addex exercise week 37
This commit is contained in:
Morten Hjorth-Jensen
@@ -8,7 +8,7 @@
|
||||
<meta charset="utf-8" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||
|
||||
<title>Exercises week 36 — Applied Data Analysis and Machine Learning</title>
|
||||
<title>Exercises week 37 — Applied Data Analysis and Machine Learning</title>
|
||||
|
||||
|
||||
|
||||
@@ -228,7 +228,7 @@
|
||||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
|
||||
<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 36</a></li>
|
||||
<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 37</a></li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
@@ -367,7 +367,7 @@ document.write(`
|
||||
|
||||
|
||||
<div id="jb-print-docs-body" class="onlyprint">
|
||||
<h1>Exercises week 36</h1>
|
||||
<h1>Exercises week 37</h1>
|
||||
<!-- Table of contents -->
|
||||
<div id="print-main-content">
|
||||
<div id="jb-print-toc">
|
||||
@@ -378,21 +378,23 @@ document.write(`
|
||||
<nav aria-label="Page">
|
||||
<ul class="visible nav section-nav flex-column">
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning goals</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ridge-regression-and-a-new-synthetic-dataset">Ridge regression and a new Synthetic Dataset</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-one-dimensional-second-order-polynomial">Simple one-dimensional second-order polynomial</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1-scale-your-data">Exercise 1, scale your data</a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a">1a)</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span></a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">2a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#b">2b)</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-calculate-the-gradients">Exercise 2, calculate the gradients</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span></a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">3a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#b">3b)</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-implementing-the-simplest-form-for-gradient-descent">Exercise 3, Implementing the simplest form for gradient descent</a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">3a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">3b)</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-4-implementing-the-simplest-form-for-gradient-descent">Exercise 4, Implementing the simplest form for gradient descent</a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">4a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">4b)</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-5-ridge-regression-and-a-new-synthetic-dataset">Exercise 5, Ridge regression and a new Synthetic Dataset</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
</div>
|
||||
@@ -406,8 +408,8 @@ document.write(`
|
||||
|
||||
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
|
||||
doconce format html exercisesweek37.do.txt -->
|
||||
<!-- dom:TITLE: Exercises week 36 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-36">
|
||||
<h1>Exercises week 36<a class="headerlink" href="#exercises-week-36" title="Link to this heading">#</a></h1>
|
||||
<!-- dom:TITLE: Exercises week 37 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-37">
|
||||
<h1>Exercises week 37<a class="headerlink" href="#exercises-week-37" title="Link to this heading">#</a></h1>
|
||||
<p><strong>Implementing gradient descent for Ridge and ordinary Least Squares Regression</strong></p>
|
||||
<p>Date: <strong>September 8-12, 2025</strong></p>
|
||||
<section id="learning-goals">
|
||||
@@ -420,53 +422,16 @@ doconce format html exercisesweek37.do.txt -->
|
||||
<li><p>Scale the data properly</p></li>
|
||||
</ol>
|
||||
</section>
|
||||
<section id="ridge-regression-and-a-new-synthetic-dataset">
|
||||
<h2>Ridge regression and a new Synthetic Dataset<a class="headerlink" href="#ridge-regression-and-a-new-synthetic-dataset" title="Link to this heading">#</a></h2>
|
||||
<p>We create a synthetic linear regression dataset with a sparse
|
||||
underlying relationship. This means we have many features but only a
|
||||
few of them actually contribute to the target. In our example, we’ll
|
||||
use 10 features with only 3 non-zero weights in the true model. This
|
||||
way, the target is generated as a linear combination of a few features
|
||||
(with known coefficients) plus some random noise. The steps we include are:</p>
|
||||
<p>Decide on the number of samples and features (e.g. 100 samples, 10 features).
|
||||
Define the <strong>true</strong> coefficient vector with mostly zeros (for sparsity). For example, we set <span class="math notranslate nohighlight">\(\hat{\boldsymbol{\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]\)</span>, meaning only features 0, 1, and 6 have a real effect on y.</p>
|
||||
<p>Then we sample feature values for <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.
|
||||
Then we compute the target values <span class="math notranslate nohighlight">\(y\)</span> using the linear combination <span class="math notranslate nohighlight">\(\boldsymbol{X}\hat{\boldsymbol{\theta}}\)</span> and add some noise (to simulate measurement error or unexplained variance).</p>
|
||||
<p>Below is the code to generate the dataset:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import numpy as np
|
||||
|
||||
# Set random seed for reproducibility
|
||||
np.random.seed(0)
|
||||
|
||||
# Define dataset size
|
||||
n_samples = 100
|
||||
n_features = 10
|
||||
|
||||
# Define true coefficients (sparse linear relationship)
|
||||
theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])
|
||||
|
||||
# Generate feature matrix X (n_samples x n_features) with random values
|
||||
X = np.random.randn(n_samples, n_features) # standard normal distribution
|
||||
|
||||
# Generate target values y with a linear combination of X and theta_true, plus noise
|
||||
noise = 0.5 * np.random.randn(n_samples) # Gaussian noise
|
||||
y = X.dot @ theta_true + noise
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>This code produces a dataset where only features 0, 1, and 6
|
||||
significantly influence <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>. The rest of the features have zero true
|
||||
coefficient. For example, feature 0 has
|
||||
a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so
|
||||
the expected relationship is:</p>
|
||||
<section id="simple-one-dimensional-second-order-polynomial">
|
||||
<h2>Simple one-dimensional second-order polynomial<a class="headerlink" href="#simple-one-dimensional-second-order-polynomial" title="Link to this heading">#</a></h2>
|
||||
<p>We start with a very simple function</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
y \approx 5 \times x_0 \;-\; 3 \times x_1 \;+\; 2 \times x_6 \;+\; \text{noise}.
|
||||
\f(x)= 2-x+5x^2,
|
||||
\]</div>
|
||||
<p>You can remove the noise if you wish to.</p>
|
||||
<p>defined for <span class="math notranslate nohighlight">\(x\in [-2,2]\)</span>. You can add noise if you wish.</p>
|
||||
<p>We are going to fit this function with a polynomial ansatz. The easiest thing is to set up a second-order polynomial and see if you can fit the above function.
|
||||
Feel free to play around with higher-order polynomials.</p>
|
||||
</section>
|
||||
<section id="exercise-1-scale-your-data">
|
||||
<h2>Exercise 1, scale your data<a class="headerlink" href="#exercise-1-scale-your-data" title="Link to this heading">#</a></h2>
|
||||
@@ -477,13 +442,14 @@ regularization. Here we will perform standardization, scaling each
|
||||
feature to have mean 0 and standard deviation 1.</p>
|
||||
<section id="a">
|
||||
<h3>1a)<a class="headerlink" href="#a" title="Link to this heading">#</a></h3>
|
||||
<p>Compute the mean and standard deviation of each column (feature) in <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.
|
||||
<p>Compute the mean and standard deviation of each column (feature) in your design/feature matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.
|
||||
Subtract the mean and divide by the standard deviation for each feature.</p>
|
||||
<p>We will also center the target <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> to mean <span class="math notranslate nohighlight">\(0\)</span>. Centering <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>
|
||||
(and each feature) means the model does not require a separate intercept
|
||||
term, the data is shifted such that the intercept is effectively 0
|
||||
. (In practice, one could include an intercept in the model and not
|
||||
penalize it, but here we simplify by centering.)</p>
|
||||
penalize it, but here we simplify by centering.)
|
||||
Choose <span class="math notranslate nohighlight">\(n=100\)</span> data points and set up <span class="math notranslate nohighlight">\(\boldsymbol{x}, \)</span>\boldsymbol{y} and the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Standardize features (zero mean, unit variance for each feature)
|
||||
@@ -507,8 +473,12 @@ nicer and ensures the regularization penalty <span class="math notranslate nohig
|
||||
same scale).</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="exercise-2-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">
|
||||
<h2>Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span><a class="headerlink" href="#exercise-2-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta" title="Link to this heading">#</a></h2>
|
||||
<section id="exercise-2-calculate-the-gradients">
|
||||
<h2>Exercise 2, calculate the gradients<a class="headerlink" href="#exercise-2-calculate-the-gradients" title="Link to this heading">#</a></h2>
|
||||
<p>Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.</p>
|
||||
</section>
|
||||
<section id="exercise-3-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">
|
||||
<h2>Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span><a class="headerlink" href="#exercise-3-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta" title="Link to this heading">#</a></h2>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Set regularization parameter, either a single value or a vector of values
|
||||
@@ -531,16 +501,16 @@ then invert this matrix and multiply by \)</span>X^T y<span class="math notransl
|
||||
for \)</span>\boldsymbol{\theta}<span class="math notranslate nohighlight">\( is a NumPy array of shape (n\)</span>_<span class="math notranslate nohighlight">\(features,) containing the
|
||||
fitted parameters \)</span>\boldsymbol{\theta}$..</p>
|
||||
<section id="id1">
|
||||
<h3>2a)<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
|
||||
<h3>3a)<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
|
||||
<p>Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span>.</p>
|
||||
</section>
|
||||
<section id="b">
|
||||
<h3>2b)<a class="headerlink" href="#b" title="Link to this heading">#</a></h3>
|
||||
<h3>3b)<a class="headerlink" href="#b" title="Link to this heading">#</a></h3>
|
||||
<p>Explore the results as function of different values of the hyperparameter <span class="math notranslate nohighlight">\(\lambda\)</span>. See for example exercise 4 from week 36.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="exercise-3-implementing-the-simplest-form-for-gradient-descent">
|
||||
<h2>Exercise 3, Implementing the simplest form for gradient descent<a class="headerlink" href="#exercise-3-implementing-the-simplest-form-for-gradient-descent" title="Link to this heading">#</a></h2>
|
||||
<section id="exercise-4-implementing-the-simplest-form-for-gradient-descent">
|
||||
<h2>Exercise 4, Implementing the simplest form for gradient descent<a class="headerlink" href="#exercise-4-implementing-the-simplest-form-for-gradient-descent" title="Link to this heading">#</a></h2>
|
||||
<p>Alternatively, we can fit the ridge regression model using gradient
|
||||
descent. This is useful to visualize the iterative convergence and is
|
||||
necessary if <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(p\)</span> are so large that the closed-form might be
|
||||
@@ -587,19 +557,68 @@ print("Gradient Descent Ridge coefficients:", theta_gdRidge)
|
||||
</div>
|
||||
</div>
|
||||
<section id="id2">
|
||||
<h3>3a)<a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
|
||||
<h3>4a)<a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
|
||||
<p>Discuss the results as function of the learning rate parameters and the number of iterations.</p>
|
||||
</section>
|
||||
<section id="id3">
|
||||
<h3>3b)<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
|
||||
<h3>4b)<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
|
||||
<p>Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="exercise-5-ridge-regression-and-a-new-synthetic-dataset">
|
||||
<h2>Exercise 5, Ridge regression and a new Synthetic Dataset<a class="headerlink" href="#exercise-5-ridge-regression-and-a-new-synthetic-dataset" title="Link to this heading">#</a></h2>
|
||||
<p>We create a synthetic linear regression dataset with a sparse
|
||||
underlying relationship. This means we have many features but only a
|
||||
few of them actually contribute to the target. In our example, we’ll
|
||||
use 10 features with only 3 non-zero weights in the true model. This
|
||||
way, the target is generated as a linear combination of a few features
|
||||
(with known coefficients) plus some random noise. The steps we include are:</p>
|
||||
<p>Decide on the number of samples and features (e.g. 100 samples, 10 features).
|
||||
Define the <strong>true</strong> coefficient vector with mostly zeros (for sparsity). For example, we set <span class="math notranslate nohighlight">\(\hat{\boldsymbol{\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]\)</span>, meaning only features 0, 1, and 6 have a real effect on y.</p>
|
||||
<p>Then we sample feature values for <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.
|
||||
Then we compute the target values <span class="math notranslate nohighlight">\(y\)</span> using the linear combination <span class="math notranslate nohighlight">\(\boldsymbol{X}\hat{\boldsymbol{\theta}}\)</span> and add some noise (to simulate measurement error or unexplained variance).</p>
|
||||
<p>Below is the code to generate the dataset:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import numpy as np
|
||||
|
||||
# Set random seed for reproducibility
|
||||
np.random.seed(0)
|
||||
|
||||
# Define dataset size
|
||||
n_samples = 100
|
||||
n_features = 10
|
||||
|
||||
# Define true coefficients (sparse linear relationship)
|
||||
theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])
|
||||
|
||||
# Generate feature matrix X (n_samples x n_features) with random values
|
||||
X = np.random.randn(n_samples, n_features) # standard normal distribution
|
||||
|
||||
# Generate target values y with a linear combination of X and theta_true, plus noise
|
||||
noise = 0.5 * np.random.randn(n_samples) # Gaussian noise
|
||||
y = X.dot @ theta_true + noise
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>This code produces a dataset where only features 0, 1, and 6
|
||||
significantly influence <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>. The rest of the features have zero true
|
||||
coefficient. For example, feature 0 has
|
||||
a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so
|
||||
the expected relationship is:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
y \approx 5 \times x_0 \;-\; 3 \times x_1 \;+\; 2 \times x_6 \;+\; \text{noise}.
|
||||
\]</div>
|
||||
<p>You can remove the noise if you wish to.</p>
|
||||
<p>Try to fit the above data set using OLS and Ridge regression with the analytical expressions and your own gradient descent codes.</p>
|
||||
<p>If everything worked correctly, the learned coefficients should be
|
||||
close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to
|
||||
generate the data. Keep in mind that due to regularization and noise,
|
||||
the learned values will not exactly equal the true ones, but they
|
||||
should be in the same ballpark. Which method (OLS or Ridge) gives the best results?</p>
|
||||
</section>
|
||||
</section>
|
||||
</section>
|
||||
|
||||
<script type="text/x-thebe-config">
|
||||
@@ -658,21 +677,23 @@ should be in the same ballpark. Which method (OLS or Ridge) gives the best resu
|
||||
<nav class="bd-toc-nav page-toc">
|
||||
<ul class="visible nav section-nav flex-column">
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning goals</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ridge-regression-and-a-new-synthetic-dataset">Ridge regression and a new Synthetic Dataset</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-one-dimensional-second-order-polynomial">Simple one-dimensional second-order polynomial</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1-scale-your-data">Exercise 1, scale your data</a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a">1a)</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span></a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">2a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#b">2b)</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-calculate-the-gradients">Exercise 2, calculate the gradients</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span></a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">3a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#b">3b)</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-implementing-the-simplest-form-for-gradient-descent">Exercise 3, Implementing the simplest form for gradient descent</a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">3a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">3b)</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-4-implementing-the-simplest-form-for-gradient-descent">Exercise 4, Implementing the simplest form for gradient descent</a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">4a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">4b)</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-5-ridge-regression-and-a-new-synthetic-dataset">Exercise 5, Ridge regression and a new Synthetic Dataset</a></li>
|
||||
</ul>
|
||||
</nav></div>
|
||||
|
||||
|
||||
Reference in New Issue
Block a user