{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "# Week 40: From Stochastic Gradient Descent to Neural networks\n", "\n", " \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", "Date: **Oct 6, 2020**\n", "\n", "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", "\n", "\n", "\n", "## Plan for week 40\n", "\n", "* Thursday: Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks. [Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureOctober1.mp4?vrtx=view-as-webpage) \n", "\n", "* Friday: Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. [Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureOctober2.mp4?vrtx=view-as-webpage) \n", "\n", "Reading suggestions for both days: [Aurelien Geron's chapter 10](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf) and Hastie et al chapter 11.\n", "For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text. \n", "\n", "## Overview video for week 40\n", "\n", "[Overview Video, from Stochastic Gradient methods to Neural Networks](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/OverviewWeek40.mp4?vrtx=view-as-webpage)\n", "\n", "## Stochastic Gradient Descent\n", "\n", "Stochastic gradient descent (SGD) and variants thereof address some of\n", "the shortcomings of the Gradient descent method discussed above.\n", "\n", "The underlying idea of SGD comes from the observation that the cost\n", "function, which we want to minimize, can almost always be written as a\n", "sum over $n$ data points $\\{\\mathbf{x}_i\\}_{i=1}^n$," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "C(\\mathbf{\\beta}) = \\sum_{i=1}^n c_i(\\mathbf{x}_i,\n", "\\mathbf{\\beta}).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computation of gradients\n", "\n", "This in turn means that the gradient can be\n", "computed as a sum over $i$-gradients" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\nabla_\\beta C(\\mathbf{\\beta}) = \\sum_i^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", "\\mathbf{\\beta}).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Stochasticity/randomness is introduced by only taking the\n", "gradient on a subset of the data called minibatches. If there are $n$\n", "data points and the size of each minibatch is $M$, there will be $n/M$\n", "minibatches. We denote these minibatches by $B_k$ where\n", "$k=1,\\cdots,n/M$.\n", "\n", "## SGD example\n", "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n", "and we choose to have $M=5$ minibathces,\n", "then each minibatch contains two data points. In particular we have\n", "$B_1 = (\\mathbf{x}_1,\\mathbf{x}_2), \\cdots, B_5 =\n", "(\\mathbf{x}_9,\\mathbf{x}_{10})$. Note that if you choose $M=1$ you\n", "have only a single batch with all data points and on the other extreme,\n", "you may choose $M=n$ resulting in a minibatch for each datapoint, i.e\n", "$B_k = \\mathbf{x}_k$.\n", "\n", "The idea is now to approximate the gradient by replacing the sum over\n", "all data points with a sum over the data points in one the minibatches\n", "picked at random in each gradient descent step" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\nabla_{\\beta}\n", "C(\\mathbf{\\beta}) = \\sum_{i=1}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", "\\mathbf{\\beta}) \\rightarrow \\sum_{i \\in B_k}^n \\nabla_\\beta\n", "c_i(\\mathbf{x}_i, \\mathbf{\\beta}).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The gradient step\n", "\n", "Thus a gradient descent step now looks like" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\beta_{j+1} = \\beta_j - \\gamma_j \\sum_{i \\in B_k}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", "\\mathbf{\\beta})\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $k$ is picked at random with equal\n", "probability from $[1,n/M]$. An iteration over the number of\n", "minibathces (n/M) is commonly referred to as an epoch. Thus it is\n", "typical to choose a number of epochs and for each epoch iterate over\n", "the number of minibatches, as exemplified in the code below.\n", "\n", "## Simple example code" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np \n", "\n", "n = 100 #100 datapoints \n", "M = 5 #size of each minibatch\n", "m = int(n/M) #number of minibatches\n", "n_epochs = 10 #number of epochs\n", "\n", "j = 0\n", "for epoch in range(1,n_epochs+1):\n", " for i in range(m):\n", " k = np.random.randint(m) #Pick the k-th minibatch at random\n", " #Compute the gradient using the data in minibatch Bk\n", " #Compute new suggestion for \n", " j += 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Taking the gradient only on a subset of the data has two important\n", "benefits. First, it introduces randomness which decreases the chance\n", "that our opmization scheme gets stuck in a local minima. Second, if\n", "the size of the minibatches are small relative to the number of\n", "datapoints ($M < n$), the computation of the gradient is much\n", "cheaper since we sum over the datapoints in the $k-th$ minibatch and not\n", "all $n$ datapoints.\n", "\n", "## When do we stop?\n", "\n", "A natural question is when do we stop the search for a new minimum?\n", "One possibility is to compute the full gradient after a given number\n", "of epochs and check if the norm of the gradient is smaller than some\n", "threshold and stop if true. However, the condition that the gradient\n", "is zero is valid also for local minima, so this would only tell us\n", "that we are close to a local/global minimum. However, we could also\n", "evaluate the cost function at this point, store the result and\n", "continue the search. If the test kicks in at a later stage we can\n", "compare the values of the cost function and keep the $\\beta$ that\n", "gave the lowest value.\n", "\n", "## Slightly different approach\n", "\n", "Another approach is to let the step length $\\gamma_j$ depend on the\n", "number of epochs in such a way that it becomes very small after a\n", "reasonable time such that we do not move at all.\n", "\n", "As an example, let $e = 0,1,2,3,\\cdots$ denote the current epoch and let $t_0, t_1 > 0$ be two fixed numbers. Furthermore, let $t = e \\cdot m + i$ where $m$ is the number of minibatches and $i=0,\\cdots,m-1$. Then the function $$\\gamma_j(t; t_0, t_1) = \\frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length $\\gamma_j (0; t_0, t_1) = t_0/t_1$ which decays in *time* $t$.\n", "\n", "In this way we can fix the number of epochs, compute $\\beta$ and\n", "evaluate the cost function at the end. Repeating the computation will\n", "give a different result since the scheme is random by design. Then we\n", "pick the final $\\beta$ that gives the lowest value of the cost\n", "function." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "import numpy as np \n", "\n", "def step_length(t,t0,t1):\n", " return t0/(t+t1)\n", "\n", "n = 100 #100 datapoints \n", "M = 5 #size of each minibatch\n", "m = int(n/M) #number of minibatches\n", "n_epochs = 500 #number of epochs\n", "t0 = 1.0\n", "t1 = 10\n", "\n", "gamma_j = t0/t1\n", "j = 0\n", "for epoch in range(1,n_epochs+1):\n", " for i in range(m):\n", " k = np.random.randint(m) #Pick the k-th minibatch at random\n", " #Compute the gradient using the data in minibatch Bk\n", " #Compute new suggestion for beta\n", " t = epoch*m+i\n", " gamma_j = step_length(t,t0,t1)\n", " j += 1\n", "\n", "print(\"gamma_j after %d epochs: %g\" % (n_epochs,gamma_j))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Program for stochastic gradient" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "\n", "# Importing various packages\n", "from math import exp, sqrt\n", "from random import random, seed\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from sklearn.linear_model import SGDRegressor\n", "\n", "m = 100\n", "x = 2*np.random.rand(m,1)\n", "y = 4+3*x+np.random.randn(m,1)\n", "\n", "X = np.c_[np.ones((m,1)), x]\n", "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n", "print(\"Own inversion\")\n", "print(theta_linreg)\n", "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n", "sgdreg.fit(x,y.ravel())\n", "print(\"sgdreg from scikit\")\n", "print(sgdreg.intercept_, sgdreg.coef_)\n", "\n", "\n", "theta = np.random.randn(2,1)\n", "eta = 0.1\n", "Niterations = 1000\n", "\n", "\n", "for iter in range(Niterations):\n", " gradients = 2.0/m*X.T @ ((X @ theta)-y)\n", " theta -= eta*gradients\n", "print(\"theta from own gd\")\n", "print(theta)\n", "\n", "xnew = np.array([[0],[2]])\n", "Xnew = np.c_[np.ones((2,1)), xnew]\n", "ypredict = Xnew.dot(theta)\n", "ypredict2 = Xnew.dot(theta_linreg)\n", "\n", "\n", "n_epochs = 50\n", "t0, t1 = 5, 50\n", "def learning_schedule(t):\n", " return t0/(t+t1)\n", "\n", "theta = np.random.randn(2,1)\n", "\n", "for epoch in range(n_epochs):\n", " for i in range(m):\n", " random_index = np.random.randint(m)\n", " xi = X[random_index:random_index+1]\n", " yi = y[random_index:random_index+1]\n", " gradients = 2 * xi.T @ ((xi @ theta)-yi)\n", " eta = learning_schedule(epoch*m+i)\n", " theta = theta - eta*gradients\n", "print(\"theta from own sdg\")\n", "print(theta)\n", "\n", "plt.plot(xnew, ypredict, \"r-\")\n", "plt.plot(xnew, ypredict2, \"b-\")\n", "plt.plot(x, y ,'ro')\n", "plt.axis([0,2.0,0, 15.0])\n", "plt.xlabel(r'$x$')\n", "plt.ylabel(r'$y$')\n", "plt.title(r'Random numbers ')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Challenge**: try to write a similar code for a Logistic Regression case.\n", "\n", "\n", "## Momentum based GD\n", "\n", "The stochastic gradient descent (SGD) is almost always used with a\n", "*momentum* or inertia term that serves as a memory of the direction we\n", "are moving in parameter space. This is typically implemented as\n", "follows" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t) \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation} \n", "\\boldsymbol{\\theta}_{t+1}= \\boldsymbol{\\theta}_t -\\mathbf{v}_{t},\n", "\\label{_auto1} \\tag{1}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have introduced a momentum parameter $\\gamma$, with\n", "$0\\le\\gamma\\le 1$, and for brevity we dropped the explicit notation to\n", "indicate the gradient is to be taken over a different mini-batch at\n", "each step. We call this algorithm gradient descent with momentum\n", "(GDM). From these equations, it is clear that $\\mathbf{v}_t$ is a\n", "running average of recently encountered gradients and\n", "$(1-\\gamma)^{-1}$ sets the characteristic time scale for the memory\n", "used in the averaging procedure. Consistent with this, when\n", "$\\gamma=0$, this just reduces down to ordinary SGD as discussed\n", "earlier. An equivalent way of writing the updates is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\Delta \\boldsymbol{\\theta}_{t+1} = \\gamma \\Delta \\boldsymbol{\\theta}_t -\\ \\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t),\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have defined $\\Delta \\boldsymbol{\\theta}_{t}= \\boldsymbol{\\theta}_t-\\boldsymbol{\\theta}_{t-1}$.\n", "\n", "## More on momentum based approaches\n", "\n", "Let us try to get more intuition from these equations. It is helpful\n", "to consider a simple physical analogy with a particle of mass $m$\n", "moving in a viscous medium with drag coefficient $\\mu$ and potential\n", "$E(\\mathbf{w})$. If we denote the particle's position by $\\mathbf{w}$,\n", "then its motion is described by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "m {d^2 \\mathbf{w} \\over dt^2} + \\mu {d \\mathbf{w} \\over dt }= -\\nabla_w E(\\mathbf{w}).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can discretize this equation in the usual way to get" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "m { \\mathbf{w}_{t+\\Delta t}-2 \\mathbf{w}_{t} +\\mathbf{w}_{t-\\Delta t} \\over (\\Delta t)^2}+\\mu {\\mathbf{w}_{t+\\Delta t}- \\mathbf{w}_{t} \\over \\Delta t} = -\\nabla_w E(\\mathbf{w}).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Rearranging this equation, we can rewrite this as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\Delta \\mathbf{w}_{t +\\Delta t}= - { (\\Delta t)^2 \\over m +\\mu \\Delta t} \\nabla_w E(\\mathbf{w})+ {m \\over m +\\mu \\Delta t} \\Delta \\mathbf{w}_t.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Momentum parameter\n", "\n", "Notice that this equation is identical to previous one if we identify\n", "the position of the particle, $\\mathbf{w}$, with the parameters\n", "$\\boldsymbol{\\theta}$. This allows us to identify the momentum\n", "parameter and learning rate with the mass of the particle and the\n", "viscous drag as:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\gamma= {m \\over m +\\mu \\Delta t }, \\qquad \\eta = {(\\Delta t)^2 \\over m +\\mu \\Delta t}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thus, as the name suggests, the momentum parameter is proportional to\n", "the mass of the particle and effectively provides inertia.\n", "Furthermore, in the large viscosity/small learning rate limit, our\n", "memory time scales as $(1-\\gamma)^{-1} \\approx m/(\\mu \\Delta t)$.\n", "\n", "Why is momentum useful? SGD momentum helps the gradient descent\n", "algorithm gain speed in directions with persistent but small gradients\n", "even in the presence of stochasticity, while suppressing oscillations\n", "in high-curvature directions. This becomes especially important in\n", "situations where the landscape is shallow and flat in some directions\n", "and narrow and steep in others. It has been argued that first-order\n", "methods (with appropriate initial conditions) can perform comparable\n", "to more expensive second order methods, especially in the context of\n", "complex deep learning models.\n", "\n", "These beneficial properties of momentum can sometimes become even more\n", "pronounced by using a slight modification of the classical momentum\n", "algorithm called Nesterov Accelerated Gradient (NAG).\n", "\n", "In the NAG algorithm, rather than calculating the gradient at the\n", "current parameters, $\\nabla_\\theta E(\\boldsymbol{\\theta}_t)$, one\n", "calculates the gradient at the expected value of the parameters given\n", "our current momentum, $\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma\n", "\\mathbf{v}_{t-1})$. This yields the NAG update rule" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma \\mathbf{v}_{t-1}) \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation} \n", "\\boldsymbol{\\theta}_{t+1}= \\boldsymbol{\\theta}_t -\\mathbf{v}_{t}.\n", "\\label{_auto2} \\tag{2}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\\gamma$.\n", "\n", "\n", "## Second moment of the gradient\n", "\n", "\n", "In stochastic gradient descent, with and without momentum, we still\n", "have to specify a schedule for tuning the learning rates $\\eta_t$\n", "as a function of time. As discussed in the context of Newton's\n", "method, this presents a number of dilemmas. The learning rate is\n", "limited by the steepest direction which can change depending on the\n", "current position in the landscape. To circumvent this problem, ideally\n", "our algorithm would keep track of curvature and take large steps in\n", "shallow, flat directions and small steps in steep, narrow directions.\n", "Second-order methods accomplish this by calculating or approximating\n", "the Hessian and normalizing the learning rate by the\n", "curvature. However, this is very computationally expensive for\n", "extremely large models. Ideally, we would like to be able to\n", "adaptively change the step size to match the landscape without paying\n", "the steep computational price of calculating or approximating\n", "Hessians.\n", "\n", "Recently, a number of methods have been introduced that accomplish\n", "this by tracking not only the gradient, but also the second moment of\n", "the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and\n", "ADAM.\n", "\n", "## RMS prop\n", "\n", "In RMS prop, in addition to keeping a running average of the first\n", "moment of the gradient, we also keep track of the second moment\n", "denoted by $\\mathbf{s}_t=\\mathbb{E}[\\mathbf{g}_t^2]$. The update rule\n", "for RMS prop is given by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", "\\mathbf{g}_t = \\nabla_\\theta E(\\boldsymbol{\\theta}) \n", "\\label{_auto3} \\tag{3}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{s}_t =\\beta \\mathbf{s}_{t-1} +(1-\\beta)\\mathbf{g}_t^2 \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\mathbf{g}_t \\over \\sqrt{\\mathbf{s}_t +\\epsilon}}, \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $\\beta$ controls the averaging time of the second moment and is\n", "typically taken to be about $\\beta=0.9$, $\\eta_t$ is a learning rate\n", "typically chosen to be $10^{-3}$, and $\\epsilon\\sim 10^{-8} $ is a\n", "small regularization constant to prevent divergences. Multiplication\n", "and division by vectors is understood as an element-wise operation. It\n", "is clear from this formula that the learning rate is reduced in\n", "directions where the norm of the gradient is consistently large. This\n", "greatly speeds up the convergence by allowing us to use a larger\n", "learning rate for flat directions.\n", "\n", "\n", "## ADAM optimizer\n", "\n", "A related algorithm is the ADAM optimizer. In ADAM, we keep a running\n", "average of both the first and second moment of the gradient and use\n", "this information to adaptively change the learning rate for different\n", "parameters. In addition to keeping a running average of the first and\n", "second moments of the gradient\n", "(i.e. $\\mathbf{m}_t=\\mathbb{E}[\\mathbf{g}_t]$ and\n", "$\\mathbf{s}_t=\\mathbb{E}[\\mathbf{g}^2_t]$, respectively), ADAM\n", "performs an additional bias correction to account for the fact that we\n", "are estimating the first two moments of the gradient using a running\n", "average (denoted by the hats in the update rule below). The update\n", "rule for ADAM is given by (where multiplication and division are once\n", "again understood to be element-wise operations below)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", "\\mathbf{g}_t = \\nabla_\\theta E(\\boldsymbol{\\theta}) \n", "\\label{_auto4} \\tag{4}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{m}_t = \\beta_1 \\mathbf{m}_{t-1} + (1-\\beta_1) \\mathbf{g}_t \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{s}_t =\\beta_2 \\mathbf{s}_{t-1} +(1-\\beta_2)\\mathbf{g}_t^2 \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\boldsymbol{\\mathbf{m}}_t={\\mathbf{m}_t \\over 1-\\beta_1^t} \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\boldsymbol{\\mathbf{s}}_t ={\\mathbf{s}_t \\over1-\\beta_2^t} \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\boldsymbol{\\mathbf{m}}_t \\over \\sqrt{\\boldsymbol{\\mathbf{s}}_t} +\\epsilon}, \\nonumber\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation} \n", "\\label{_auto5} \\tag{5}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $\\beta_1$ and $\\beta_2$ set the memory lifetime of the first and\n", "second moment and are typically taken to be $0.9$ and $0.99$\n", "respectively, and $\\eta$ and $\\epsilon$ are identical to RMSprop.\n", "\n", "Like in RMSprop, the effective step size of a parameter depends on the\n", "magnitude of its gradient squared. To understand this better, let us\n", "rewrite this expression in terms of the variance\n", "$\\boldsymbol{\\sigma}_t^2 = \\boldsymbol{\\mathbf{s}}_t -\n", "(\\boldsymbol{\\mathbf{m}}_t)^2$. Consider a single parameter $\\theta_t$. The\n", "update rule for this parameter is given by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\Delta \\theta_{t+1}= -\\eta_t { \\boldsymbol{m}_t \\over \\sqrt{\\sigma_t^2 + m_t^2 }+\\epsilon}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Practical tips\n", "\n", "* **Randomize the data when making mini-batches**. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.\n", "\n", "* **Transform your inputs**. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.\n", "\n", "* **Monitor the out-of-sample performance.** Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This *early stopping* significantly improves performance in many settings.\n", "\n", "* **Adaptive optimization methods don't always have good generalization.** Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.\n", "\n", "Geron's text, see chapter 11, has several interesting discussions.\n", "\n", "\n", "\n", "## Automatic differentiation\n", "\n", "[Automatic differentiation (AD)](https://en.wikipedia.org/wiki/Automatic_differentiation), \n", "also called algorithmic\n", "differentiation or computational differentiation,is a set of\n", "techniques to numerically evaluate the derivative of a function\n", "specified by a computer program. AD exploits the fact that every\n", "computer program, no matter how complicated, executes a sequence of\n", "elementary arithmetic operations (addition, subtraction,\n", "multiplication, division, etc.) and elementary functions (exp, log,\n", "sin, cos, etc.). By applying the chain rule repeatedly to these\n", "operations, derivatives of arbitrary order can be computed\n", "automatically, accurately to working precision, and using at most a\n", "small constant factor more arithmetic operations than the original\n", "program.\n", "\n", "Automatic differentiation is neither:\n", "\n", "* Symbolic differentiation, nor\n", "\n", "* Numerical differentiation (the method of finite differences).\n", "\n", "Symbolic differentiation can lead to inefficient code and faces the\n", "difficulty of converting a computer program into a single expression,\n", "while numerical differentiation can introduce round-off errors in the\n", "discretization process and cancellation\n", "\n", "\n", "\n", "Python has tools for so-called **automatic differentiation**.\n", "Consider the following example" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "f(x) = \\sin\\left(2\\pi x + x^2\\right)\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which has the following derivative" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "f'(x) = \\cos\\left(2\\pi x + x^2\\right)\\left(2\\pi + 2x\\right)\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using **autograd** we have" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The max absolute difference is: 1.77636e-15\n" ] } ], "source": [ "import autograd.numpy as np\n", "\n", "# To do elementwise differentiation:\n", "from autograd import elementwise_grad as egrad \n", "\n", "# To plot:\n", "import matplotlib.pyplot as plt \n", "\n", "\n", "def f(x):\n", " return np.sin(2*np.pi*x + x**2)\n", "\n", "def f_grad_analytic(x):\n", " return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)\n", "\n", "# Do the comparison:\n", "x = np.linspace(0,1,1000)\n", "\n", "f_grad = egrad(f)\n", "\n", "computed = f_grad(x)\n", "analytic = f_grad_analytic(x)\n", "\n", "plt.title('Derivative computed from Autograd compared with the analytical derivative')\n", "plt.plot(x,computed,label='autograd')\n", "plt.plot(x,analytic,label='analytic')\n", "\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.legend()\n", "\n", "plt.show()\n", "\n", "print(\"The max absolute difference is: %g\"%(np.max(np.abs(computed - analytic))))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## Using autograd\n", "\n", "Here we\n", "experiment with what kind of functions Autograd is capable\n", "of finding the gradient of. The following Python functions are just\n", "meant to illustrate what Autograd can do, but please feel free to\n", "experiment with other, possibly more complicated, functions as well." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The gradient of f1 evaluated at a = 1 using autograd is: 3\n", "The gradient of f1 evaluated at a = 1 by finding the analytic expression is: 3\n" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "\n", "def f1(x):\n", " return x**3 + 1\n", "\n", "f1_grad = grad(f1)\n", "\n", "# Remember to send in float as argument to the computed gradient from Autograd!\n", "a = 1.0\n", "\n", "# See the evaluated gradient at a using autograd:\n", "print(\"The gradient of f1 evaluated at a = %g using autograd is: %g\"%(a,f1_grad(a)))\n", "\n", "# Compare with the analytical derivative, that is f1'(x) = 3*x**2 \n", "grad_analytical = 3*a**2\n", "print(\"The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g\"%(a,grad_analytical))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Autograd with more complicated functions\n", "\n", "To differentiate with respect to two (or more) arguments of a Python\n", "function, Autograd need to know at which variable the function if\n", "being differentiated with respect to." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Evaluating at x1 = 1, x2 = 3\n", "------------------------------\n", "The derivative of f2 w.r.t x1: 12\n", "The analytical derivative of f2 w.r.t x1: 12\n", "\n", "The derivative of f2 w.r.t x2: -4\n", "The analytical derivative of f2 w.r.t x2: -4\n" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "def f2(x1,x2):\n", " return 3*x1**3 + x2*(x1 - 5) + 1\n", "\n", "# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1\n", "f2_grad_x1 = grad(f2,0)\n", "\n", "# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad\n", "f2_grad_x2 = grad(f2,1)\n", "\n", "x1 = 1.0\n", "x2 = 3.0 \n", "\n", "print(\"Evaluating at x1 = %g, x2 = %g\"%(x1,x2))\n", "print(\"-\"*30)\n", "\n", "# Compare with the analytical derivatives:\n", "\n", "# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:\n", "f2_grad_x1_analytical = 9*x1**2 + x2\n", "\n", "# Derivative of f2 w.r.t x2 is: x1 - 5:\n", "f2_grad_x2_analytical = x1 - 5\n", "\n", "# See the evaluated derivations:\n", "print(\"The derivative of f2 w.r.t x1: %g\"%( f2_grad_x1(x1,x2) ))\n", "print(\"The analytical derivative of f2 w.r.t x1: %g\"%( f2_grad_x1(x1,x2) ))\n", "\n", "print()\n", "\n", "print(\"The derivative of f2 w.r.t x2: %g\"%( f2_grad_x2(x1,x2) ))\n", "print(\"The analytical derivative of f2 w.r.t x2: %g\"%( f2_grad_x2(x1,x2) ))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.\n", "\n", "\n", "## More complicated functions using the elements of their arguments directly" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The computed gradient of f3 is: [ 2. 3. 5. 7. 88.]\n", "The analytical gradient of f3 is: [ 2. 3. 5. 7. 88.]\n" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "def f3(x): # Assumes x is an array of length 5 or higher\n", " return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2\n", "\n", "f3_grad = grad(f3)\n", "\n", "x = np.linspace(0,4,5)\n", "\n", "# Print the computed gradient:\n", "print(\"The computed gradient of f3 is: \", f3_grad(x))\n", "\n", "# The analytical gradient is: (2, 3, 5, 7, 22*x[4])\n", "f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])\n", "\n", "# Print the analytical gradient:\n", "print(\"The analytical gradient of f3 is: \", f3_grad_analytical)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that in this case, when sending an array as input argument, the\n", "output from Autograd is another array. This is the true gradient of\n", "the function, as opposed to the function in the previous example. By\n", "using arrays to represent the variables, the output from Autograd\n", "might be easier to work with, as the output is closer to what one\n", "could expect form a gradient-evaluting function.\n", "\n", "\n", "## Functions using mathematical functions from Numpy" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The computed derivative of f4 at x = 2.7 is: 13.8759\n", "The analytical gradient of f4 at x = 2.7 is: 13.8759\n" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "def f4(x):\n", " return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)\n", "\n", "f4_grad = grad(f4)\n", "\n", "x = 2.7\n", "\n", "# Print the computed derivative:\n", "print(\"The computed derivative of f4 at x = %g is: %g\"%(x,f4_grad(x)))\n", "\n", "# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi\n", "f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi\n", "\n", "# Print the analytical gradient:\n", "print(\"The analytical gradient of f4 at x = %g is: %g\"%(x,f4_grad_analytical))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## More autograd" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The computed derivative of f5 at x = 2.7 is: 5.4\n" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "def f5(x):\n", " if x >= 0:\n", " return x**2\n", " else:\n", " return -3*x + 1\n", "\n", "f5_grad = grad(f5)\n", "\n", "x = 2.7\n", "\n", "# Print the computed derivative:\n", "print(\"The computed derivative of f5 at x = %g is: %g\"%(x,f5_grad(x)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## And with loops" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "9\n", " \n", "<\n", "<\n", "<\n", "!\n", "!\n", "C\n", "O\n", "D\n", "E\n", "_\n", "B\n", "L\n", "O\n", "C\n", "K\n", " \n", " \n", "p\n", "y\n", "c\n", "o\n", "d" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The analytical derivative of f6 at x = 2.7 is: 37732.5\n" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9\n", "# The analytical derivative is: sum(i*x**(i-1)) \n", "f6_grad_analytical = 0\n", "for i in range(10):\n", " f6_grad_analytical += i*x**(i-1)\n", "\n", "print(\"The analytical derivative of f6 at x = %g is: %g\"%(x,f6_grad_analytical))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Using recursion" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The computed derivative of f7 at n = 2 is: 1\n", "The analytical derivative of f7 at n = 2 is: 1\n" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "\n", "def f7(n): # Assume that n is an integer\n", " if n == 1 or n == 0:\n", " return 1\n", " else:\n", " return n*f7(n-1)\n", "\n", "f7_grad = grad(f7)\n", "\n", "n = 2.0\n", "\n", "print(\"The computed derivative of f7 at n = %d is: %g\"%(n,f7_grad(n)))\n", "\n", "# The function f7 is an implementation of the factorial of n.\n", "# By using the product rule, one can find that the derivative is:\n", "\n", "f7_grad_analytical = 0\n", "for i in range(int(n)-1):\n", " tmp = 1\n", " for k in range(int(n)-1):\n", " if k != i:\n", " tmp *= (n - k)\n", " f7_grad_analytical += tmp\n", "\n", "print(\"The analytical derivative of f7 at n = %d is: %g\"%(n,f7_grad_analytical))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.\n", "\n", "## Unsupported functions\n", "Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.\n", "\n", "Assigning a value to the variable being differentiated with respect to" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "ename": "TypeError", "evalue": "'ArrayBox' object does not support item assignment", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 9\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m8.4\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 11\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"The derivative of f8 is:\"\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mf8_grad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/autograd/wrap_util.py\u001b[0m in \u001b[0;36mnary_f\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margnum\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 20\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0munary_operator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0munary_f\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mnary_op_args\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mnary_op_kwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 21\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_f\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 22\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_operator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/autograd/differential_operators.py\u001b[0m in \u001b[0;36mgrad\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m 23\u001b[0m \u001b[0marguments\u001b[0m \u001b[0;32mas\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbut\u001b[0m \u001b[0mreturns\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mgradient\u001b[0m \u001b[0minstead\u001b[0m\u001b[0;34m.\u001b[0m \u001b[0mThe\u001b[0m \u001b[0mfunction\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 24\u001b[0m should be scalar-valued. 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f8 is:\",f8_grad(x))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.\n", "\n", "## The syntax a.dot(b) when finding the dot product" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "ename": "AttributeError", "evalue": "'ArrayBox' object has no attribute 'dot'", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 9\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1.0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m0.0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 11\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"The derivative of f9 is:\"\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mf9_grad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/autograd/wrap_util.py\u001b[0m in \u001b[0;36mnary_f\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margnum\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 20\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0munary_operator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0munary_f\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mnary_op_args\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mnary_op_kwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 21\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_f\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 22\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_operator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/autograd/differential_operators.py\u001b[0m in \u001b[0;36mgrad\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m 23\u001b[0m \u001b[0marguments\u001b[0m \u001b[0;32mas\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbut\u001b[0m \u001b[0mreturns\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mgradient\u001b[0m \u001b[0minstead\u001b[0m\u001b[0;34m.\u001b[0m \u001b[0mThe\u001b[0m \u001b[0mfunction\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 24\u001b[0m should be scalar-valued. The gradient has the same type as the argument.\"\"\"\n\u001b[0;32m---> 25\u001b[0;31m \u001b[0mvjp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mans\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_make_vjp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 26\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mvspace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mans\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msize\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 27\u001b[0m raise TypeError(\"Grad only applies to real scalar-output functions. \"\n", "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/autograd/core.py\u001b[0m in \u001b[0;36mmake_vjp\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m 8\u001b[0m \u001b[0;32mdef\u001b[0m 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elements\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1.0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2.0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 6\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[0mf9_grad\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgrad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf9\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mAttributeError\u001b[0m: 'ArrayBox' object has no attribute 'dot'" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "def f9(a): # Assume a is an array with 2 elements\n", " b = np.array([1.0,2.0])\n", " return a.dot(b)\n", "\n", "f9_grad = grad(f9)\n", "\n", "x = np.array([1.0,0.0])\n", "\n", "print(\"The derivative of f9 is:\",f9_grad(x))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here we are told that the 'dot' function does not belong to Autograd's\n", "version of a Numpy array. To overcome this, an alternative syntax\n", "which also computed the dot product can be used:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The gradient of f9 is: [1. 2.]\n" ] } ], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", "def f9_alternative(x): # Assume a is an array with 2 elements\n", " b = np.array([1.0,2.0])\n", " return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2\n", "\n", "f9_alternative_grad = grad(f9_alternative)\n", "\n", "x = np.array([3.0,0.0])\n", "\n", "print(\"The gradient of f9 is:\",f9_alternative_grad(x))\n", "\n", "# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively\n", "# w.r.t x is (b_1, b_2)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Recommended to avoid\n", "The documentation recommends to avoid inplace operations such as" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "a += b\n", "a -= b\n", "a*= b\n", "a /=b" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Neural networks\n", "\n", "Artificial neural networks are computational systems that can learn to\n", "perform tasks by considering examples, generally without being\n", "programmed with any task-specific rules. It is supposed to mimic a\n", "biological system, wherein neurons interact by sending signals in the\n", "form of mathematical functions between layers. All layers can contain\n", "an arbitrary number of neurons, and each connection is represented by\n", "a weight variable.\n", "\n", "\n", "## Artificial neurons\n", "\n", "The field of artificial neural networks has a long history of\n", "development, and is closely connected with the advancement of computer\n", "science and computers in general. A model of artificial neurons was\n", "first developed by McCulloch and Pitts in 1943 to study signal\n", "processing in the brain and has later been refined by others. The\n", "general idea is to mimic neural networks in the human brain, which is\n", "composed of billions of neurons that communicate with each other by\n", "sending electrical signals. Each neuron accumulates its incoming\n", "signals, which must exceed an activation threshold to yield an\n", "output. If the threshold is not overcome, the neuron remains inactive,\n", "i.e. has zero output.\n", "\n", "This behaviour has inspired a simple mathematical model for an artificial neuron." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " y = f\\left(\\sum_{i=1}^n w_ix_i\\right) = f(u)\n", "\\label{artificialNeuron} \\tag{6}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here, the output $y$ of the neuron is the value of its activation function, which have as input\n", "a weighted sum of signals $x_i, \\dots ,x_n$ received by $n$ other neurons.\n", "\n", "Conceptually, it is helpful to divide neural networks into four\n", "categories:\n", "1. general purpose neural networks for supervised learning,\n", "\n", "2. neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),\n", "\n", "3. neural networks for sequential data such as Recurrent Neural Networks (RNNs), and\n", "\n", "4. neural networks for unsupervised learning such as Deep Boltzmann Machines.\n", "\n", "In natural science, DNNs and CNNs have already found numerous\n", "applications. In statistical physics, they have been applied to detect\n", "phase transitions in 2D Ising and Potts models, lattice gauge\n", "theories, and different phases of polymers, or solving the\n", "Navier-Stokes equation in weather forecasting. Deep learning has also\n", "found interesting applications in quantum physics. Various quantum\n", "phase transitions can be detected and studied using DNNs and CNNs,\n", "topological phases, and even non-equilibrium many-body\n", "localization. Representing quantum states as DNNs quantum state\n", "tomography are among some of the impressive achievements to reveal the\n", "potential of DNNs to facilitate the study of quantum systems.\n", "\n", "In quantum information theory, it has been shown that one can perform\n", "gate decompositions with the help of neural. \n", "\n", "The applications are not limited to the natural sciences. There is a\n", "plethora of applications in essentially all disciplines, from the\n", "humanities to life science and medicine.\n", "\n", "## Neural network types\n", "\n", "An artificial neural network (ANN), is a computational model that\n", "consists of layers of connected neurons, or nodes or units. We will\n", "refer to these interchangeably as units or nodes, and sometimes as\n", "neurons.\n", "\n", "It is supposed to mimic a biological nervous system by letting each\n", "neuron interact with other neurons by sending signals in the form of\n", "mathematical functions between layers. A wide variety of different\n", "ANNs have been developed, but most of them consist of an input layer,\n", "an output layer and eventual layers in-between, called *hidden\n", "layers*. All layers can contain an arbitrary number of nodes, and each\n", "connection between two nodes is associated with a weight variable.\n", "\n", "Neural networks (also called neural nets) are neural-inspired\n", "nonlinear models for supervised learning. As we will see, neural nets\n", "can be viewed as natural, more powerful extensions of supervised\n", "learning methods such as linear and logistic regression and soft-max\n", "methods we discussed earlier.\n", "\n", "\n", "## Feed-forward neural networks\n", "\n", "The feed-forward neural network (FFNN) was the first and simplest type\n", "of ANNs that were devised. In this network, the information moves in\n", "only one direction: forward through the layers.\n", "\n", "Nodes are represented by circles, while the arrows display the\n", "connections between the nodes, including the direction of information\n", "flow. Additionally, each arrow corresponds to a weight variable\n", "(figure to come). We observe that each node in a layer is connected\n", "to *all* nodes in the subsequent layer, making this a so-called\n", "*fully-connected* FFNN.\n", "\n", "\n", "\n", "## Convolutional Neural Network\n", "\n", "A different variant of FFNNs are *convolutional neural networks*\n", "(CNNs), which have a connectivity pattern inspired by the animal\n", "visual cortex. Individual neurons in the visual cortex only respond to\n", "stimuli from small sub-regions of the visual field, called a receptive\n", "field. This makes the neurons well-suited to exploit the strong\n", "spatially local correlation present in natural images. The response of\n", "each neuron can be approximated mathematically as a convolution\n", "operation. (figure to come)\n", "\n", "Convolutional neural networks emulate the behaviour of neurons in the\n", "visual cortex by enforcing a *local* connectivity pattern between\n", "nodes of adjacent layers: Each node in a convolutional layer is\n", "connected only to a subset of the nodes in the previous layer, in\n", "contrast to the fully-connected FFNN. Often, CNNs consist of several\n", "convolutional layers that learn local features of the input, with a\n", "fully-connected layer at the end, which gathers all the local data and\n", "produces the outputs. They have wide applications in image and video\n", "recognition.\n", "\n", "## Recurrent neural networks\n", "\n", "So far we have only mentioned ANNs where information flows in one\n", "direction: forward. *Recurrent neural networks* on the other hand,\n", "have connections between nodes that form directed *cycles*. This\n", "creates a form of internal memory which are able to capture\n", "information on what has been calculated before; the output is\n", "dependent on the previous computations. Recurrent NNs make use of\n", "sequential information by performing the same task for every element\n", "in a sequence, where each element depends on previous elements. An\n", "example of such information is sentences, making recurrent NNs\n", "especially well-suited for handwriting and speech recognition.\n", "\n", "## Other types of networks\n", "\n", "There are many other kinds of ANNs that have been developed. One type\n", "that is specifically designed for interpolation in multidimensional\n", "space is the radial basis function (RBF) network. RBFs are typically\n", "made up of three layers: an input layer, a hidden layer with\n", "non-linear radial symmetric activation functions and a linear output\n", "layer (''linear'' here means that each node in the output layer has a\n", "linear activation function). The layers are normally fully-connected\n", "and there are no cycles, thus RBFs can be viewed as a type of\n", "fully-connected FFNN. They are however usually treated as a separate\n", "type of NN due the unusual activation functions.\n", "\n", "## Multilayer perceptrons\n", "\n", "One uses often so-called fully-connected feed-forward neural networks\n", "with three or more layers (an input layer, one or more hidden layers\n", "and an output layer) consisting of neurons that have non-linear\n", "activation functions.\n", "\n", "Such networks are often called *multilayer perceptrons* (MLPs).\n", "\n", "## Why multilayer perceptrons?\n", "\n", "According to the *Universal approximation theorem*, a feed-forward\n", "neural network with just a single hidden layer containing a finite\n", "number of neurons can approximate a continuous multidimensional\n", "function to arbitrary accuracy, assuming the activation function for\n", "the hidden layer is a **non-constant, bounded and\n", "monotonically-increasing continuous function**.\n", "\n", "Note that the requirements on the activation function only applies to\n", "the hidden layer, the output nodes are always assumed to be linear, so\n", "as to not restrict the range of output values.\n", "\n", "\n", "## Mathematical model\n", "\n", "The output $y$ is produced via the activation function $f$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "y = f\\left(\\sum_{i=1}^n w_ix_i + b_i\\right) = f(z),\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This function receives $x_i$ as inputs.\n", "Here the activation $z=(\\sum_{i=1}^n w_ix_i+b_i)$. \n", "In an FFNN of such neurons, the *inputs* $x_i$ are the *outputs* of\n", "the neurons in the preceding layer. Furthermore, an MLP is\n", "fully-connected, which means that each neuron receives a weighted sum\n", "of the outputs of *all* neurons in the previous layer.\n", "\n", "## Mathematical model\n", "\n", "First, for each node $i$ in the first hidden layer, we calculate a weighted sum $z_i^1$ of the input coordinates $x_j$," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation} z_i^1 = \\sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1\n", "\\label{_auto6} \\tag{7}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here $b_i$ is the so-called bias which is normally needed in\n", "case of zero activation weights or inputs. How to fix the biases and\n", "the weights will be discussed below. The value of $z_i^1$ is the\n", "argument to the activation function $f_i$ of each node $i$, The\n", "variable $M$ stands for all possible inputs to a given node $i$ in the\n", "first layer. We define the output $y_i^1$ of all neurons in layer 1 as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " y_i^1 = f(z_i^1) = f\\left(\\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\\right)\n", "\\label{outputLayer1} \\tag{8}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we assume that all nodes in the same layer have identical\n", "activation functions, hence the notation $f$. In general, we could assume in the more general case that different layers have different activation functions.\n", "In this case we would identify these functions with a superscript $l$ for the $l$-th layer," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " y_i^l = f^l(u_i^l) = f^l\\left(\\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\\right)\n", "\\label{generalLayer} \\tag{9}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $N_l$ is the number of nodes in layer $l$. When the output of\n", "all the nodes in the first hidden layer are computed, the values of\n", "the subsequent layer can be calculated and so forth until the output\n", "is obtained.\n", "\n", "\n", "\n", "## Mathematical model\n", "\n", "The output of neuron $i$ in layer 2 is thus," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " y_i^2 = f^2\\left(\\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\\right) \n", "\\label{_auto7} \\tag{10}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation} \n", " = f^2\\left[\\sum_{j=1}^N w_{ij}^2f^1\\left(\\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\\right) + b_i^2\\right]\n", "\\label{outputLayer2} \\tag{11}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have substituted $y_k^1$ with the inputs $x_k$. Finally, the ANN output reads" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " y_i^3 = f^3\\left(\\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\\right) \n", "\\label{_auto8} \\tag{12}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation} \n", " = f_3\\left[\\sum_{j} w_{ij}^3 f^2\\left(\\sum_{k} w_{jk}^2 f^1\\left(\\sum_{m} w_{km}^1 x_m + b_k^1\\right) + b_j^2\\right)\n", " + b_1^3\\right]\n", "\\label{_auto9} \\tag{13}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Mathematical model\n", "\n", "We can generalize this expression to an MLP with $l$ hidden\n", "layers. The complete functional form is," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", "y^{l+1}_i = f^{l+1}\\left[\\!\\sum_{j=1}^{N_l} w_{ij}^3 f^l\\left(\\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\\left(\\dots f^1\\left(\\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\\right)\\dots\\right)+b_k^2\\right)+b_1^3\\right] \n", "\\label{completeNN} \\tag{14}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which illustrates a basic property of MLPs: The only independent\n", "variables are the input values $x_n$.\n", "\n", "## Mathematical model\n", "\n", "This confirms that an MLP, despite its quite convoluted mathematical\n", "form, is nothing more than an analytic function, specifically a\n", "mapping of real-valued vectors $\\hat{x} \\in \\mathbb{R}^n \\rightarrow\n", "\\hat{y} \\in \\mathbb{R}^m$.\n", "\n", "Furthermore, the flexibility and universality of an MLP can be\n", "illustrated by realizing that the expression is essentially a nested\n", "sum of scaled activation functions of the form" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " f(x) = c_1 f(c_2 x + c_3) + c_4\n", "\\label{_auto10} \\tag{15}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the parameters $c_i$ are weights and biases. By adjusting these\n", "parameters, the activation functions can be shifted up and down or\n", "left and right, change slope or be rescaled which is the key to the\n", "flexibility of a neural network.\n", "\n", "### Matrix-vector notation\n", "\n", "We can introduce a more convenient notation for the activations in an A NN. \n", "\n", "Additionally, we can represent the biases and activations\n", "as layer-wise column vectors $\\hat{b}_l$ and $\\hat{y}_l$, so that the $i$-th element of each vector \n", "is the bias $b_i^l$ and activation $y_i^l$ of node $i$ in layer $l$ respectively. \n", "\n", "We have that $\\mathrm{W}_l$ is an $N_{l-1} \\times N_l$ matrix, while $\\hat{b}_l$ and $\\hat{y}_l$ are $N_l \\times 1$ column vectors. \n", "With this notation, the sum becomes a matrix-vector multiplication, and we can write\n", "the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " \\hat{y}_2 = f_2(\\mathrm{W}_2 \\hat{y}_{1} + \\hat{b}_{2}) = \n", " f_2\\left(\\left[\\begin{array}{ccc}\n", " w^2_{11} &w^2_{12} &w^2_{13} \\\\\n", " w^2_{21} &w^2_{22} &w^2_{23} \\\\\n", " w^2_{31} &w^2_{32} &w^2_{33} \\\\\n", " \\end{array} \\right] \\cdot\n", " \\left[\\begin{array}{c}\n", " y^1_1 \\\\\n", " y^1_2 \\\\\n", " y^1_3 \\\\\n", " \\end{array}\\right] + \n", " \\left[\\begin{array}{c}\n", " b^2_1 \\\\\n", " b^2_2 \\\\\n", " b^2_3 \\\\\n", " \\end{array}\\right]\\right).\n", "\\label{_auto11} \\tag{16}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Matrix-vector notation and activation\n", "\n", "The activation of node $i$ in layer 2 is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " y^2_i = f_2\\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\\Bigr) = \n", " f_2\\left(\\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\\right).\n", "\\label{_auto12} \\tag{17}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is not just a convenient and compact notation, but also a useful\n", "and intuitive way to think about MLPs: The output is calculated by a\n", "series of matrix-vector multiplications and vector additions that are\n", "used as input to the activation functions. For each operation\n", "$\\mathrm{W}_l \\hat{y}_{l-1}$ we move forward one layer.\n", "\n", "\n", "### Activation functions\n", "\n", "A property that characterizes a neural network, other than its\n", "connectivity, is the choice of activation function(s). As described\n", "in, the following restrictions are imposed on an activation function\n", "for a FFNN to fulfill the universal approximation theorem\n", "\n", " * Non-constant\n", "\n", " * Bounded\n", "\n", " * Monotonically-increasing\n", "\n", " * Continuous\n", "\n", "### Activation functions, Logistic and Hyperbolic ones\n", "\n", "The second requirement excludes all linear functions. Furthermore, in\n", "a MLP with only linear activation functions, each layer simply\n", "performs a linear transformation of its inputs.\n", "\n", "Regardless of the number of layers, the output of the NN will be\n", "nothing but a linear function of the inputs. Thus we need to introduce\n", "some kind of non-linearity to the NN to be able to fit non-linear\n", "functions Typical examples are the logistic *Sigmoid*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "f(x) = \\frac{1}{1 + e^{-x}},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and the *hyperbolic tangent* function" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "f(x) = \\tanh(x)\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Relevance\n", "\n", "The *sigmoid* function are more biologically plausible because the\n", "output of inactive neurons are zero. Such activation function are\n", "called *one-sided*. However, it has been shown that the hyperbolic\n", "tangent performs better than the sigmoid for training MLPs. has\n", "become the most popular for *deep neural networks*" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "\"\"\"The sigmoid function (or the logistic curve) is a \n", "function that takes any real number, z, and outputs a number (0,1).\n", "It is useful in neural networks for assigning weights on a relative scale.\n", "The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n", "\n", "import numpy\n", "import matplotlib.pyplot as plt\n", "import math as mt\n", "\n", "z = numpy.arange(-5, 5, .1)\n", "sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n", "sigma = sigma_fn(z)\n", "\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "ax.plot(z, sigma)\n", "ax.set_ylim([-0.1, 1.1])\n", "ax.set_xlim([-5,5])\n", "ax.grid(True)\n", "ax.set_xlabel('z')\n", "ax.set_title('sigmoid function')\n", "\n", "plt.show()\n", "\n", "\"\"\"Step Function\"\"\"\n", "z = numpy.arange(-5, 5, .02)\n", "step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n", "step = step_fn(z)\n", "\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "ax.plot(z, step)\n", "ax.set_ylim([-0.5, 1.5])\n", "ax.set_xlim([-5,5])\n", "ax.grid(True)\n", "ax.set_xlabel('z')\n", "ax.set_title('step function')\n", "\n", "plt.show()\n", "\n", "\"\"\"Sine Function\"\"\"\n", "z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n", "t = numpy.sin(z)\n", "\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "ax.plot(z, t)\n", "ax.set_ylim([-1.0, 1.0])\n", "ax.set_xlim([-2*mt.pi,2*mt.pi])\n", "ax.grid(True)\n", "ax.set_xlabel('z')\n", "ax.set_title('sine function')\n", "\n", "plt.show()\n", "\n", "\"\"\"Plots a graph of the squashing function used by a rectified linear\n", "unit\"\"\"\n", "z = numpy.arange(-2, 2, .1)\n", "zero = numpy.zeros(len(z))\n", "y = numpy.max([zero, z], axis=0)\n", "\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "ax.plot(z, y)\n", "ax.set_ylim([-2.0, 2.0])\n", "ax.set_xlim([-2.0, 2.0])\n", "ax.grid(True)\n", "ax.set_xlabel('z')\n", "ax.set_title('Rectified linear unit')\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The multilayer perceptron (MLP)\n", "\n", "The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of\n", "1. A neural network with one or more layers of nodes between the input and the output nodes.\n", "\n", "2. The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.\n", "\n", "3. The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.\n", "\n", "As a convention it is normal to call a network with one layer of input units, one layer of hidden\n", "units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.\n", "\n", "For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.\n", "Hereafter we will call the various entities of a layer for nodes.\n", "There are also no connections within a single layer.\n", "\n", "The number of input nodes does not need to equal the number of output\n", "nodes. This applies also to the hidden layers. Each layer may have its\n", "own number of nodes and activation functions.\n", "\n", "The hidden layers have their name from the fact that they are not\n", "linked to observables and as we will see below when we define the\n", "so-called activation $\\hat{z}$, we can think of this as a basis\n", "expansion of the original inputs $\\hat{x}$. The difference however\n", "between neural networks and say linear regression is that now these\n", "basis functions (which will correspond to the weights in the network)\n", "are learned from data. This results in an important difference between\n", "neural networks and deep learning approaches on one side and methods\n", "like logistic regression or linear regression and their modifications on the other side.\n", "\n", "\n", "## From one to many layers, the universal approximation theorem\n", "\n", "\n", "A neural network with only one layer, what we called the simple\n", "perceptron, is best suited if we have a standard binary model with\n", "clear (linear) boundaries between the outcomes. As such it could\n", "equally well be replaced by standard linear regression or logistic\n", "regression. Networks with one or more hidden layers approximate\n", "systems with more complex boundaries.\n", "\n", "As stated earlier, \n", "an important theorem in studies of neural networks, restated without\n", "proof here, is the [universal approximation\n", "theorem](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf).\n", "\n", "It states that a feed-forward network with a single hidden layer\n", "containing a finite number of neurons can approximate continuous\n", "functions on compact subsets of real functions. The theorem thus\n", "states that simple neural networks can represent a wide variety of\n", "interesting functions when given appropriate parameters. It is the\n", "multilayer feedforward architecture itself which gives neural networks\n", "the potential of being universal approximators.\n", "\n", "\n", "## Deriving the back propagation code for a multilayer perceptron model\n", "\n", "\n", "\n", "As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.\n", "The unknowwn quantities are our weights $w_{ij}$ and we need to find an algorithm for changing them so that our errors are as small as possible.\n", "This leads us to the famous [back propagation algorithm](https://www.nature.com/articles/323533a0).\n", "\n", "The questions we want to ask are how do changes in the biases and the\n", "weights in our network change the cost function and how can we use the\n", "final output to modify the weights?\n", "\n", "To derive these equations let us start with a plain regression problem\n", "and define our cost function as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "{\\cal C}(\\hat{W}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the $t_i$s are our $n$ targets (the values we want to\n", "reproduce), while the outputs of the network after having propagated\n", "all inputs $\\hat{x}$ are given by $y_i$. Below we will demonstrate\n", "how the basic equations arising from the back propagation algorithm\n", "can be modified in order to study classification problems with $K$\n", "classes.\n", "\n", "## Definitions\n", "\n", "With our definition of the targets $\\hat{t}$, the outputs of the\n", "network $\\hat{y}$ and the inputs $\\hat{x}$ we\n", "define now the activation $z_j^l$ of node/neuron/unit $j$ of the\n", "$l$-th layer as a function of the bias, the weights which add up from\n", "the previous layer $l-1$ and the forward passes/outputs\n", "$\\hat{a}^{l-1}$ from the previous layer as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "z_j^l = \\sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $b_k^l$ are the biases from layer $l$. Here $M_{l-1}$\n", "represents the total number of nodes/neurons/units of layer $l-1$. The\n", "figure here illustrates this equation. We can rewrite this in a more\n", "compact form as the matrix-vector products we discussed earlier," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{z}^l = \\left(\\hat{W}^l\\right)^T\\hat{a}^{l-1}+\\hat{b}^l.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With the activation values $\\hat{z}^l$ we can in turn define the\n", "output of layer $l$ as $\\hat{a}^l = f(\\hat{z}^l)$ where $f$ is our\n", "activation function. In the examples here we will use the sigmoid\n", "function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers\n", "and their nodes. It means we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "a_j^l = f(z_j^l) = \\frac{1}{1+\\exp{-(z_j^l)}}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Derivatives and the chain rule\n", "\n", "From the definition of the activation $z_j^l$ we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\partial z_j^l}{\\partial w_{ij}^l} = a_i^{l-1},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\partial z_j^l}{\\partial a_i^{l-1}} = w_{ji}^l.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With our definition of the activation function we have that (note that this function depends only on $z_j^l$)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\partial a_j^l}{\\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Derivative of the cost function\n", "\n", "With these definitions we can now compute the derivative of the cost function in terms of the weights.\n", "\n", "Let us specialize to the output layer $l=L$. Our cost function is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "{\\cal C}(\\hat{W^L}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2=\\frac{1}{2}\\sum_{i=1}^n\\left(a_i^L - t_i\\right)^2,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The derivative of this function with respect to the weights is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\left(a_j^L - t_j\\right)\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The last partial derivative can easily be computed and reads (by applying the chain rule)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}} = \\frac{\\partial a_j^L}{\\partial z_{j}^{L}}\\frac{\\partial z_j^L}{\\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Bringing it together, first back propagation equation\n", "\n", "We have thus" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\left(a_j^L - t_j\\right)a_j^L(1-a_j^L)a_k^{L-1},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Defining" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\delta_j^L = a_j^L(1-a_j^L)\\left(a_j^L - t_j\\right) = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and using the Hadamard product of two vectors we can write this as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{\\delta}^L = f'(\\hat{z}^L)\\circ\\frac{\\partial {\\cal C}}{\\partial (\\hat{a}^L)}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is an important expression. The second term on the right handside\n", "measures how fast the cost function is changing as a function of the $j$th\n", "output activation. If, for example, the cost function doesn't depend\n", "much on a particular output node $j$, then $\\delta_j^L$ will be small,\n", "which is what we would expect. The first term on the right, measures\n", "how fast the activation function $f$ is changing at a given activation\n", "value $z_j^L$.\n", "\n", "Notice that everything in the above equations is easily computed. In\n", "particular, we compute $z_j^L$ while computing the behaviour of the\n", "network, and it is only a small additional overhead to compute\n", "$f'(z^L_j)$. The exact form of the derivative with respect to the\n", "output depends on the form of the cost function.\n", "However, provided the cost function is known there should be little\n", "trouble in calculating" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With the definition of $\\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Derivatives in terms of $z_j^L$\n", "\n", "It is also easy to see that our previous equation can be written as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\delta_j^L =\\frac{\\partial {\\cal C}}{\\partial z_j^L}= \\frac{\\partial {\\cal C}}{\\partial a_j^L}\\frac{\\partial a_j^L}{\\partial z_j^L},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L}\\frac{\\partial b_j^L}{\\partial z_j^L}=\\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That is, the error $\\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias. \n", "## Bringing it together\n", "\n", "We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are\n", "\n", "**The starting equations.**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1},\n", "\\label{_auto13} \\tag{18}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", "\\label{_auto14} \\tag{19}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", "\\label{_auto15} \\tag{20}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An interesting consequence of the above equations is that when the\n", "activation $a_k^{L-1}$ is small, the gradient term, that is the\n", "derivative of the cost function with respect to the weights, will also\n", "tend to be small. We say then that the weight learns slowly, meaning\n", "that it changes slowly when we minimize the weights via say gradient\n", "descent. In this case we say the system learns slowly.\n", "\n", "Another interesting feature is that is when the activation function,\n", "represented by the sigmoid function here, is rather flat when we move towards\n", "its end values $0$ and $1$ (see the above Python codes). In these\n", "cases, the derivatives of the activation function will also be close\n", "to zero, meaning again that the gradients will be small and the\n", "network learns slowly again.\n", "\n", "\n", "\n", "We need a fourth equation and we are set. We are going to propagate\n", "backwards in order to the determine the weights and biases. In order\n", "to do so we need to represent the error in the layer before the final\n", "one $L-1$ in terms of the errors in the final output layer.\n", "\n", "## Final back propagating equation\n", "\n", "We have that (replacing $L$ with a general layer $l$)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\delta_j^l =\\frac{\\partial {\\cal C}}{\\partial z_j^l}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\delta_j^l =\\sum_k \\frac{\\partial {\\cal C}}{\\partial z_k^{l+1}}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}}=\\sum_k \\delta_k^{l+1}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and recalling that" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "z_j^{l+1} = \\sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with $M_l$ being the number of nodes in layer $l$, we obtain" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\delta_j^l =\\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is our final equation.\n", "\n", "We are now ready to set up the algorithm for back propagation and learning the weights and biases.\n", "\n", "## Setting up the Back propagation algorithm\n", "\n", "\n", "\n", "The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.\n", "\n", "First, we set up the input data $\\hat{x}$ and the activations\n", "$\\hat{z}_1$ of the input layer and compute the activation function and\n", "the pertinent outputs $\\hat{a}^1$.\n", "\n", "\n", "\n", "Secondly, we perform then the feed forward till we reach the output\n", "layer and compute all $\\hat{z}_l$ of the input layer and compute the\n", "activation function and the pertinent outputs $\\hat{a}^l$ for\n", "$l=2,3,\\dots,L$.\n", "\n", "\n", "\n", "Thereafter we compute the ouput error $\\hat{\\delta}^L$ by computing all" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Then we compute the back propagate error for each $l=L-1,L-2,\\dots,2$ as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,2$ and update the weights and biases according to the rules" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "w_{jk}^l\\leftarrow = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The parameter $\\eta$ is the learning parameter discussed in connection with the gradient descent methods.\n", "Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.8" } }, "nbformat": 4, "nbformat_minor": 4 }