Morten Hjorth-Jensen
766 lines
42 KiB
HTML
766 lines
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Applied Data Analysis and Machine Learning
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Review of Statistics with Resampling Techniques and Linear Algebra</span></p>
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<li class="toctree-l1"><a class="reference internal" href="statistics.html">1. Elements of Probability Theory and Statistical Data Analysis</a></li>
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<li class="toctree-l1"><a class="reference internal" href="linalg.html">2. Linear Algebra, Handling of Arrays and more Python Features</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">From Regression to Support Vector Machines</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter1.html">3. Linear Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter2.html">4. Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter3.html">5. Resampling Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter4.html">6. Logistic Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapteroptimization.html">7. Optimization, the central part of any Machine Learning algortithm</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter6.html">9. Decision trees, overarching aims</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter7.html">10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Dimensionality Reduction</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter8.html">11. Basic ideas of the Principal Component Analysis (PCA)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter9.html">13. Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter10.html">14. Building a Feed Forward Neural Network</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter11.html">15. Solving Differential Equations with Deep Learning</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter12.html">16. Convolutional Neural Networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter13.html">17. Recurrent neural networks: Overarching view</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek34.html">Exercises week 34</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week34.html">Week 34: Introduction to the course, Logistics and Practicalities</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek35.html">Exercises week 35</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 37</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week37.html">Week 37: Gradient descent methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek38.html">Exercises week 38</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week38.html">Week 38: Statistical analysis, bias-variance tradeoff and resampling methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek39.html">Exercises week 39</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week39.html">Week 39: Resampling methods and logistic regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week40.html">Week 40: Gradient descent methods (continued) and start Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project1.html">Project 1 on Machine Learning, deadline October 6 (midnight), 2025</a></li>
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<h1>Exercises week 37</h1>
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|
||
<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-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-using-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">Exercise 3, using 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-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>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="searchbox"></div>
|
||
<article class="bd-article">
|
||
|
||
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
|
||
doconce format html exercisesweek37.do.txt -->
|
||
<!-- 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">
|
||
<h2>Learning goals<a class="headerlink" href="#learning-goals" title="Link to this heading">#</a></h2>
|
||
<p>After having completed these exercises you will have:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression</p></li>
|
||
<li><p>Be able to compare the analytical expressions for OLS and Ridge regression with the gradient descent approach</p></li>
|
||
<li><p>Explore the role of the learning rate in the gradient descent approach and the hyperparameter <span class="math notranslate nohighlight">\(\lambda\)</span> in Ridge regression</p></li>
|
||
<li><p>Scale the data properly</p></li>
|
||
</ol>
|
||
</section>
|
||
<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">
|
||
\[
|
||
f(x)= 2-x+5x^2,
|
||
\]</div>
|
||
<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>
|
||
<p>Before fitting a regression model, it is good practice to normalize or
|
||
standardize the features. This ensures all features are on a
|
||
comparable scale, which is especially important when using
|
||
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 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.)
|
||
Choose <span class="math notranslate nohighlight">\(n=100\)</span> data points and set up <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>, <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> 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-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Standardize features (zero mean, unit variance for each feature)</span>
|
||
<span class="n">X_mean</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">X_std</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">X_std</span><span class="p">[</span><span class="n">X_std</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># safeguard to avoid division by zero for constant features</span>
|
||
<span class="n">X_norm</span> <span class="o">=</span> <span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="n">X_mean</span><span class="p">)</span> <span class="o">/</span> <span class="n">X_std</span>
|
||
|
||
<span class="c1"># Center the target to zero mean (optional, to simplify intercept handling)</span>
|
||
<span class="n">y_mean</span> <span class="o">=</span> <span class="err">?</span>
|
||
<span class="n">y_centered</span> <span class="o">=</span> <span class="err">?</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Fill in the necessary details. Do we need to center the <span class="math notranslate nohighlight">\(y\)</span>-values?</p>
|
||
<p>After this preprocessing, each column of <span class="math notranslate nohighlight">\(\boldsymbol{X}_{\mathrm{norm}}\)</span> has mean zero and standard deviation <span class="math notranslate nohighlight">\(1\)</span>
|
||
and <span class="math notranslate nohighlight">\(\boldsymbol{y}_{\mathrm{centered}}\)</span> has mean 0. This makes the optimization landscape
|
||
nicer and ensures the regularization penalty <span class="math notranslate nohighlight">\(\lambda \sum_j
|
||
\theta_j^2\)</span> in Ridge regression treats each coefficient fairly (since features are on the
|
||
same scale).</p>
|
||
</section>
|
||
</section>
|
||
<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-using-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">
|
||
<h2>Exercise 3, using 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-using-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-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set regularization parameter, either a single value or a vector of values</span>
|
||
<span class="c1"># Note that lambda is a python keyword. The lambda keyword is used to create small, single-expression functions without a formal name. These are often called "anonymous functions" or "lambda functions."</span>
|
||
<span class="n">lam</span> <span class="o">=</span> <span class="err">?</span>
|
||
|
||
|
||
<span class="c1"># Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y</span>
|
||
<span class="n">I</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n_features</span><span class="p">)</span>
|
||
<span class="n">theta_closed_formRidge</span> <span class="o">=</span> <span class="err">?</span>
|
||
<span class="n">theta_closed_formOLS</span> <span class="o">=</span> <span class="err">?</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Closed-form Ridge coefficients:"</span><span class="p">,</span> <span class="n">theta_closed_form</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Closed-form OLS coefficients:"</span><span class="p">,</span> <span class="n">theta_closed_form</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>This computes the Ridge and OLS regression coefficients directly. The identity
|
||
matrix <span class="math notranslate nohighlight">\(I\)</span> has the same size as <span class="math notranslate nohighlight">\(X^T X\)</span>. It adds <span class="math notranslate nohighlight">\(\lambda\)</span> to the diagonal of <span class="math notranslate nohighlight">\(X^T X\)</span> for Ridge regression. We
|
||
then invert this matrix and multiply by <span class="math notranslate nohighlight">\(X^T y\)</span>. The result
|
||
for <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> is a NumPy array of shape (n<span class="math notranslate nohighlight">\(\_\)</span>features,) containing the
|
||
fitted parameters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span>.</p>
|
||
<section id="id1">
|
||
<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>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-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
|
||
too slow or memory-intensive. We derive the gradients from the cost
|
||
functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to
|
||
the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> and set up (using the template below) your own gradient descent code for OLS and Ridge regression.</p>
|
||
<p>Below is a template code for gradient descent implementation of ridge:</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Gradient descent parameters, learning rate eta first</span>
|
||
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.1</span>
|
||
<span class="c1"># Then number of iterations</span>
|
||
<span class="n">num_iters</span> <span class="o">=</span> <span class="mi">1000</span>
|
||
|
||
<span class="c1"># Initialize weights for gradient descent</span>
|
||
<span class="n">theta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_features</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Gradient descent loop</span>
|
||
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iters</span><span class="p">):</span>
|
||
<span class="c1"># Compute gradients for OSL and Ridge</span>
|
||
<span class="n">grad_OLS</span> <span class="o">=</span> <span class="err">?</span>
|
||
<span class="n">grad_Ridge</span> <span class="o">=</span> <span class="err">?</span>
|
||
<span class="c1"># Update parameters theta</span>
|
||
<span class="n">theta_gdOLS</span> <span class="o">=</span> <span class="err">?</span>
|
||
<span class="n">theta_gdRidge</span> <span class="o">=</span> <span class="err">?</span>
|
||
|
||
<span class="c1"># After the loop, theta contains the fitted coefficients</span>
|
||
<span class="n">theta_gdOLS</span> <span class="o">=</span> <span class="err">?</span>
|
||
<span class="n">theta_gdRidge</span> <span class="o">=</span> <span class="err">?</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Gradient Descent OLS coefficients:"</span><span class="p">,</span> <span class="n">theta_gdOLS</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Gradient Descent Ridge coefficients:"</span><span class="p">,</span> <span class="n">theta_gdRidge</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<section id="id2">
|
||
<h3>4a)<a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
|
||
<p>Write first a gradient descent code for OLS only using the above template.
|
||
Discuss the results as function of the learning rate parameters and the number of iterations</p>
|
||
</section>
|
||
<section id="id3">
|
||
<h3>4b)<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
|
||
<p>Write then a similar code for Ridge regression using the above template.
|
||
Try to add a stopping parameter as function of the number iterations and the difference between the new and old <span class="math notranslate nohighlight">\(\theta\)</span> values. 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-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
<span class="c1"># Set random seed for reproducibility</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Define dataset size</span>
|
||
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">n_features</span> <span class="o">=</span> <span class="mi">10</span>
|
||
|
||
<span class="c1"># Define true coefficients (sparse linear relationship)</span>
|
||
<span class="n">theta_true</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">5.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
|
||
|
||
<span class="c1"># Generate feature matrix X (n_samples x n_features) with random values</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)</span> <span class="c1"># standard normal distribution</span>
|
||
|
||
<span class="c1"># Generate target values y with a linear combination of X and theta_true, plus noise</span>
|
||
<span class="n">noise</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="c1"># Gaussian noise</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">dot</span> <span class="o">@</span> <span class="n">theta_true</span> <span class="o">+</span> <span class="n">noise</span>
|
||
</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
|
||
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning goals</a></li>
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<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>
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