": "f8b98ecd", "metadata": { "editable": true }, @@ -1449,7 +1819,7 @@ }, { "cell_type": "markdown", - "id": "44b313c8", + "id": "c41121c9", "metadata": { "editable": true }, @@ -1461,7 +1831,7 @@ }, { "cell_type": "markdown", - "id": "b56c85b9", + "id": "0c9cde87", "metadata": { "editable": true }, @@ -1473,7 +1843,7 @@ }, { "cell_type": "markdown", - "id": "5bcc6bd2", + "id": "9079853e", "metadata": { "editable": true }, @@ -1484,7 +1854,7 @@ }, { "cell_type": "markdown", - "id": "41fc9f01", + "id": "1b2340aa", "metadata": { "editable": true }, @@ -1496,7 +1866,7 @@ }, { "cell_type": "markdown", - "id": "8151719b", + "id": "1c63eff7", "metadata": { "editable": true }, @@ -1506,7 +1876,7 @@ }, { "cell_type": "markdown", - "id": "bb75b0ad", + "id": "e05e89e4", "metadata": { "editable": true }, @@ -1518,7 +1888,7 @@ }, { "cell_type": "markdown", - "id": "3c71fd46", + "id": "b3cbe567", "metadata": { "editable": true }, @@ -1528,7 +1898,7 @@ }, { "cell_type": "markdown", - "id": "1d835a18", + "id": "5d2f1096", "metadata": { "editable": true }, @@ -1549,7 +1919,7 @@ }, { "cell_type": "markdown", - "id": "77dcc8c3", + "id": "4c4f3846", "metadata": { "editable": true }, @@ -1562,7 +1932,7 @@ }, { "cell_type": "markdown", - "id": "21161d57", + "id": "57d24251", "metadata": { "editable": true }, @@ -1574,7 +1944,7 @@ }, { "cell_type": "markdown", - "id": "e87e09a9", + "id": "caff3ad3", "metadata": { "editable": true }, @@ -1591,7 +1961,7 @@ }, { "cell_type": "markdown", - "id": "1a98c681", + "id": "67133da8", "metadata": { "editable": true }, @@ -1607,7 +1977,7 @@ }, { "cell_type": "markdown", - "id": "8b337277", + "id": "d2d2d644", "metadata": { "editable": true }, @@ -1629,7 +1999,7 @@ }, { "cell_type": "markdown", - "id": "af77b83f", + "id": "897d1ca3", "metadata": { "editable": true }, @@ -1647,7 +2017,7 @@ }, { "cell_type": "markdown", - "id": "bc924f77", + "id": "549532b3", "metadata": { "editable": true }, @@ -1667,7 +2037,7 @@ }, { "cell_type": "markdown", - "id": "86e5ab5e", + "id": "f014a3a2", "metadata": { "editable": true }, @@ -1681,7 +2051,7 @@ }, { "cell_type": "markdown", - "id": "949f359d", + "id": "67bed63f", "metadata": { "editable": true }, @@ -1693,7 +2063,7 @@ }, { "cell_type": "markdown", - "id": "0ba26be3", + "id": "3014fe59", "metadata": { "editable": true }, @@ -1705,7 +2075,7 @@ }, { "cell_type": "markdown", - "id": "4fb9b2a2", + "id": "a99d9c1c", "metadata": { "editable": true }, @@ -1717,7 +2087,7 @@ }, { "cell_type": "markdown", - "id": "8711e597", + "id": "907f9915", "metadata": { "editable": true }, @@ -1729,7 +2099,7 @@ }, { "cell_type": "markdown", - "id": "49e6e73d", + "id": "551eb7db", "metadata": { "editable": true }, @@ -1740,7 +2110,7 @@ }, { "cell_type": "markdown", - "id": "ca5bb491", + "id": "ea8ae470", "metadata": { "editable": true }, @@ -1752,7 +2122,7 @@ }, { "cell_type": "markdown", - "id": "5e19d7bf", + "id": "9f5d78fd", "metadata": { "editable": true }, @@ -1766,7 +2136,7 @@ }, { "cell_type": "markdown", - "id": "f79d952e", + "id": "8291642f", "metadata": { "editable": true }, @@ -1777,7 +2147,7 @@ }, { "cell_type": "markdown", - "id": "13e9862f", + "id": "ee0e74ec", "metadata": { "editable": true }, @@ -1789,7 +2159,7 @@ }, { "cell_type": "markdown", - "id": "5693500e", + "id": "3699f7e5", "metadata": { "editable": true }, @@ -1811,7 +2181,7 @@ }, { "cell_type": "markdown", - "id": "65a5e1e7", + "id": "16cdd781", "metadata": { "editable": true }, @@ -1835,7 +2205,7 @@ }, { "cell_type": "markdown", - "id": "27686255", + "id": "779881a9", "metadata": { "editable": true }, @@ -1851,7 +2221,7 @@ }, { "cell_type": "markdown", - "id": "f3dfc1e2", + "id": "0724c747", "metadata": { "editable": true }, @@ -1867,7 +2237,7 @@ }, { "cell_type": "markdown", - "id": "045d399c", + "id": "02a113cb", "metadata": { "editable": true }, @@ -1881,7 +2251,7 @@ }, { "cell_type": "markdown", - "id": "4e75ee41", + "id": "e013ce1a", "metadata": { "editable": true }, @@ -1899,7 +2269,7 @@ }, { "cell_type": "markdown", - "id": "ddbb28ab", + "id": "1baa5b4e", "metadata": { "editable": true }, @@ -1920,7 +2290,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "dae38b6c", + "id": "8aab54bd", "metadata": { "collapsed": false, "editable": true @@ -1980,7 +2350,7 @@ }, { "cell_type": "markdown", - "id": "ca5a343a", + "id": "26c6e4f1", "metadata": { "editable": true }, @@ -1991,7 +2361,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "08d97c1e", + "id": "226c13ec", "metadata": { "collapsed": false, "editable": true @@ -2055,7 +2425,7 @@ }, { "cell_type": "markdown", - "id": "727d8fc3", + "id": "6895dbfe", "metadata": { "editable": true }, @@ -2070,7 +2440,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "4e41c003", + "id": "f8d01982", "metadata": { "collapsed": false, "editable": true @@ -2154,7 +2524,7 @@ }, { "cell_type": "markdown", - "id": "fe00db52", + "id": "cffe8367", "metadata": { "editable": true }, @@ -2165,7 +2535,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "8f22105b", + "id": "b57871a9", "metadata": { "collapsed": false, "editable": true @@ -2243,7 +2613,7 @@ }, { "cell_type": "markdown", - "id": "8956bf7a", + "id": "37b273c1", "metadata": { "editable": true }, @@ -2256,7 +2626,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "044275ef", + "id": "9ff0eb69", "metadata": { "collapsed": false, "editable": true @@ -2300,7 +2670,7 @@ }, { "cell_type": "markdown", - "id": "353b50b3", + "id": "e7d143b6", "metadata": { "editable": true }, @@ -2311,7 +2681,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "fdc8debd", + "id": "9b5e2d1d", "metadata": { "collapsed": false, "editable": true @@ -2370,7 +2740,7 @@ }, { "cell_type": "markdown", - "id": "b738f1b8", + "id": "8b6fb13f", "metadata": { "editable": true }, @@ -2380,7 +2750,7 @@ }, { "cell_type": "markdown", - "id": "65ce93ba", + "id": "06c3f4bb", "metadata": { "editable": true }, @@ -2391,7 +2761,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "604d7286", + "id": "4abf9ccd", "metadata": { "collapsed": false, "editable": true @@ -2456,7 +2826,7 @@ }, { "cell_type": "markdown", - "id": "e663a714", + "id": "18d42e29", "metadata": { "editable": true }, @@ -2467,7 +2837,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "749fa687", + "id": "03415114", "metadata": { "collapsed": false, "editable": true @@ -2537,7 +2907,7 @@ }, { "cell_type": "markdown", - "id": "8801fcd5", + "id": "41120d8f", "metadata": { "editable": true }, @@ -2554,7 +2924,7 @@ }, { "cell_type": "markdown", - "id": "8ea68725", + "id": "16eb2a88", "metadata": { "editable": true }, @@ -2582,7 +2952,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "04811786", + "id": "a6df3a5c", "metadata": { "collapsed": false, "editable": true @@ -2602,7 +2972,7 @@ }, { "cell_type": "markdown", - "id": "b0e7cc2c", + "id": "fc15d89b", "metadata": { "editable": true }, @@ -2618,7 +2988,7 @@ }, { "cell_type": "markdown", - "id": "f8a8132d", + "id": "4e4b5ee0", "metadata": { "editable": true }, @@ -2638,7 +3008,7 @@ }, { "cell_type": "markdown", - "id": "03eca41f", + "id": "4455b9a0", "metadata": { "editable": true }, @@ -2665,7 +3035,7 @@ }, { "cell_type": "markdown", - "id": "710e8f88", + "id": "9592eb20", "metadata": { "editable": true }, @@ -2678,7 +3048,7 @@ }, { "cell_type": "markdown", - "id": "5d3df9bf", + "id": "e3022ea8", "metadata": { "editable": true }, @@ -2690,7 +3060,7 @@ }, { "cell_type": "markdown", - "id": "be0fd5f1", + "id": "246a28bf", "metadata": { "editable": true }, @@ -2705,7 +3075,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "2a0924bb", + "id": "d132a060", "metadata": { "collapsed": false, "editable": true @@ -2732,7 +3102,7 @@ }, { "cell_type": "markdown", - "id": "d116f448", + "id": "9e70a01f", "metadata": { "editable": true }, @@ -2746,7 +3116,7 @@ }, { "cell_type": "markdown", - "id": "41caea07", + "id": "2c0ad6b4", "metadata": { "editable": true }, @@ -2758,7 +3128,7 @@ }, { "cell_type": "markdown", - "id": "1fa96f7c", + "id": "3c6081b8", "metadata": { "editable": true }, @@ -2775,7 +3145,7 @@ }, { "cell_type": "markdown", - "id": "70038d6a", + "id": "845af933", "metadata": { "editable": true }, @@ -2787,7 +3157,7 @@ }, { "cell_type": "markdown", - "id": "852a77d0", + "id": "564afbbd", "metadata": { "editable": true }, @@ -2797,7 +3167,7 @@ }, { "cell_type": "markdown", - "id": "fc4afaaf", + "id": "c4088263", "metadata": { "editable": true }, @@ -2809,7 +3179,7 @@ }, { "cell_type": "markdown", - "id": "94b18ced", + "id": "96983e3d", "metadata": { "editable": true }, @@ -2819,7 +3189,7 @@ }, { "cell_type": "markdown", - "id": "d7a95314", + "id": "91d029d7", "metadata": { "editable": true }, @@ -2831,7 +3201,7 @@ }, { "cell_type": "markdown", - "id": "eaf6a485", + "id": "20d351f6", "metadata": { "editable": true }, @@ -2842,7 +3212,7 @@ }, { "cell_type": "markdown", - "id": "3d9442a2", + "id": "7a8e79fd", "metadata": { "editable": true }, @@ -2854,7 +3224,7 @@ }, { "cell_type": "markdown", - "id": "e4aeef17", + "id": "4ec7ad68", "metadata": { "editable": true }, @@ -2864,7 +3234,7 @@ }, { "cell_type": "markdown", - "id": "4ce9dee9", + "id": "df09c13b", "metadata": { "editable": true }, @@ -2876,7 +3246,7 @@ }, { "cell_type": "markdown", - "id": "752ce099", + "id": "bb2a9c1f", "metadata": { "editable": true }, @@ -2886,7 +3256,7 @@ }, { "cell_type": "markdown", - "id": "7cad5229", + "id": "b3507e2d", "metadata": { "editable": true }, @@ -2898,7 +3268,7 @@ }, { "cell_type": "markdown", - "id": "46f1aaf9", + "id": "2010542f", "metadata": { "editable": true }, @@ -2908,7 +3278,7 @@ }, { "cell_type": "markdown", - "id": "7d25a9fb", + "id": "e03a1590", "metadata": { "editable": true }, @@ -2920,7 +3290,7 @@ }, { "cell_type": "markdown", - "id": "57b4c7d9", + "id": "71872755", "metadata": { "editable": true }, @@ -2930,7 +3300,7 @@ }, { "cell_type": "markdown", - "id": "fb833214", + "id": "167238dc", "metadata": { "editable": true }, @@ -2942,7 +3312,7 @@ }, { "cell_type": "markdown", - "id": "5fa29cd3", + "id": "38d0cc0f", "metadata": { "editable": true }, @@ -2952,7 +3322,7 @@ }, { "cell_type": "markdown", - "id": "6c0e668d", + "id": "e9e1beb9", "metadata": { "editable": true }, @@ -2964,7 +3334,7 @@ }, { "cell_type": "markdown", - "id": "9d928664", + "id": "9a8576e4", "metadata": { "editable": true }, @@ -2974,7 +3344,7 @@ }, { "cell_type": "markdown", - "id": "65434b84", + "id": "937d703f", "metadata": { "editable": true }, @@ -2986,7 +3356,7 @@ }, { "cell_type": "markdown", - "id": "127c9817", + "id": "e4723b95", "metadata": { "editable": true }, @@ -2996,7 +3366,7 @@ }, { "cell_type": "markdown", - "id": "46f45c10", + "id": "6df6f6d8", "metadata": { "editable": true }, @@ -3008,7 +3378,7 @@ }, { "cell_type": "markdown", - "id": "4fbaa69a", + "id": "39bdaf00", "metadata": { "editable": true }, @@ -3020,7 +3390,7 @@ }, { "cell_type": "markdown", - "id": "25f1abd4", + "id": "e4584236", "metadata": { "editable": true }, @@ -3032,7 +3402,7 @@ }, { "cell_type": "markdown", - "id": "9fd5ef9e", + "id": "d0c5d728", "metadata": { "editable": true }, @@ -3042,7 +3412,7 @@ }, { "cell_type": "markdown", - "id": "f1cb8e35", + "id": "9b637fd2", "metadata": { "editable": true }, @@ -3054,7 +3424,7 @@ }, { "cell_type": "markdown", - "id": "c0c5100a", + "id": "9627e6fb", "metadata": { "editable": true }, @@ -3067,7 +3437,7 @@ }, { "cell_type": "markdown", - "id": "c80e55cb", + "id": "662fe97e", "metadata": { "editable": true }, @@ -3079,7 +3449,7 @@ }, { "cell_type": "markdown", - "id": "a47f5c5e", + "id": "fa2d5cb5", "metadata": { "editable": true }, @@ -3093,7 +3463,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "e093186c", + "id": "a530b3ba", "metadata": { "collapsed": false, "editable": true @@ -3190,7 +3560,7 @@ }, { "cell_type": "markdown", - "id": "de555fff", + "id": "58cc6d9d", "metadata": { "editable": true }, @@ -3211,7 +3581,7 @@ }, { "cell_type": "markdown", - "id": "72178d39", + "id": "6a2d3f87", "metadata": { "editable": true }, @@ -3223,7 +3593,7 @@ }, { "cell_type": "markdown", - "id": "8e5d822b", + "id": "9eab78a9", "metadata": { "editable": true }, @@ -3233,7 +3603,7 @@ }, { "cell_type": "markdown", - "id": "e9218f82", + "id": "50d7f5c3", "metadata": { "editable": true }, @@ -3245,7 +3615,7 @@ }, { "cell_type": "markdown", - "id": "2223d1b1", + "id": "4c96e589", "metadata": { "editable": true }, @@ -3255,7 +3625,7 @@ }, { "cell_type": "markdown", - "id": "e5474a5b", + "id": "b57830ea", "metadata": { "editable": true }, @@ -3267,7 +3637,7 @@ }, { "cell_type": "markdown", - "id": "691295ed", + "id": "05325bc6", "metadata": { "editable": true }, @@ -3285,7 +3655,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "e243cef5", + "id": "d95b15bc", "metadata": { "collapsed": false, "editable": true @@ -3361,7 +3731,7 @@ }, { "cell_type": "markdown", - "id": "ef2eaa7a", + "id": "b88ebede", "metadata": { "editable": true }, @@ -3375,7 +3745,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "546e3504", + "id": "47036b16", "metadata": { "collapsed": false, "editable": true @@ -3464,7 +3834,7 @@ }, { "cell_type": "markdown", - "id": "f6787352", + "id": "52faee2f", "metadata": { "editable": true }, diff --git a/doc/LectureNotes/week37.ipynb b/doc/LectureNotes/week37.ipynb index 9038066ab..8e275c56f 100644 --- a/doc/LectureNotes/week37.ipynb +++ b/doc/LectureNotes/week37.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "311a2385", + "id": "53d0b4e7", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "9e4484dc", + "id": "0c919844", "metadata": { "editable": true }, @@ -29,7 +29,7 @@ }, { "cell_type": "markdown", - "id": "a24010ae", + "id": "a5769b3d", "metadata": { "editable": true }, @@ -52,7 +52,7 @@ }, { "cell_type": "markdown", - "id": "4a291d59", + "id": "21f2937f", "metadata": { "editable": true }, @@ -69,7 +69,7 @@ }, { "cell_type": "markdown", - "id": "85c747e2", + "id": "da32a24e", "metadata": { "editable": true }, @@ -79,7 +79,7 @@ }, { "cell_type": "markdown", - "id": "6580dfe2", + "id": "2b7f8433", "metadata": { "editable": true }, @@ -103,7 +103,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "c2ddcfe5", + "id": "2a8c5baa", "metadata": { "collapsed": false, "editable": true @@ -117,7 +117,7 @@ }, { "cell_type": "markdown", - "id": "e1e8a5b2", + "id": "47d8423d", "metadata": { "editable": true }, @@ -128,7 +128,7 @@ }, { "cell_type": "markdown", - "id": "c8a5100b", + "id": "08d37b91", "metadata": { "editable": true }, @@ -140,7 +140,7 @@ }, { "cell_type": "markdown", - "id": "b026883e", + "id": "a3c412ef", "metadata": { "editable": true }, @@ -150,7 +150,7 @@ }, { "cell_type": "markdown", - "id": "3a2f7b75", + "id": "3f1b2071", "metadata": { "editable": true }, @@ -162,7 +162,7 @@ }, { "cell_type": "markdown", - "id": "6380eed5", + "id": "07536cbd", "metadata": { "editable": true }, @@ -176,7 +176,7 @@ }, { "cell_type": "markdown", - "id": "c5d3766d", + "id": "34287135", "metadata": { "editable": true }, @@ -192,7 +192,7 @@ }, { "cell_type": "markdown", - "id": "1d313807", + "id": "c1063bfb", "metadata": { "editable": true }, @@ -202,7 +202,7 @@ }, { "cell_type": "markdown", - "id": "bee64882", + "id": "d327d2e3", "metadata": { "editable": true }, @@ -214,7 +214,7 @@ }, { "cell_type": "markdown", - "id": "7ffe8d02", + "id": "b2c9a9cd", "metadata": { "editable": true }, @@ -224,7 +224,7 @@ }, { "cell_type": "markdown", - "id": "97225362", + "id": "062a7534", "metadata": { "editable": true }, @@ -236,7 +236,7 @@ }, { "cell_type": "markdown", - "id": "9fe2a0b3", + "id": "b1b15536", "metadata": { "editable": true }, @@ -250,7 +250,7 @@ }, { "cell_type": "markdown", - "id": "2e678439", + "id": "a259e250", "metadata": { "editable": true }, @@ -260,7 +260,7 @@ }, { "cell_type": "markdown", - "id": "5f45e358", + "id": "002197c1", "metadata": { "editable": true }, @@ -271,7 +271,7 @@ }, { "cell_type": "markdown", - "id": "1713ee43", + "id": "55e8dca9", "metadata": { "editable": true }, @@ -286,7 +286,7 @@ }, { "cell_type": "markdown", - "id": "671ea0fc", + "id": "a97ffaec", "metadata": { "editable": true }, @@ -296,7 +296,7 @@ }, { "cell_type": "markdown", - "id": "7df56d17", + "id": "b59a4220", "metadata": { "editable": true }, @@ -308,7 +308,7 @@ }, { "cell_type": "markdown", - "id": "5887c657", + "id": "ba5dcc08", "metadata": { "editable": true }, @@ -320,7 +320,7 @@ }, { "cell_type": "markdown", - "id": "5a012ac0", + "id": "d8907fed", "metadata": { "editable": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "markdown", - "id": "cf1fd4f4", + "id": "728d5b78", "metadata": { "editable": true }, @@ -348,7 +348,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "4417d3aa", + "id": "02d7e401", "metadata": { "collapsed": false, "editable": true @@ -407,7 +407,7 @@ }, { "cell_type": "markdown", - "id": "7d39d005", + "id": "4dd147c2", "metadata": { "editable": true }, @@ -419,7 +419,7 @@ }, { "cell_type": "markdown", - "id": "45a85d32", + "id": "75ca7f80", "metadata": { "editable": true }, @@ -431,7 +431,7 @@ }, { "cell_type": "markdown", - "id": "31d267ea", + "id": "5b897c75", "metadata": { "editable": true }, @@ -441,7 +441,7 @@ }, { "cell_type": "markdown", - "id": "f8f50b02", + "id": "46aa12f6", "metadata": { "editable": true }, @@ -455,7 +455,7 @@ }, { "cell_type": "markdown", - "id": "ac21d44c", + "id": "8ac05816", "metadata": { "editable": true }, @@ -465,7 +465,7 @@ }, { "cell_type": "markdown", - "id": "aae5aaa1", + "id": "cee76d94", "metadata": { "editable": true }, @@ -477,7 +477,7 @@ }, { "cell_type": "markdown", - "id": "319922a5", + "id": "88cf9577", "metadata": { "editable": true }, @@ -488,7 +488,7 @@ }, { "cell_type": "markdown", - "id": "724078a1", + "id": "0108d67e", "metadata": { "editable": true }, @@ -503,7 +503,7 @@ }, { "cell_type": "markdown", - "id": "dbc443e3", + "id": "1e307469", "metadata": { "editable": true }, @@ -517,7 +517,7 @@ }, { "cell_type": "markdown", - "id": "2ea2bf50", + "id": "c8dc7485", "metadata": { "editable": true }, @@ -528,7 +528,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "9f431da1", + "id": "2909407a", "metadata": { "collapsed": false, "editable": true @@ -589,7 +589,7 @@ }, { "cell_type": "markdown", - "id": "8aa155a9", + "id": "d25693ff", "metadata": { "editable": true }, @@ -611,7 +611,7 @@ }, { "cell_type": "markdown", - "id": "03bd2e44", + "id": "78b0bf65", "metadata": { "editable": true }, @@ -626,7 +626,7 @@ }, { "cell_type": "markdown", - "id": "0e101e2d", + "id": "9ee803d8", "metadata": { "editable": true }, @@ -637,7 +637,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "09ecede4", + "id": "ac420f7a", "metadata": { "collapsed": false, "editable": true @@ -703,7 +703,7 @@ }, { "cell_type": "markdown", - "id": "3489dbbc", + "id": "c548d574", "metadata": { "editable": true }, @@ -714,7 +714,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "426eaa39", + "id": "687e9d89", "metadata": { "collapsed": false, "editable": true @@ -788,7 +788,7 @@ }, { "cell_type": "markdown", - "id": "6220214d", + "id": "27a27a67", "metadata": { "editable": true }, @@ -807,7 +807,7 @@ }, { "cell_type": "markdown", - "id": "bf86ac65", + "id": "a12c19b2", "metadata": { "editable": true }, @@ -828,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "4ac61edb", + "id": "b8436434", "metadata": { "editable": true }, @@ -844,7 +844,7 @@ }, { "cell_type": "markdown", - "id": "0058008d", + "id": "f00e1864", "metadata": { "editable": true }, @@ -858,7 +858,7 @@ }, { "cell_type": "markdown", - "id": "f994e1e2", + "id": "ea91af30", "metadata": { "editable": true }, @@ -886,7 +886,7 @@ }, { "cell_type": "markdown", - "id": "842a8611", + "id": "bc9502a0", "metadata": { "editable": true }, @@ -918,7 +918,7 @@ }, { "cell_type": "markdown", - "id": "90bd121a", + "id": "6a236a2a", "metadata": { "editable": true }, @@ -935,7 +935,7 @@ }, { "cell_type": "markdown", - "id": "5cd81303", + "id": "29dc562b", "metadata": { "editable": true }, @@ -948,7 +948,7 @@ }, { "cell_type": "markdown", - "id": "60e085a9", + "id": "6a34f155", "metadata": { "editable": true }, @@ -961,7 +961,7 @@ }, { "cell_type": "markdown", - "id": "fef0100e", + "id": "0afd8cd8", "metadata": { "editable": true }, @@ -974,7 +974,7 @@ }, { "cell_type": "markdown", - "id": "aaba7f05", + "id": "f0b27e71", "metadata": { "editable": true }, @@ -988,7 +988,7 @@ }, { "cell_type": "markdown", - "id": "038b47ae", + "id": "3b04b9c6", "metadata": { "editable": true }, @@ -1010,7 +1010,7 @@ }, { "cell_type": "markdown", - "id": "0ad42833", + "id": "05eca708", "metadata": { "editable": true }, @@ -1025,7 +1025,7 @@ }, { "cell_type": "markdown", - "id": "64b15ba2", + "id": "473025f4", "metadata": { "editable": true }, @@ -1037,7 +1037,7 @@ }, { "cell_type": "markdown", - "id": "49c6adb0", + "id": "26e0b288", "metadata": { "editable": true }, @@ -1050,7 +1050,7 @@ }, { "cell_type": "markdown", - "id": "82873545", + "id": "091efee5", "metadata": { "editable": true }, @@ -1064,7 +1064,7 @@ }, { "cell_type": "markdown", - "id": "35a8e70d", + "id": "22c5f80e", "metadata": { "editable": true }, @@ -1075,7 +1075,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "6aa32b90", + "id": "102b1658", "metadata": { "collapsed": false, "editable": true @@ -1100,7 +1100,7 @@ }, { "cell_type": "markdown", - "id": "6e20f534", + "id": "79448e46", "metadata": { "editable": true }, @@ -1116,7 +1116,7 @@ }, { "cell_type": "markdown", - "id": "71745d3e", + "id": "dbc8b940", "metadata": { "editable": true }, @@ -1137,7 +1137,7 @@ }, { "cell_type": "markdown", - "id": "bad95be2", + "id": "b63ae18d", "metadata": { "editable": true }, @@ -1157,7 +1157,7 @@ }, { "cell_type": "markdown", - "id": "40b4d87e", + "id": "c5ee074e", "metadata": { "editable": true }, @@ -1176,7 +1176,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "1208bbec", + "id": "cfc48413", "metadata": { "collapsed": false, "editable": true @@ -1211,7 +1211,7 @@ }, { "cell_type": "markdown", - "id": "b83b5ed1", + "id": "fbb8c0eb", "metadata": { "editable": true }, @@ -1224,7 +1224,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "1f669db6", + "id": "cc1e51cd", "metadata": { "collapsed": false, "editable": true @@ -1301,7 +1301,7 @@ }, { "cell_type": "markdown", - "id": "3e9ed564", + "id": "22a23ea0", "metadata": { "editable": true }, @@ -1316,7 +1316,377 @@ }, { "cell_type": "markdown", - "id": "9c0ac318", + "id": "f0258497", + "metadata": { + "editable": true + }, + "source": [ + "## SGD vs Full-Batch GD: Convergence Speed and Memory Comparison" + ] + }, + { + "cell_type": "markdown", + "id": "69f1d941", + "metadata": { + "editable": true + }, + "source": [ + "### Theoretical Convergence Speed and convex optimization\n", + "\n", + "Consider minimizing an empirical cost function" + ] + }, + { + "cell_type": "markdown", + "id": "876e1d2b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\theta) =\\frac{1}{N}\\sum_{i=1}^N l_i(\\theta),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "67381fd3", + "metadata": { + "editable": true + }, + "source": [ + "where each $l_i(\\theta)$ is a\n", + "differentiable loss term. Gradient Descent (GD) updates parameters\n", + "using the full gradient $\\nabla C(\\theta)$, while Stochastic Gradient\n", + "Descent (SGD) uses a single sample (or mini-batch) gradient $\\nabla\n", + "l_i(\\theta)$ selected at random. In equation form, one GD step is:" + ] + }, + { + "cell_type": "markdown", + "id": "f08ec530", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_{t+1} = \\theta_t-\\eta \\nabla C(\\theta_t) =\\theta_t -\\eta \\frac{1}{N}\\sum_{i=1}^N \\nabla l_i(\\theta_t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3d99c5ee", + "metadata": { + "editable": true + }, + "source": [ + "whereas one SGD step is:" + ] + }, + { + "cell_type": "markdown", + "id": "03ecb7f9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_{t+1} = \\theta_t -\\eta \\nabla l_{i_t}(\\theta_t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a6db0c6c", + "metadata": { + "editable": true + }, + "source": [ + "with $i_t$ randomly chosen. On smooth convex problems, GD and SGD both\n", + "converge to the global minimum, but their rates differ. GD can take\n", + "larger, more stable steps since it uses the exact gradient, achieving\n", + "an error that decreases on the order of $O(1/t)$ per iteration for\n", + "convex objectives (and even exponentially fast for strongly convex\n", + "cases). In contrast, plain SGD has more variance in each step, leading\n", + "to sublinear convergence in expectation – typically $O(1/\\sqrt{t})$\n", + "for general convex objectives (\\thetaith appropriate diminishing step\n", + "sizes) . Intuitively, GD’s trajectory is smoother and more\n", + "predictable, while SGD’s path oscillates due to noise but costs far\n", + "less per iteration, enabling many more updates in the same time." + ] + }, + { + "cell_type": "markdown", + "id": "167f76aa", + "metadata": { + "editable": true + }, + "source": [ + "### Strongly Convex Case\n", + "\n", + "If $C(\\theta)$ is strongly convex and $L$-smooth (so GD enjoys linear\n", + "convergence), the gap $C(\\theta_t)-C(\\theta^*)$ for GD shrinks as" + ] + }, + { + "cell_type": "markdown", + "id": "d8d5cb23", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\theta_t) - C(\\theta^* ) \\le \\Big(1 - \\frac{\\mu}{L}\\Big)^t [C(\\theta_0)-C(\\theta^*)],\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e127c141", + "metadata": { + "editable": true + }, + "source": [ + "a geometric (linear) convergence per iteration . Achieving an\n", + "$\\epsilon$-accurate solution thus takes on the order of\n", + "$\\log(1/\\epsilon)$ iterations for GD. However, each GD iteration costs\n", + "$O(N)$ gradient evaluations. SGD cannot exploit strong convexity to\n", + "obtain a linear rate – instead, with a properly decaying step size\n", + "(e.g. $\\eta_t = \\frac{1}{\\mu t}$) or iterate averaging, SGD attains an\n", + "$O(1/t)$ convergence rate in expectation . For example, one result\n", + "of Moulines and Bach 2011, see shows that with $\\eta_t = \\Theta(1/t)$," + ] + }, + { + "cell_type": "markdown", + "id": "7ea2c03c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathbb{E}[C(\\theta_t) - C(\\theta^*)] = O(1/t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e2e33c54", + "metadata": { + "editable": true + }, + "source": [ + "for strongly convex, smooth $F$ . This $1/t$ rate is slower per\n", + "iteration than GD’s exponential decay, but each SGD iteration is $N$\n", + "times cheaper. In fact, to reach error $\\epsilon$, plain SGD needs on\n", + "the order of $T=O(1/\\epsilon)$ iterations (sub-linear convergence),\n", + "while GD needs $O(\\log(1/\\epsilon))$ iterations. When accounting for\n", + "cost-per-iteration, GD requires $O(N \\log(1/\\epsilon))$ total gradient\n", + "computations versus SGD’s $O(1/\\epsilon)$ single-sample\n", + "computations. In large-scale regimes (huge $N$), SGD can be\n", + "faster in wall-clock time because $N \\log(1/\\epsilon)$ may far exceed\n", + "$1/\\epsilon$ for reasonable accuracy levels. In other words,\n", + "with millions of data points, one epoch of GD (one full gradient) is\n", + "extremely costly, whereas SGD can make $N$ cheap updates in the time\n", + "GD makes one – often yielding a good solution faster in practice, even\n", + "though SGD’s asymptotic error decays more slowly. As one lecture\n", + "succinctly puts it: “SGD can be super effective in terms of iteration\n", + "cost and memory, but SGD is slow to converge and can’t adapt to strong\n", + "convexity” . Thus, the break-even point depends on $N$ and the desired\n", + "accuracy: for moderate accuracy on very large $N$, SGD’s cheaper\n", + "updates win; for extremely high precision (very small $\\epsilon$) on a\n", + "modest $N$, GD’s fast convergence per step can be advantageous." + ] + }, + { + "cell_type": "markdown", + "id": "88f943e6", + "metadata": { + "editable": true + }, + "source": [ + "### Non-Convex Problems\n", + "\n", + "In non-convex optimization (e.g. deep neural networks), neither GD nor\n", + "SGD guarantees global minima, but SGD often displays faster progress\n", + "in finding useful minima. Theoretical results here are weaker, usually\n", + "showing convergence to a stationary point $\\theta$ ($|\\nabla C|$ is\n", + "small) in expectation. For example, GD might require $O(1/\\epsilon^2)$\n", + "iterations to ensure $|\\nabla C(\\theta)| < \\epsilon$, and SGD typically has\n", + "similar polynomial complexity (often worse due to gradient\n", + "noise). However, a noteworthy difference is that SGD’s stochasticity\n", + "can help escape saddle points or poor local minima. Random gradient\n", + "fluctuations act like implicit noise, helping the iterate “jump” out\n", + "of flat saddle regions where full-batch GD could stagnate . In fact,\n", + "research has shown that adding noise to GD can guarantee escaping\n", + "saddle points in polynomial time, and the inherent noise in SGD often\n", + "serves this role. Empirically, this means SGD can sometimes find a\n", + "lower loss basin faster, whereas full-batch GD might get “stuck” near\n", + "saddle points or need a very small learning rate to navigate complex\n", + "error surfaces . Overall, in modern high-dimensional machine learning,\n", + "SGD (or mini-batch SGD) is the workhorse for large non-convex problems\n", + "because it converges to good solutions much faster in practice,\n", + "despite the lack of a linear convergence guarantee. Full-batch GD is\n", + "rarely used on large neural networks, as it would require tiny steps\n", + "to avoid divergence and is extremely slow per iteration ." + ] + }, + { + "cell_type": "markdown", + "id": "aa4a8927", + "metadata": { + "editable": true + }, + "source": [ + "## Memory Usage and Scalability\n", + "\n", + "A major advantage of SGD is its memory efficiency in handling large\n", + "datasets. Full-batch GD requires access to the entire training set for\n", + "each iteration, which often means the whole dataset (or a large\n", + "subset) must reside in memory to compute $\\nabla C(\\theta)$ . This results\n", + "in memory usage that scales linearly with the dataset size $N$. For\n", + "instance, if each training sample is large (e.g. high-dimensional\n", + "features), computing a full gradient may require storing a substantial\n", + "portion of the data or all intermediate gradients until they are\n", + "aggregated. In contrast, SGD needs only a single (or a small\n", + "mini-batch of) training example(s) in memory at any time . The\n", + "algorithm processes one sample (or mini-batch) at a time and\n", + "immediately updates the model, discarding that sample before moving to\n", + "the next. This streaming approach means that memory footprint is\n", + "essentially independent of $N$ (apart from storing the model\n", + "parameters themselves). As one source notes, gradient descent\n", + "“requires more memory than SGD” because it “must store the entire\n", + "dataset for each iteration,” whereas SGD “only needs to store the\n", + "current training example” . In practical terms, if you have a dataset\n", + "of size, say, 1 million examples, full-batch GD would need memory for\n", + "all million every step, while SGD could be implemented to load just\n", + "one example at a time – a crucial benefit if data are too large to fit\n", + "in RAM or GPU memory. This scalability makes SGD suitable for\n", + "large-scale learning: as long as you can stream data from disk, SGD\n", + "can handle arbitrarily large datasets with fixed memory. In fact, SGD\n", + "“does not need to remember which examples were visited” in the past,\n", + "allowing it to run in an online fashion on infinite data streams\n", + ". Full-batch GD, on the other hand, would require multiple passes\n", + "through a giant dataset per update (or a complex distributed memory\n", + "system), which is often infeasible.\n", + "\n", + "There is also a secondary memory effect: computing a full-batch\n", + "gradient in deep learning requires storing all intermediate\n", + "activations for backpropagation across the entire batch. A very large\n", + "batch (approaching the full dataset) might exhaust GPU memory due to\n", + "the need to hold activation gradients for thousands or millions of\n", + "examples simultaneously. SGD/minibatches mitigate this by splitting\n", + "the workload – e.g. with a mini-batch of size 32 or 256, memory use\n", + "stays bounded, whereas a full-batch (size = $N$) forward/backward pass\n", + "could not even be executed if $N$ is huge. Techniques like gradient\n", + "accumulation exist to simulate large-batch GD by summing many\n", + "small-batch gradients – but these still process data in manageable\n", + "chunks to avoid memory overflow. In summary, memory complexity for GD\n", + "grows with $N$, while for SGD it remains $O(1)$ w.r.t. dataset size\n", + "(only the model and perhaps a mini-batch reside in memory) . This is a\n", + "key reason why batch GD “does not scale” to very large data and why\n", + "virtually all large-scale machine learning algorithms rely on\n", + "stochastic or mini-batch methods." + ] + }, + { + "cell_type": "markdown", + "id": "72d0192b", + "metadata": { + "editable": true + }, + "source": [ + "## Empirical Evidence: Convergence Time and Memory in Practice\n", + "\n", + "Empirical studies strongly support the theoretical trade-offs\n", + "above. In large-scale machine learning tasks, SGD often converges to a\n", + "good solution much faster in wall-clock time than full-batch GD, and\n", + "it uses far less memory. For example, Bottou & Bousquet (2008)\n", + "analyzed learning time under a fixed computational budget and\n", + "concluded that when data is abundant, it’s better to use a faster\n", + "(even if less precise) optimization method to process more examples in\n", + "the same time . This analysis showed that for large-scale problems,\n", + "processing more data with SGD yields lower error than spending the\n", + "time to do exact (batch) optimization on fewer data . In other words,\n", + "if you have a time budget, it’s often optimal to accept slightly\n", + "slower convergence per step (as with SGD) in exchange for being able\n", + "to use many more training samples in that time. This phenomenon is\n", + "borne out by experiments:" + ] + }, + { + "cell_type": "markdown", + "id": "44fcd423", + "metadata": { + "editable": true + }, + "source": [ + "### Deep Neural Networks\n", + "\n", + "In modern deep learning, full-batch GD is so slow that it is rarely\n", + "attempted; instead, mini-batch SGD is standard. A recent study\n", + "demonstrated that it is possible to train a ResNet-50 on ImageNet\n", + "using full-batch gradient descent, but it required careful tuning\n", + "(e.g. gradient clipping, tiny learning rates) and vast computational\n", + "resources – and even then, each full-batch update was extremely\n", + "expensive.\n", + "\n", + "Using a huge batch\n", + "(closer to full GD) tends to slow down convergence if the learning\n", + "rate is not scaled up, and often encounters optimization difficulties\n", + "(plateaus) that small batches avoid.\n", + "Empirically, small or medium\n", + "batch SGD finds minima in fewer clock hours because it can rapidly\n", + "loop over the data with gradient noise aiding exploration." + ] + }, + { + "cell_type": "markdown", + "id": "8de0942f", + "metadata": { + "editable": true + }, + "source": [ + "### Memory constraints\n", + "\n", + "From a memory standpoint, practitioners note that batch GD becomes\n", + "infeasible on large data. For example, if one tried to do full-batch\n", + "training on a dataset that doesn’t fit in RAM or GPU memory, the\n", + "program would resort to heavy disk I/O or simply crash. SGD\n", + "circumvents this by processing mini-batches. Even in cases where data\n", + "does fit in memory, using a full batch can spike memory usage due to\n", + "storing all gradients. One empirical observation is that mini-batch\n", + "training has a “lower, fluctuating usage pattern” of memory, whereas\n", + "full-batch loading “quickly consumes memory (often exceeding limits)”\n", + ". This is especially relevant for graph neural networks or other\n", + "models where a “batch” may include a huge chunk of a graph: full-batch\n", + "gradient computation can exhaust GPU memory, whereas mini-batch\n", + "methods keep memory usage manageable .\n", + "\n", + "In summary, SGD converges faster than full-batch GD in terms of actual\n", + "training time for large-scale problems, provided we measure\n", + "convergence as reaching a good-enough solution. Theoretical bounds\n", + "show SGD needs more iterations, but because it performs many more\n", + "updates per unit time (and requires far less memory), it often\n", + "achieves lower loss in a given time frame than GD. Full-batch GD might\n", + "take slightly fewer iterations in theory, but each iteration is so\n", + "costly that it is “slower… especially for large datasets” . Meanwhile,\n", + "memory scaling strongly favors SGD: GD’s memory cost grows with\n", + "dataset size, making it impractical beyond a point, whereas SGD’s\n", + "memory use is modest and mostly constant w.r.t. $N$ . These\n", + "differences have made SGD (and mini-batch variants) the de facto\n", + "choice for training large machine learning models, from logistic\n", + "regression on millions of examples to deep neural networks with\n", + "billions of parameters. The consensus in both research and practice is\n", + "that for large-scale or high-dimensional tasks, SGD-type methods\n", + "converge quicker per unit of computation and handle memory constraints\n", + "better than standard full-batch gradient descent ." + ] + }, + { + "cell_type": "markdown", + "id": "f08a4bbe", "metadata": { "editable": true }, @@ -1347,7 +1717,7 @@ }, { "cell_type": "markdown", - "id": "d8f518c4", + "id": "dc1fa30f", "metadata": { "editable": true }, @@ -1369,7 +1739,7 @@ }, { "cell_type": "markdown", - "id": "3dcb89bd", + "id": "1fbfcb5e", "metadata": { "editable": true }, @@ -1389,7 +1759,7 @@ }, { "cell_type": "markdown", - "id": "8f258bc2", + "id": "83d5dfc2", "metadata": { "editable": true }, @@ -1405,7 +1775,7 @@ }, { "cell_type": "markdown", - "id": "2a3715f8", + "id": "4cf425f2", "metadata": { "editable": true }, @@ -1425,7 +1795,7 @@ }, { "cell_type": "markdown", - "id": "a1d9578a", + "id": "a8de083c", "metadata": { "editable": true }, @@ -1437,7 +1807,7 @@ }, { "cell_type": "markdown", - "id": "b6b5bc5e", + "id": "f8b98ecd", "metadata": { "editable": true }, @@ -1449,7 +1819,7 @@ }, { "cell_type": "markdown", - "id": "44b313c8", + "id": "c41121c9", "metadata": { "editable": true }, @@ -1461,7 +1831,7 @@ }, { "cell_type": "markdown", - "id": "b56c85b9", + "id": "0c9cde87", "metadata": { "editable": true }, @@ -1473,7 +1843,7 @@ }, { "cell_type": "markdown", - "id": "5bcc6bd2", + "id": "9079853e", "metadata": { "editable": true }, @@ -1484,7 +1854,7 @@ }, { "cell_type": "markdown", - "id": "41fc9f01", + "id": "1b2340aa", "metadata": { "editable": true }, @@ -1496,7 +1866,7 @@ }, { "cell_type": "markdown", - "id": "8151719b", + "id": "1c63eff7", "metadata": { "editable": true }, @@ -1506,7 +1876,7 @@ }, { "cell_type": "markdown", - "id": "bb75b0ad", + "id": "e05e89e4", "metadata": { "editable": true }, @@ -1518,7 +1888,7 @@ }, { "cell_type": "markdown", - "id": "3c71fd46", + "id": "b3cbe567", "metadata": { "editable": true }, @@ -1528,7 +1898,7 @@ }, { "cell_type": "markdown", - "id": "1d835a18", + "id": "5d2f1096", "metadata": { "editable": true }, @@ -1549,7 +1919,7 @@ }, { "cell_type": "markdown", - "id": "77dcc8c3", + "id": "4c4f3846", "metadata": { "editable": true }, @@ -1562,7 +1932,7 @@ }, { "cell_type": "markdown", - "id": "21161d57", + "id": "57d24251", "metadata": { "editable": true }, @@ -1574,7 +1944,7 @@ }, { "cell_type": "markdown", - "id": "e87e09a9", + "id": "caff3ad3", "metadata": { "editable": true }, @@ -1591,7 +1961,7 @@ }, { "cell_type": "markdown", - "id": "1a98c681", + "id": "67133da8", "metadata": { "editable": true }, @@ -1607,7 +1977,7 @@ }, { "cell_type": "markdown", - "id": "8b337277", + "id": "d2d2d644", "metadata": { "editable": true }, @@ -1629,7 +1999,7 @@ }, { "cell_type": "markdown", - "id": "af77b83f", + "id": "897d1ca3", "metadata": { "editable": true }, @@ -1647,7 +2017,7 @@ }, { "cell_type": "markdown", - "id": "bc924f77", + "id": "549532b3", "metadata": { "editable": true }, @@ -1667,7 +2037,7 @@ }, { "cell_type": "markdown", - "id": "86e5ab5e", + "id": "f014a3a2", "metadata": { "editable": true }, @@ -1681,7 +2051,7 @@ }, { "cell_type": "markdown", - "id": "949f359d", + "id": "67bed63f", "metadata": { "editable": true }, @@ -1693,7 +2063,7 @@ }, { "cell_type": "markdown", - "id": "0ba26be3", + "id": "3014fe59", "metadata": { "editable": true }, @@ -1705,7 +2075,7 @@ }, { "cell_type": "markdown", - "id": "4fb9b2a2", + "id": "a99d9c1c", "metadata": { "editable": true }, @@ -1717,7 +2087,7 @@ }, { "cell_type": "markdown", - "id": "8711e597", + "id": "907f9915", "metadata": { "editable": true }, @@ -1729,7 +2099,7 @@ }, { "cell_type": "markdown", - "id": "49e6e73d", + "id": "551eb7db", "metadata": { "editable": true }, @@ -1740,7 +2110,7 @@ }, { "cell_type": "markdown", - "id": "ca5bb491", + "id": "ea8ae470", "metadata": { "editable": true }, @@ -1752,7 +2122,7 @@ }, { "cell_type": "markdown", - "id": "5e19d7bf", + "id": "9f5d78fd", "metadata": { "editable": true }, @@ -1766,7 +2136,7 @@ }, { "cell_type": "markdown", - "id": "f79d952e", + "id": "8291642f", "metadata": { "editable": true }, @@ -1777,7 +2147,7 @@ }, { "cell_type": "markdown", - "id": "13e9862f", + "id": "ee0e74ec", "metadata": { "editable": true }, @@ -1789,7 +2159,7 @@ }, { "cell_type": "markdown", - "id": "5693500e", + "id": "3699f7e5", "metadata": { "editable": true }, @@ -1811,7 +2181,7 @@ }, { "cell_type": "markdown", - "id": "65a5e1e7", + "id": "16cdd781", "metadata": { "editable": true }, @@ -1835,7 +2205,7 @@ }, { "cell_type": "markdown", - "id": "27686255", + "id": "779881a9", "metadata": { "editable": true }, @@ -1851,7 +2221,7 @@ }, { "cell_type": "markdown", - "id": "f3dfc1e2", + "id": "0724c747", "metadata": { "editable": true }, @@ -1867,7 +2237,7 @@ }, { "cell_type": "markdown", - "id": "045d399c", + "id": "02a113cb", "metadata": { "editable": true }, @@ -1881,7 +2251,7 @@ }, { "cell_type": "markdown", - "id": "4e75ee41", + "id": "e013ce1a", "metadata": { "editable": true }, @@ -1899,7 +2269,7 @@ }, { "cell_type": "markdown", - "id": "ddbb28ab", + "id": "1baa5b4e", "metadata": { "editable": true }, @@ -1920,7 +2290,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "dae38b6c", + "id": "8aab54bd", "metadata": { "collapsed": false, "editable": true @@ -1980,7 +2350,7 @@ }, { "cell_type": "markdown", - "id": "ca5a343a", + "id": "26c6e4f1", "metadata": { "editable": true }, @@ -1991,7 +2361,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "08d97c1e", + "id": "226c13ec", "metadata": { "collapsed": false, "editable": true @@ -2055,7 +2425,7 @@ }, { "cell_type": "markdown", - "id": "727d8fc3", + "id": "6895dbfe", "metadata": { "editable": true }, @@ -2070,7 +2440,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "4e41c003", + "id": "f8d01982", "metadata": { "collapsed": false, "editable": true @@ -2154,7 +2524,7 @@ }, { "cell_type": "markdown", - "id": "fe00db52", + "id": "cffe8367", "metadata": { "editable": true }, @@ -2165,7 +2535,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "8f22105b", + "id": "b57871a9", "metadata": { "collapsed": false, "editable": true @@ -2243,7 +2613,7 @@ }, { "cell_type": "markdown", - "id": "8956bf7a", + "id": "37b273c1", "metadata": { "editable": true }, @@ -2256,7 +2626,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "044275ef", + "id": "9ff0eb69", "metadata": { "collapsed": false, "editable": true @@ -2300,7 +2670,7 @@ }, { "cell_type": "markdown", - "id": "353b50b3", + "id": "e7d143b6", "metadata": { "editable": true }, @@ -2311,7 +2681,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "fdc8debd", + "id": "9b5e2d1d", "metadata": { "collapsed": false, "editable": true @@ -2370,7 +2740,7 @@ }, { "cell_type": "markdown", - "id": "b738f1b8", + "id": "8b6fb13f", "metadata": { "editable": true }, @@ -2380,7 +2750,7 @@ }, { "cell_type": "markdown", - "id": "65ce93ba", + "id": "06c3f4bb", "metadata": { "editable": true }, @@ -2391,7 +2761,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "604d7286", + "id": "4abf9ccd", "metadata": { "collapsed": false, "editable": true @@ -2456,7 +2826,7 @@ }, { "cell_type": "markdown", - "id": "e663a714", + "id": "18d42e29", "metadata": { "editable": true }, @@ -2467,7 +2837,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "749fa687", + "id": "03415114", "metadata": { "collapsed": false, "editable": true @@ -2537,7 +2907,7 @@ }, { "cell_type": "markdown", - "id": "8801fcd5", + "id": "41120d8f", "metadata": { "editable": true }, @@ -2554,7 +2924,7 @@ }, { "cell_type": "markdown", - "id": "8ea68725", + "id": "16eb2a88", "metadata": { "editable": true }, @@ -2582,7 +2952,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "04811786", + "id": "a6df3a5c", "metadata": { "collapsed": false, "editable": true @@ -2602,7 +2972,7 @@ }, { "cell_type": "markdown", - "id": "b0e7cc2c", + "id": "fc15d89b", "metadata": { "editable": true }, @@ -2618,7 +2988,7 @@ }, { "cell_type": "markdown", - "id": "f8a8132d", + "id": "4e4b5ee0", "metadata": { "editable": true }, @@ -2638,7 +3008,7 @@ }, { "cell_type": "markdown", - "id": "03eca41f", + "id": "4455b9a0", "metadata": { "editable": true }, @@ -2665,7 +3035,7 @@ }, { "cell_type": "markdown", - "id": "710e8f88", + "id": "9592eb20", "metadata": { "editable": true }, @@ -2678,7 +3048,7 @@ }, { "cell_type": "markdown", - "id": "5d3df9bf", + "id": "e3022ea8", "metadata": { "editable": true }, @@ -2690,7 +3060,7 @@ }, { "cell_type": "markdown", - "id": "be0fd5f1", + "id": "246a28bf", "metadata": { "editable": true }, @@ -2705,7 +3075,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "2a0924bb", + "id": "d132a060", "metadata": { "collapsed": false, "editable": true @@ -2732,7 +3102,7 @@ }, { "cell_type": "markdown", - "id": "d116f448", + "id": "9e70a01f", "metadata": { "editable": true }, @@ -2746,7 +3116,7 @@ }, { "cell_type": "markdown", - "id": "41caea07", + "id": "2c0ad6b4", "metadata": { "editable": true }, @@ -2758,7 +3128,7 @@ }, { "cell_type": "markdown", - "id": "1fa96f7c", + "id": "3c6081b8", "metadata": { "editable": true }, @@ -2775,7 +3145,7 @@ }, { "cell_type": "markdown", - "id": "70038d6a", + "id": "845af933", "metadata": { "editable": true }, @@ -2787,7 +3157,7 @@ }, { "cell_type": "markdown", - "id": "852a77d0", + "id": "564afbbd", "metadata": { "editable": true }, @@ -2797,7 +3167,7 @@ }, { "cell_type": "markdown", - "id": "fc4afaaf", + "id": "c4088263", "metadata": { "editable": true }, @@ -2809,7 +3179,7 @@ }, { "cell_type": "markdown", - "id": "94b18ced", + "id": "96983e3d", "metadata": { "editable": true }, @@ -2819,7 +3189,7 @@ }, { "cell_type": "markdown", - "id": "d7a95314", + "id": "91d029d7", "metadata": { "editable": true }, @@ -2831,7 +3201,7 @@ }, { "cell_type": "markdown", - "id": "eaf6a485", + "id": "20d351f6", "metadata": { "editable": true }, @@ -2842,7 +3212,7 @@ }, { "cell_type": "markdown", - "id": "3d9442a2", + "id": "7a8e79fd", "metadata": { "editable": true }, @@ -2854,7 +3224,7 @@ }, { "cell_type": "markdown", - "id": "e4aeef17", + "id": "4ec7ad68", "metadata": { "editable": true }, @@ -2864,7 +3234,7 @@ }, { "cell_type": "markdown", - "id": "4ce9dee9", + "id": "df09c13b", "metadata": { "editable": true }, @@ -2876,7 +3246,7 @@ }, { "cell_type": "markdown", - "id": "752ce099", + "id": "bb2a9c1f", "metadata": { "editable": true }, @@ -2886,7 +3256,7 @@ }, { "cell_type": "markdown", - "id": "7cad5229", + "id": "b3507e2d", "metadata": { "editable": true }, @@ -2898,7 +3268,7 @@ }, { "cell_type": "markdown", - "id": "46f1aaf9", + "id": "2010542f", "metadata": { "editable": true }, @@ -2908,7 +3278,7 @@ }, { "cell_type": "markdown", - "id": "7d25a9fb", + "id": "e03a1590", "metadata": { "editable": true }, @@ -2920,7 +3290,7 @@ }, { "cell_type": "markdown", - "id": "57b4c7d9", + "id": "71872755", "metadata": { "editable": true }, @@ -2930,7 +3300,7 @@ }, { "cell_type": "markdown", - "id": "fb833214", + "id": "167238dc", "metadata": { "editable": true }, @@ -2942,7 +3312,7 @@ }, { "cell_type": "markdown", - "id": "5fa29cd3", + "id": "38d0cc0f", "metadata": { "editable": true }, @@ -2952,7 +3322,7 @@ }, { "cell_type": "markdown", - "id": "6c0e668d", + "id": "e9e1beb9", "metadata": { "editable": true }, @@ -2964,7 +3334,7 @@ }, { "cell_type": "markdown", - "id": "9d928664", + "id": "9a8576e4", "metadata": { "editable": true }, @@ -2974,7 +3344,7 @@ }, { "cell_type": "markdown", - "id": "65434b84", + "id": "937d703f", "metadata": { "editable": true }, @@ -2986,7 +3356,7 @@ }, { "cell_type": "markdown", - "id": "127c9817", + "id": "e4723b95", "metadata": { "editable": true }, @@ -2996,7 +3366,7 @@ }, { "cell_type": "markdown", - "id": "46f45c10", + "id": "6df6f6d8", "metadata": { "editable": true }, @@ -3008,7 +3378,7 @@ }, { "cell_type": "markdown", - "id": "4fbaa69a", + "id": "39bdaf00", "metadata": { "editable": true }, @@ -3020,7 +3390,7 @@ }, { "cell_type": "markdown", - "id": "25f1abd4", + "id": "e4584236", "metadata": { "editable": true }, @@ -3032,7 +3402,7 @@ }, { "cell_type": "markdown", - "id": "9fd5ef9e", + "id": "d0c5d728", "metadata": { "editable": true }, @@ -3042,7 +3412,7 @@ }, { "cell_type": "markdown", - "id": "f1cb8e35", + "id": "9b637fd2", "metadata": { "editable": true }, @@ -3054,7 +3424,7 @@ }, { "cell_type": "markdown", - "id": "c0c5100a", + "id": "9627e6fb", "metadata": { "editable": true }, @@ -3067,7 +3437,7 @@ }, { "cell_type": "markdown", - "id": "c80e55cb", + "id": "662fe97e", "metadata": { "editable": true }, @@ -3079,7 +3449,7 @@ }, { "cell_type": "markdown", - "id": "a47f5c5e", + "id": "fa2d5cb5", "metadata": { "editable": true }, @@ -3093,7 +3463,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "e093186c", + "id": "a530b3ba", "metadata": { "collapsed": false, "editable": true @@ -3190,7 +3560,7 @@ }, { "cell_type": "markdown", - "id": "de555fff", + "id": "58cc6d9d", "metadata": { "editable": true }, @@ -3211,7 +3581,7 @@ }, { "cell_type": "markdown", - "id": "72178d39", + "id": "6a2d3f87", "metadata": { "editable": true }, @@ -3223,7 +3593,7 @@ }, { "cell_type": "markdown", - "id": "8e5d822b", + "id": "9eab78a9", "metadata": { "editable": true }, @@ -3233,7 +3603,7 @@ }, { "cell_type": "markdown", - "id": "e9218f82", + "id": "50d7f5c3", "metadata": { "editable": true }, @@ -3245,7 +3615,7 @@ }, { "cell_type": "markdown", - "id": "2223d1b1", + "id": "4c96e589", "metadata": { "editable": true }, @@ -3255,7 +3625,7 @@ }, { "cell_type": "markdown", - "id": "e5474a5b", + "id": "b57830ea", "metadata": { "editable": true }, @@ -3267,7 +3637,7 @@ }, { "cell_type": "markdown", - "id": "691295ed", + "id": "05325bc6", "metadata": { "editable": true }, @@ -3285,7 +3655,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "e243cef5", + "id": "d95b15bc", "metadata": { "collapsed": false, "editable": true @@ -3361,7 +3731,7 @@ }, { "cell_type": "markdown", - "id": "ef2eaa7a", + "id": "b88ebede", "metadata": { "editable": true }, @@ -3375,7 +3745,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "546e3504", + "id": "47036b16", "metadata": { "collapsed": false, "editable": true @@ -3464,7 +3834,7 @@ }, { "cell_type": "markdown", - "id": "f6787352", + "id": "52faee2f", "metadata": { "editable": true },