Morten Hjorth-Jensen
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Applied Data Analysis and Machine Learning
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About the course
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Teachers and Grading
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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2. Linear Algebra, Handling of Arrays and more Python Features
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From Regression to Support Vector Machines
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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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7. Optimization, the central part of any Machine Learning algortithm
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8. Support Vector Machines, overarching aims
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<p aria-level="2" class="caption" role="heading">
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Decision Trees, Ensemble Methods and Boosting
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9. Decision trees, overarching aims
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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Dimensionality Reduction
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11. Basic ideas of the Principal Component Analysis (PCA)
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12. Clustering and Unsupervised Learning
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Deep Learning Methods
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13. Neural networks
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14. Building a Feed Forward Neural Network
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15. Solving Differential Equations with Deep Learning
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16. Convolutional Neural Networks
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17. Recurrent neural networks: Overarching view
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Weekly material, notes and exercises
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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Exercises week 35
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Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
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Exercises week 36
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Week 36: Statistical interpretation of Linear Regression and Resampling techniques
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Exercises week 37
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Week 37: Statistical interpretations and Resampling Methods
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Exercises week 38
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Week 38: Logistic Regression and Optimization
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Exercises week 39
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Week 39: Optimization and Gradient Methods
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<a class="reference internal" href="week40.html">
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Week 40: Gradient descent methods (continued) and start Neural networks
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Exercises week 41
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Week 41 Neural networks and constructing a neural network code
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Projects
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Project 1 on Machine Learning, deadline October 9 (midnight), 2023
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<h1>Exercises week 41</h1>
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<h1>Exercises week 41<a class="headerlink" href="#exercises-week-41" title="Permalink to this headline">¶</a></h1>
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<p><strong>October 9-13, 2023</strong></p>
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<p>Date: <strong>Deadline is Sunday October 15 at midnight</strong></p>
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<div class="section" id="overarching-aims-of-the-exercises-this-week">
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<h2>Overarching aims of the exercises this week<a class="headerlink" href="#overarching-aims-of-the-exercises-this-week" title="Permalink to this headline">¶</a></h2>
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<p>The aim of the exercises this week is to get started with implementing
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gradient methods of relevance for project 2. This exercise will also
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be continued next week with the addition of automatic differentation.
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||
Everything you develop here will be used in project 2.</p>
|
||
<p>In order to get started, we will now replace in our standard ordinary
|
||
least squares (OLS) and Ridge regression codes (from project 1) the
|
||
matrix inversion algorithm with our own gradient descent (GD) and SGD
|
||
codes. You can use the Franke function or the terrain data from
|
||
project 1. <strong>However, we recommend using a simpler function like</strong>
|
||
<span class="math notranslate nohighlight">\(f(x)=a_0+a_1x+a_2x^2\)</span> or higher-order one-dimensional polynomials.
|
||
You can obviously test your final codes against for example the Franke
|
||
function.</p>
|
||
<p>You should include in your analysis of the GD and SGD codes the following elements</p>
|
||
<ol class="simple">
|
||
<li><p>A plain gradient descent with a fixed learning rate (you will need to tune it) using the analytical expression of the gradients</p></li>
|
||
<li><p>Add momentum to the plain GD code and compare convergence with a fixed learning rate (you may need to tune the learning rate), again using the analytical expression of the gradients.</p></li>
|
||
<li><p>Repeat these steps for stochastic gradient descent with mini batches and a given number of epochs. Use a tunable learning rate as discussed in the lectures from week 39. Discuss the results as functions of the various parameters (size of batches, number of epochs etc)</p></li>
|
||
<li><p>Implement the Adagrad method in order to tune the learning rate. Do this with and without momentum for plain gradient descent and SGD.</p></li>
|
||
<li><p>Add RMSprop and Adam to your library of methods for tuning the learning rate.</p></li>
|
||
</ol>
|
||
<p>The lecture notes from “weeks 39 and 40 contain more information and code examples. Feel free to use these examples.</p>
|
||
<p>In summary, you should
|
||
perform an analysis of the results for OLS and Ridge regression as
|
||
function of the chosen learning rates, the number of mini-batches and
|
||
epochs as well as algorithm for scaling the learning rate. You can
|
||
also compare your own results with those that can be obtained using
|
||
for example <strong>Scikit-Learn</strong>’s various SGD options. Discuss your
|
||
results. For Ridge regression you need now to study the results as functions of the hyper-parameter <span class="math notranslate nohighlight">\(\lambda\)</span> and
|
||
the learning rate <span class="math notranslate nohighlight">\(\eta\)</span>. Discuss your results.</p>
|
||
<p>You will need your SGD code for the setup of the Neural Network and
|
||
Logistic Regression codes. You will find the Python <a class="reference external" href="https://seaborn.pydata.org/generated/seaborn.heatmap.html">Seaborn
|
||
package</a>
|
||
useful when plotting the results as function of the learning rate
|
||
<span class="math notranslate nohighlight">\(\eta\)</span> and the hyper-parameter <span class="math notranslate nohighlight">\(\lambda\)</span> when you use Ridge
|
||
regression.</p>
|
||
<p>We recommend reading chapter 8 on optimization from the textbook of <a class="reference external" href="https://www.deeplearningbook.org/">Goodfellow, Bengio and Courville</a>. This chapter contains many useful insights and discussions on the optimization part of machine learning.</p>
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