After having completed these exercises you will have:
Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression
Be able to compare the analytical expressions for OLS and Rudge regression with the gradient descent approach
Be able to compare the analytical expressions for OLS and Ridge regression with the gradient descent approach
Explore the role of the learning rate in the gradient descent approach and the hyperparameter \(\lambda\) in Ridge regression
Scale the data properly
Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.
-# Set regularization parameter, either a single value or a vector of values
@@ -537,13 +537,14 @@ theta = np.zeros(n_features)
cost_history = np.zeros(num_iters)
# Gradient descent loop
-m = n_samples # number of examples
+m = n_samples # number of data points
for t in range(num_iters):
# Compute prediction error
error = X_norm.dot(theta) - y_centered
# Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring
cost_OLS = ?
cost_Ridge = ?
+ # You could add a history for both methods (optional)
cost_history[t] = ?
# Compute gradients for OSL and Ridge
grad_OLS = ?
@@ -567,7 +568,7 @@ print("Gradient Descent Ridge coefficients:", theta_gdRidge)
4b)#
-Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?
+Try to add a stopping parameter as function of the number iterations and the difference between the new and old \(\theta\) values. How would you define a stopping criterion?
@@ -697,7 +698,7 @@ should be in the same ballpark. Which method (OLS or Ridge) gives the best resu
- Exercise 2, calculate the gradients
-- Exercise 3, use 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 \(\boldsymbol{\theta}\)
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 1ad914449..4c7aa149b 100644
--- a/doc/LectureNotes/_build/html/searchindex.js
+++ b/doc/LectureNotes/_build/html/searchindex.js
@@ -1 +1 @@
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"exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[17, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[15, "exercise-2-setting-up-a-github-repository"]], "Exercise 2, calculate the gradients": [[18, "exercise-2-calculate-the-gradients"]], "Exercise 2: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[16, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[15, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[17, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[15, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3, use the analytical 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[[26, "other-popular-texts"]], "Other techniques": [[11, "other-techniques"]], "Other types of networks": [[12, "other-types-of-networks"]], "Other ways of visualizing the trees": [[9, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[26, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[26, "overview-of-first-week"]], "Own code for Ordinary Least Squares": [[26, "own-code-for-ordinary-least-squares"], [27, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[26, "pandas-ai"]], "Part a : Ordinary Least Square (OLS) for the Runge function": [[21, "part-a-ordinary-least-square-ols-for-the-runge-function"]], "Part b: Adding Ridge regression for the Runge function": [[21, "part-b-adding-ridge-regression-for-the-runge-function"]], "Part c: Writing your own gradient descent code": [[21, "part-c-writing-your-own-gradient-descent-code"]], "Part d: Including momentum and 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"adding-error-analysis-and-training-set-up"], [27, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[1, "adjust-hyperparameters"]], "Algorithms for Setting up Decision Trees": [[9, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[10, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[4, "an-extrapolation-example"]], "An optimization/minimization problem": [[26, "an-optimization-minimization-problem"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[27, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And what about using neural networks?": [[26, "and-what-about-using-neural-networks"]], "Another Example, now with a polynomial fit": [[28, "another-example-now-with-a-polynomial-fit"]], "Another example, the moons again": [[9, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[19, null]], "Autocorrelation function": [[23, "autocorrelation-function"]], "Automatic differentiation": [[13, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[27, "back-to-ridge-and-lasso-regression"], [28, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[11, "back-to-the-cancer-data"]], "Background literature": [[21, "background-literature"]], "Bagging": [[10, "bagging"]], "Bagging Examples": [[10, "bagging-examples"]], "Basic Matrix Features": [[20, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[11, null]], "Basic math of the SVD": [[5, "basic-math-of-the-svd"], [27, "basic-math-of-the-svd"], [28, "basic-math-of-the-svd"]], "Basics": [[7, "basics"]], "Basics of a tree": [[9, "basics-of-a-tree"]], "Batch Normalization": [[1, "batch-normalization"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[5, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[10, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[6, "bootstrap"]], "Bringing it together, first back propagation equation": [[12, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[1, null]], "Building a tree, regression": [[9, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[1, "building-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[3, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[9, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Choose cost function and optimizer": [[1, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[11, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[14, null]], "Code for SVD and Inversion of Matrices": [[5, "code-for-svd-and-inversion-of-matrices"]], "Codes and Approaches": [[14, "codes-and-approaches"]], "Codes for the SVD": [[5, "codes-for-the-svd"], [27, "codes-for-the-svd"], [28, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[15, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[1, "collect-and-pre-process-data"]], "Communication channels": [[26, "communication-channels"]], "Compare Bagging on Trees with Random Forests": [[10, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[2, "comparing-with-a-numerical-scheme"]], "Comparison with OLS": [[28, "comparison-with-ols"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conditions on convex functions": [[28, "conditions-on-convex-functions"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convex function": [[28, "convex-function"]], "Convex functions": [[13, "convex-functions"], [28, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[3, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[3, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[12, "convolutional-neural-network"]], "Convolutional Neural Networks": [[3, null]], "Correlation Function and Design/Feature Matrix": [[27, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[11, "correlation-matrix"], [27, "correlation-matrix"]], "Correlation Matrix with Pandas": [[27, "correlation-matrix-with-pandas"]], "Course Format": [[26, "course-format"]], "Course setting": [[22, null]], "Covariance Matrix Examples": [[27, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[27, "covariance-and-correlation-matrix"]], "Cross-validation": [[6, "cross-validation"]], "Deadlines for projects (tentative)": [[26, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[9, null]], "Deep learning methods": [[26, "deep-learning-methods"]], "Define model and architecture": [[1, "define-model-and-architecture"]], "Defining the cost function": [[1, "defining-the-cost-function"]], "Deliverables": [[15, "deliverables"], [16, "deliverables"]], "Derivatives and the chain rule": [[12, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[27, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[5, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[16, "deriving-and-implementing-ordinary-least-squares"]], "Deriving and Implementing Ridge Regression": [[17, "deriving-and-implementing-ridge-regression"]], "Deriving the Lasso Regression Equations": [[27, "deriving-the-lasso-regression-equations"], [28, "deriving-the-lasso-regression-equations"], [28, "id6"]], "Deriving the Ridge Regression Equations": [[27, "deriving-the-ridge-regression-equations"], [28, "deriving-the-ridge-regression-equations"], [28, "id3"]], "Deriving the back propagation code for a multilayer perceptron model": [[12, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[1, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[11, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[8, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[9, "disadvantages"]], "Discriminative Modeling": [[26, "discriminative-modeling"]], "Domains and probabilities": [[23, "domains-and-probabilities"]], "Dropout": [[1, "dropout"]], "Economy-size SVD": [[27, "economy-size-svd"], [28, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[23, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[10, null]], "Entropy and the ID3 algorithm": [[9, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[26, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[1, "evaluate-model-performance-on-test-data"]], "Example 2": [[27, "example-2"]], "Example 3": [[27, "example-3"]], "Example 4": [[27, "example-4"]], "Example Matrix": [[27, "example-matrix"], [28, "example-matrix"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[26, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[26, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[27, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[27, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[2, "example-exponential-decay"]], "Example: Population growth": [[2, "example-population-growth"]], "Example: The diffusion equation": [[2, "example-the-diffusion-equation"]], "Example: binary classification problem": [[1, "example-binary-classification-problem"]], "Examples": [[26, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[7, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Exercise 1 - Choice of model and degrees of freedom": [[17, "exercise-1-choice-of-model-and-degrees-of-freedom"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[16, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[15, "exercise-1-github-setup"]], "Exercise 1, scale your data": [[18, "exercise-1-scale-your-data"]], "Exercise 1: Setting up various Python environments": [[0, "exercise-1-setting-up-various-python-environments"]], "Exercise 2 - Deriving the expression for OLS": [[16, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[17, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[15, "exercise-2-setting-up-a-github-repository"]], "Exercise 2, calculate the gradients": [[18, "exercise-2-calculate-the-gradients"]], "Exercise 2: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[16, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[15, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[17, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[15, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters \\boldsymbol{\\theta}": [[18, "exercise-3-using-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta"]], "Exercise 3: Normalizing our data": [[0, "exercise-3-normalizing-our-data"]], "Exercise 4 - Fitting a polynomial": [[16, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - Implementing Ridge Regression": [[17, "exercise-4-implementing-ridge-regression"]], "Exercise 4 - Testing multiple hyperparameters": [[17, "exercise-4-testing-multiple-hyperparameters"]], "Exercise 4 - The train-test split": [[15, "exercise-4-the-train-test-split"]], "Exercise 4, Implementing the simplest form for gradient descent": [[18, "exercise-4-implementing-the-simplest-form-for-gradient-descent"]], "Exercise 4: Adding Ridge Regression": [[0, "exercise-4-adding-ridge-regression"]], "Exercise 5 - Comparing your code with sklearn": [[16, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5, Ridge regression and a new Synthetic Dataset": [[18, "exercise-5-ridge-regression-and-a-new-synthetic-dataset"]], "Exercise 5: Analytical exercises": [[0, "exercise-5-analytical-exercises"]], "Exercise: Cross-validation as resampling techniques, adding more complexity": [[6, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[6, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[6, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function with resampling": [[6, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[6, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function with resampling": [[6, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[0, "exercises"]], "Exercises and Projects": [[6, "exercises-and-projects"]], "Exercises week 34": [[15, null]], "Exercises week 35": [[16, null]], "Exercises week 36": [[17, null]], "Exercises week 37": [[18, null]], "Expectation values": [[23, "expectation-values"]], "Extending to more than one variable": [[28, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[26, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[27, "fixing-the-singularity"], [28, "fixing-the-singularity"]], "Format for electronic delivery of report and programs": [[21, "format-for-electronic-delivery-of-report-and-programs"]], "Frequently used scaling functions": [[27, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[28, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[12, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[27, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[3, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[5, "further-properties-important-for-our-analyses-later"], [27, "further-properties-important-for-our-analyses-later"], [28, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[20, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[26, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[26, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [26, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[26, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[10, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[10, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[1, "gradient-clipping"]], "Gradient Descent Example": [[28, "id1"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], 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"installing-r-c-cython-numba-etc"]], "Instructor information": [[24, "instructor-information"]], "Interpretations and optimizing our parameters": [[26, "interpretations-and-optimizing-our-parameters"], [26, "id2"], [26, "id3"], [27, "interpretations-and-optimizing-our-parameters"], [27, "id1"], [27, "id2"]], "Interpreting the Ridge results": [[27, "interpreting-the-ridge-results"], [28, "interpreting-the-ridge-results"], [28, "id4"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [27, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [19, "introduction"], [20, "introduction"]], "Introduction to numerical projects": [[21, "introduction-to-numerical-projects"]], "Iterative Fitting, Classification and AdaBoost": [[10, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and 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[[26, "other-popular-texts"]], "Other techniques": [[11, "other-techniques"]], "Other types of networks": [[12, "other-types-of-networks"]], "Other ways of visualizing the trees": [[9, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[26, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[26, "overview-of-first-week"]], "Own code for Ordinary Least Squares": [[26, "own-code-for-ordinary-least-squares"], [27, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[26, "pandas-ai"]], "Part a : Ordinary Least Square (OLS) for the Runge function": [[21, "part-a-ordinary-least-square-ols-for-the-runge-function"]], "Part b: Adding Ridge regression for the Runge function": [[21, "part-b-adding-ridge-regression-for-the-runge-function"]], "Part c: Writing your own gradient descent code": [[21, "part-c-writing-your-own-gradient-descent-code"]], "Part d: Including momentum and 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"first": [4, 12, 26, 28], "fit": [0, 10, 15, 16, 26, 28], "fix": [27, 28], "forc": 3, "forest": 10, "form": 18, "format": [21, 26], "formula": 18, "forward": [1, 2, 12], "foster": 26, "fourier": 3, "frank": 6, "freedom": [0, 17, 27], "frequent": 27, "frequentist": [0, 26], "from": [5, 10, 12, 26, 27, 28], "full": 2, "function": [0, 1, 6, 7, 8, 10, 11, 12, 13, 21, 23, 26, 27, 28], "further": [3, 5, 27, 28], "g": 21, "gan": 4, "gaussian": 20, "gd": 13, "gener": [4, 9, 26], "geometr": [11, 28], "gini": 9, "github": 15, "goal": [15, 16, 17, 18], "good": [0, 26], "grade": [24, 26], "gradient": [1, 2, 10, 13, 18, 21, 28], "growth": 2, "h": 21, "ha": 19, "handl": [20, 26], "hessian": [27, 28], "hidden": 2, "hous": [], "how": 16, "hyperparamet": [1, 17], "hyperplan": 8, "i": [0, 1, 26], "id3": 9, "idea": 11, "ideal": 28, "ii": 26, "illustr": 28, "implement": [1, 16, 17, 18], "implic": [5, 27, 28], "import": [5, 20, 26, 27, 28], "improv": 1, "includ": [13, 21], "increment": 11, "index": 9, "inform": 24, "input": 2, "instal": [19, 21, 26], "instructor": 24, "interpret": [5, 11, 26, 27, 28], "introduc": [11, 13, 27], "introduct": [0, 6, 19, 20, 21, 26], "invers": [5, 20], "invert": [27, 28], "iter": 10, "its": 27, "jacobian": 27, "jax": 13, "julia": 26, "jungl": 10, "kera": [1, 3], "kernel": [8, 11], "lab": 28, "lagrangian": 8, "lasso": [5, 6, 21, 27, 28], "last": 27, "later": [5, 27, 28], "layer": [1, 2, 3, 12], "learn": [0, 1, 2, 11, 13, 14, 15, 16, 17, 18, 19, 21, 26, 27, 28], "least": [5, 6, 16, 21, 26, 27, 28], "lectur": [26, 28], "level": 10, "librari": [19, 26], "likelihood": 7, "limit": [1, 13, 23, 28], "linear": [0, 8, 13, 15, 20, 26, 27, 28], "link": [5, 11, 25, 27], "literatur": 21, "logist": [7, 26], "loss": [27, 28], "lu": 20, "machin": [0, 8, 13, 19, 21, 26, 28], "main": [23, 26], "make": [0, 9, 10, 27], "mani": [10, 12], "mass": 26, "materi": [21, 26, 27, 28], "math": [5, 27, 28], "mathemat": [3, 5, 8, 27, 28], "matric": [5, 20, 26], "matrix": [1, 5, 11, 12, 16, 20, 26, 27, 28], "matter": 0, "max": 27, "mean": [0, 27, 28], "meet": [5, 10, 23, 26, 27], "mercer": 8, "method": [6, 9, 10, 13, 21, 26, 28], "midnight": 21, "min": 27, "minim": 26, "ml": 26, "mlp": 12, "mnist": [3, 4], "model": [0, 1, 4, 6, 12, 15, 17, 26], "momentum": [13, 21], "mondai": 28, "moon": [8, 9], "more": [3, 6, 20, 21, 26, 27, 28], "multilay": 12, "multipl": [1, 3, 17], "multipli": 8, "need": [21, 26], "network": [1, 2, 3, 4, 7, 12, 26], "neural": [1, 2, 3, 4, 7, 12, 26], "new": [4, 18], "newton": 28, "non": 8, "normal": [0, 1], "notat": 12, "note": [21, 27, 28], "now": [1, 9, 13, 28], "nuclear": [0, 26], "numba": 26, "number": [0, 2, 23, 27], "numer": [2, 21, 23], "numpi": [20, 26], "object": 3, "obtain": 11, "octob": 21, "od": 2, "off": [6, 21], "ol": [5, 6, 15, 16, 18, 21, 28], "one": [2, 12, 18, 28], "oper": 20, "optim": [1, 8, 13, 18, 19, 26, 27, 28], "order": [13, 18], "ordinari": [5, 6, 16, 21, 26, 27, 28], "organ": [0, 26], "oslo": 25, "other": [4, 9, 11, 12, 20, 21, 26], "our": [0, 4, 5, 11, 13, 21, 26, 27, 28], "outcom": [19, 26], "output": 2, "overarch": [0, 4, 8, 9, 26, 27], "overview": [10, 26], "own": [0, 10, 11, 21, 26, 27], "packag": [20, 26], "panda": [26, 27], "paramet": [26, 27], "paramt": 18, "part": [13, 19, 21, 28], "partial": 2, "pass": 1, "pca": 11, "pdf": 23, "perceptron": 12, "perform": [1, 9], "period": 3, "perspect": 1, "plan": [27, 28], "plethora": 26, "point": 4, "poisson": 2, "polynomi": [3, 16, 18, 28], "popul": 2, "popular": 26, "practic": [13, 24, 26], "pre": [1, 3], "preambl": 21, "predict": 4, "preprocess": 27, "prerequisit": [3, 19, 26], "princip": 11, "principl": 3, "pro": 9, "probabl": [5, 23], "problem": [1, 2, 13, 26, 27, 28], "procedur": [9, 26], "process": [1, 3], "program": [2, 13, 21, 28], "project": [6, 21, 24, 26], "prop": 13, "propag": [1, 12], "properti": [5, 23, 27, 28], "python": [0, 9, 15, 19, 20, 26], "quick": 8, "r": 26, "random": [10, 11, 23], "raphson": 28, "rate": 21, "read": [9, 26, 27], "real": [6, 26], "recommend": [26, 27], "recurr": [4, 12], "reduc": [0, 27], "reduct": 3, "refer": 21, "reformul": 2, "regress": [0, 5, 6, 7, 9, 10, 13, 15, 17, 18, 21, 26, 27, 28], "regular": 1, "relat": [], "relev": [25, 27], "relu": 1, "remark": 3, "remind": [6, 8, 26, 27, 28], "replac": 13, "report": 21, "repositori": 15, "requir": [2, 19], "resampl": [6, 21], "rescal": [6, 27], "residu": [27, 28], "resourc": 2, "result": [27, 28], "revisit": [13, 28], "rewrit": [26, 27], "ridg": [0, 5, 6, 17, 18, 21, 27, 28], "rm": 13, "rule": 12, "rung": 21, "same": 13, "sampl": 11, "scale": [17, 18, 27], "schedul": 26, "schemat": 9, "scheme": 2, "scienc": 26, "scikit": [0, 1, 11, 26, 27, 28], "second": [13, 18], "semest": 24, "sensit": 28, "septemb": 28, "session": 28, "set": [0, 2, 3, 9, 12, 15, 22, 26, 27, 28], "setup": 15, "sgd": 13, "should": 1, "similar": 13, "simpl": [0, 4, 9, 13, 18, 26, 27, 28], "simplest": 18, "singl": 10, "singular": [5, 11, 27, 28], "size": [27, 28], "sklearn": 16, "soft": 8, "softmax": 1, "softwar": [21, 26], "solv": [2, 28], "solver": 13, "some": [13, 20, 27, 28], "specifi": 2, "split": [0, 15, 27], "squar": [0, 5, 6, 10, 16, 21, 26, 27, 28], "standard": [13, 27], "state": 0, "statist": [5, 6, 19, 23, 26], "steepest": [10, 13, 28], "stochast": [13, 21, 23], "strongli": 26, "suggest": 26, "summari": [24, 26], "superposit": 3, "supervis": 1, "support": 8, "svd": [5, 27, 28], "synthet": 18, "systemat": 3, "t": 27, "take": 16, "taken": 26, "teach": 24, "teacher": [24, 26], "technic": 27, "techniqu": [6, 11, 21], "technologi": 19, "tensorflow": [1, 3], "tent": [24, 26], "test": [0, 1, 15, 17, 27], "text": 26, "textbook": [25, 26], "than": 28, "theorem": [5, 8, 11, 12, 23], "theori": 23, "theta": 18, "thi": 26, "tip": 13, "togeth": 12, "tool": [21, 26], "top": 1, "topic": 26, "toward": 11, "trade": [6, 21], "tradeoff": 6, "train": [0, 1, 4, 15, 26, 27], "transform": 3, "tree": [9, 10], "tuesdai": 28, "tune": 1, "two": [3, 8, 19], "type": [2, 4, 12, 26], "uio": 26, "univers": [12, 25], "unsupervis": 14, "up": [0, 2, 9, 12, 15, 26, 27, 28], "updat": 21, "us": [0, 1, 2, 3, 7, 13, 16, 18, 19, 21, 26, 27, 28], "v": 3, "valid": [6, 21], "valu": [5, 11, 23, 27, 28], "variabl": [23, 28], "varianc": [6, 21], "variou": 0, "vector": [8, 12, 16, 20, 26, 27], "versu": 26, "view": [0, 4, 10, 27], "virtual": 15, "visual": [1, 9], "wai": [9, 21], "wave": 2, "we": 26, "wednesdai": 28, "week": [15, 16, 17, 18, 26, 27, 28], "weekli": [], "what": [0, 26, 27, 28], "which": 1, "why": 26, "wisconsin": 7, "write": [4, 11, 21, 28], "x": 27, "xgboost": 10, "yet": 28, "your": [0, 10, 16, 18, 21, 27]}})
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb b/doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb
index d94c3398b..23a5a9d27 100644
--- a/doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "markdown",
- "id": "1b638450",
+ "id": "1b941c35",
"metadata": {
"editable": true
},
@@ -14,7 +14,7 @@
},
{
"cell_type": "markdown",
- "id": "60060788",
+ "id": "dc05b096",
"metadata": {
"editable": true
},
@@ -27,7 +27,7 @@
},
{
"cell_type": "markdown",
- "id": "7cdd88e4",
+ "id": "2cf07405",
"metadata": {
"editable": true
},
@@ -37,7 +37,7 @@
"After having completed these exercises you will have:\n",
"1. Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression\n",
"\n",
- "2. Be able to compare the analytical expressions for OLS and Rudge regression with the gradient descent approach\n",
+ "2. Be able to compare the analytical expressions for OLS and Ridge regression with the gradient descent approach\n",
"\n",
"3. Explore the role of the learning rate in the gradient descent approach and the hyperparameter $\\lambda$ in Ridge regression\n",
"\n",
@@ -46,7 +46,7 @@
},
{
"cell_type": "markdown",
- "id": "0328fa2a",
+ "id": "3c139edb",
"metadata": {
"editable": true
},
@@ -58,7 +58,7 @@
},
{
"cell_type": "markdown",
- "id": "ac760265",
+ "id": "aad4cfac",
"metadata": {
"editable": true
},
@@ -70,7 +70,7 @@
},
{
"cell_type": "markdown",
- "id": "8d4b0753",
+ "id": "6682282f",
"metadata": {
"editable": true
},
@@ -83,7 +83,7 @@
},
{
"cell_type": "markdown",
- "id": "4517d311",
+ "id": "89e2f4c4",
"metadata": {
"editable": true
},
@@ -99,7 +99,7 @@
},
{
"cell_type": "markdown",
- "id": "da1834a9",
+ "id": "b06d4e53",
"metadata": {
"editable": true
},
@@ -120,7 +120,7 @@
{
"cell_type": "code",
"execution_count": 1,
- "id": "590b2fb0",
+ "id": "63796480",
"metadata": {
"collapsed": false,
"editable": true
@@ -140,7 +140,7 @@
},
{
"cell_type": "markdown",
- "id": "24dd92fc",
+ "id": "80748600",
"metadata": {
"editable": true
},
@@ -156,7 +156,7 @@
},
{
"cell_type": "markdown",
- "id": "4ba80e16",
+ "id": "92751e5f",
"metadata": {
"editable": true
},
@@ -168,18 +168,18 @@
},
{
"cell_type": "markdown",
- "id": "be65f56f",
+ "id": "aedfbd7a",
"metadata": {
"editable": true
},
"source": [
- "## Exercise 3, use 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 $\\boldsymbol{\\theta}$"
]
},
{
"cell_type": "code",
"execution_count": 2,
- "id": "c265677e",
+ "id": "5d1288fa",
"metadata": {
"collapsed": false,
"editable": true
@@ -200,7 +200,7 @@
},
{
"cell_type": "markdown",
- "id": "b989efb9",
+ "id": "628f5e89",
"metadata": {
"editable": true
},
@@ -214,7 +214,7 @@
},
{
"cell_type": "markdown",
- "id": "17b08af7",
+ "id": "f115ba4e",
"metadata": {
"editable": true
},
@@ -226,7 +226,7 @@
},
{
"cell_type": "markdown",
- "id": "6acd4708",
+ "id": "a9b5189c",
"metadata": {
"editable": true
},
@@ -238,7 +238,7 @@
},
{
"cell_type": "markdown",
- "id": "50579422",
+ "id": "a3969ff6",
"metadata": {
"editable": true
},
@@ -258,7 +258,7 @@
{
"cell_type": "code",
"execution_count": 3,
- "id": "c57cc917",
+ "id": "34d87303",
"metadata": {
"collapsed": false,
"editable": true
@@ -277,13 +277,14 @@
"cost_history = np.zeros(num_iters)\n",
"\n",
"# Gradient descent loop\n",
- "m = n_samples # number of examples\n",
+ "m = n_samples # number of data points\n",
"for t in range(num_iters):\n",
" # Compute prediction error\n",
" error = X_norm.dot(theta) - y_centered \n",
" # Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring\n",
" cost_OLS = ?\n",
" cost_Ridge = ?\n",
+ " # You could add a history for both methods (optional)\n",
" cost_history[t] = ?\n",
" # Compute gradients for OSL and Ridge\n",
" grad_OLS = ?\n",
@@ -301,7 +302,7 @@
},
{
"cell_type": "markdown",
- "id": "e2654903",
+ "id": "989f70bb",
"metadata": {
"editable": true
},
@@ -313,19 +314,19 @@
},
{
"cell_type": "markdown",
- "id": "7e7e7de6",
+ "id": "370b2dad",
"metadata": {
"editable": true
},
"source": [
"### 4b)\n",
"\n",
- "Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?"
+ "Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\\theta$ values. How would you define a stopping criterion?"
]
},
{
"cell_type": "markdown",
- "id": "75542708",
+ "id": "ef197cd7",
"metadata": {
"editable": true
},
@@ -351,7 +352,7 @@
{
"cell_type": "code",
"execution_count": 4,
- "id": "06077986",
+ "id": "4ccc2f65",
"metadata": {
"collapsed": false,
"editable": true
@@ -380,7 +381,7 @@
},
{
"cell_type": "markdown",
- "id": "86d46505",
+ "id": "00e279ef",
"metadata": {
"editable": true
},
@@ -394,7 +395,7 @@
},
{
"cell_type": "markdown",
- "id": "f1d2f848",
+ "id": "c910b3f4",
"metadata": {
"editable": true
},
@@ -406,7 +407,7 @@
},
{
"cell_type": "markdown",
- "id": "f1c91e84",
+ "id": "89e6e040",
"metadata": {
"editable": true
},
diff --git a/doc/LectureNotes/exercisesweek37.ipynb b/doc/LectureNotes/exercisesweek37.ipynb
index e6ebd7459..25296c4e0 100644
--- a/doc/LectureNotes/exercisesweek37.ipynb
+++ b/doc/LectureNotes/exercisesweek37.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "markdown",
- "id": "1b638450",
+ "id": "1b941c35",
"metadata": {
"editable": true
},
@@ -14,7 +14,7 @@
},
{
"cell_type": "markdown",
- "id": "60060788",
+ "id": "dc05b096",
"metadata": {
"editable": true
},
@@ -27,7 +27,7 @@
},
{
"cell_type": "markdown",
- "id": "7cdd88e4",
+ "id": "2cf07405",
"metadata": {
"editable": true
},
@@ -37,7 +37,7 @@
"After having completed these exercises you will have:\n",
"1. Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression\n",
"\n",
- "2. Be able to compare the analytical expressions for OLS and Rudge regression with the gradient descent approach\n",
+ "2. Be able to compare the analytical expressions for OLS and Ridge regression with the gradient descent approach\n",
"\n",
"3. Explore the role of the learning rate in the gradient descent approach and the hyperparameter $\\lambda$ in Ridge regression\n",
"\n",
@@ -46,7 +46,7 @@
},
{
"cell_type": "markdown",
- "id": "0328fa2a",
+ "id": "3c139edb",
"metadata": {
"editable": true
},
@@ -58,7 +58,7 @@
},
{
"cell_type": "markdown",
- "id": "ac760265",
+ "id": "aad4cfac",
"metadata": {
"editable": true
},
@@ -70,7 +70,7 @@
},
{
"cell_type": "markdown",
- "id": "8d4b0753",
+ "id": "6682282f",
"metadata": {
"editable": true
},
@@ -83,7 +83,7 @@
},
{
"cell_type": "markdown",
- "id": "4517d311",
+ "id": "89e2f4c4",
"metadata": {
"editable": true
},
@@ -99,7 +99,7 @@
},
{
"cell_type": "markdown",
- "id": "da1834a9",
+ "id": "b06d4e53",
"metadata": {
"editable": true
},
@@ -120,7 +120,7 @@
{
"cell_type": "code",
"execution_count": 1,
- "id": "590b2fb0",
+ "id": "63796480",
"metadata": {
"collapsed": false,
"editable": true
@@ -140,7 +140,7 @@
},
{
"cell_type": "markdown",
- "id": "24dd92fc",
+ "id": "80748600",
"metadata": {
"editable": true
},
@@ -156,7 +156,7 @@
},
{
"cell_type": "markdown",
- "id": "4ba80e16",
+ "id": "92751e5f",
"metadata": {
"editable": true
},
@@ -168,18 +168,18 @@
},
{
"cell_type": "markdown",
- "id": "be65f56f",
+ "id": "aedfbd7a",
"metadata": {
"editable": true
},
"source": [
- "## Exercise 3, use 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 $\\boldsymbol{\\theta}$"
]
},
{
"cell_type": "code",
"execution_count": 2,
- "id": "c265677e",
+ "id": "5d1288fa",
"metadata": {
"collapsed": false,
"editable": true
@@ -200,7 +200,7 @@
},
{
"cell_type": "markdown",
- "id": "b989efb9",
+ "id": "628f5e89",
"metadata": {
"editable": true
},
@@ -214,7 +214,7 @@
},
{
"cell_type": "markdown",
- "id": "17b08af7",
+ "id": "f115ba4e",
"metadata": {
"editable": true
},
@@ -226,7 +226,7 @@
},
{
"cell_type": "markdown",
- "id": "6acd4708",
+ "id": "a9b5189c",
"metadata": {
"editable": true
},
@@ -238,7 +238,7 @@
},
{
"cell_type": "markdown",
- "id": "50579422",
+ "id": "a3969ff6",
"metadata": {
"editable": true
},
@@ -258,7 +258,7 @@
{
"cell_type": "code",
"execution_count": 3,
- "id": "c57cc917",
+ "id": "34d87303",
"metadata": {
"collapsed": false,
"editable": true
@@ -277,13 +277,14 @@
"cost_history = np.zeros(num_iters)\n",
"\n",
"# Gradient descent loop\n",
- "m = n_samples # number of examples\n",
+ "m = n_samples # number of data points\n",
"for t in range(num_iters):\n",
" # Compute prediction error\n",
" error = X_norm.dot(theta) - y_centered \n",
" # Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring\n",
" cost_OLS = ?\n",
" cost_Ridge = ?\n",
+ " # You could add a history for both methods (optional)\n",
" cost_history[t] = ?\n",
" # Compute gradients for OSL and Ridge\n",
" grad_OLS = ?\n",
@@ -301,7 +302,7 @@
},
{
"cell_type": "markdown",
- "id": "e2654903",
+ "id": "989f70bb",
"metadata": {
"editable": true
},
@@ -313,19 +314,19 @@
},
{
"cell_type": "markdown",
- "id": "7e7e7de6",
+ "id": "370b2dad",
"metadata": {
"editable": true
},
"source": [
"### 4b)\n",
"\n",
- "Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?"
+ "Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\\theta$ values. How would you define a stopping criterion?"
]
},
{
"cell_type": "markdown",
- "id": "75542708",
+ "id": "ef197cd7",
"metadata": {
"editable": true
},
@@ -351,7 +352,7 @@
{
"cell_type": "code",
"execution_count": 4,
- "id": "06077986",
+ "id": "4ccc2f65",
"metadata": {
"collapsed": false,
"editable": true
@@ -380,7 +381,7 @@
},
{
"cell_type": "markdown",
- "id": "86d46505",
+ "id": "00e279ef",
"metadata": {
"editable": true
},
@@ -394,7 +395,7 @@
},
{
"cell_type": "markdown",
- "id": "f1d2f848",
+ "id": "c910b3f4",
"metadata": {
"editable": true
},
@@ -406,7 +407,7 @@
},
{
"cell_type": "markdown",
- "id": "f1c91e84",
+ "id": "89e6e040",
"metadata": {
"editable": true
},
diff --git a/doc/Programs/Regression/ooreg.py b/doc/Programs/Regression/ooreg.py
index 7121a99d9..26b842305 100644
--- a/doc/Programs/Regression/ooreg.py
+++ b/doc/Programs/Regression/ooreg.py
@@ -30,8 +30,8 @@ class LinearRegression:
return X_bias @ self.weights
class RidgeRegression:
- def __init__(self, alpha=1.0):
- self.alpha = alpha
+ def __init__(self, theta=1.0):
+ self.theta = theta
self.weights = None
def fit(self, X, y):
@@ -39,15 +39,15 @@ class RidgeRegression:
n = X_bias.shape[1]
I = np.eye(n)
I[0, 0] = 0
- self.weights = np.linalg.inv(X_bias.T @ X_bias + self.alpha * I) @ X_bias.T @ y
+ self.weights = np.linalg.pinv(X_bias.T @ X_bias + self.theta * I) @ X_bias.T @ y
def predict(self, X):
X_bias = np.c_[np.ones((X.shape[0], 1)), X]
return X_bias @ self.weights
class LassoRegression:
- def __init__(self, alpha=1.0, max_iter=1000, tol=1e-4):
- self.alpha = alpha
+ def __init__(self, theta=1.0, max_iter=1000, tol=1e-4):
+ self.theta = theta
self.max_iter = max_iter
self.tol = tol
self.weights = None
@@ -65,10 +65,10 @@ class LassoRegression:
if j == 0:
self.weights[j] = rho / np.sum(X_bias[:, j] ** 2)
else:
- if rho < -self.alpha / 2:
- self.weights[j] = (rho + self.alpha / 2) / np.sum(X_bias[:, j] ** 2)
- elif rho > self.alpha / 2:
- self.weights[j] = (rho - self.alpha / 2) / np.sum(X_bias[:, j] ** 2)
+ if rho < -self.theta / 2:
+ self.weights[j] = (rho + self.theta / 2) / np.sum(X_bias[:, j] ** 2)
+ elif rho > self.theta / 2:
+ self.weights[j] = (rho - self.theta / 2) / np.sum(X_bias[:, j] ** 2)
else:
self.weights[j] = 0
if np.linalg.norm(self.weights - weights_old, ord=1) < self.tol:
@@ -79,11 +79,11 @@ class LassoRegression:
return X_bias @ self.weights
class KernelRidgeRegression:
- def __init__(self, alpha=1.0, gamma=0.1):
- self.alpha = alpha
+ def __init__(self, theta=1.0, gamma=0.1):
+ self.theta = theta
self.gamma = gamma
self.X_train = None
- self.alpha_vec = None
+ self.theta_vec = None
def _rbf_kernel(self, X1, X2):
dists = np.sum((X1[:, np.newaxis] - X2[np.newaxis, :]) ** 2, axis=2)
@@ -93,11 +93,11 @@ class KernelRidgeRegression:
self.X_train = X
K = self._rbf_kernel(X, X)
n = K.shape[0]
- self.alpha_vec = np.linalg.inv(K + self.alpha * np.eye(n)) @ y
+ self.theta_vec = np.linalg.pinv(K + self.theta * np.eye(n)) @ y
def predict(self, X):
K = self._rbf_kernel(X, self.X_train)
- return K @ self.alpha_vec
+ return K @ self.theta_vec
if __name__ == "__main__":
np.random.seed(42)
@@ -106,9 +106,9 @@ if __name__ == "__main__":
models = {
"linear": LinearRegression(),
- "ridge": RidgeRegression(alpha=1.0),
- "lasso": LassoRegression(alpha=0.1),
- "kernel_ridge": KernelRidgeRegression(alpha=1.0, gamma=5.0)
+ "ridge": RidgeRegression(theta=1.0),
+ "lasso": LassoRegression(theta=0.1),
+ "kernel_ridge": KernelRidgeRegression(theta=1.0, gamma=5.0)
}
for name, model in models.items():
diff --git a/doc/Programs/Regression/predictions_kernel_ridge.csv b/doc/Programs/Regression/predictions_kernel_ridge.csv
new file mode 100644
index 000000000..14a72e49b
--- /dev/null
+++ b/doc/Programs/Regression/predictions_kernel_ridge.csv
@@ -0,0 +1,101 @@
+X,True Y,Predicted Y
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diff --git a/doc/Programs/Regression/predictions_lasso.csv b/doc/Programs/Regression/predictions_lasso.csv
new file mode 100644
index 000000000..d5dfb46cb
--- /dev/null
+++ b/doc/Programs/Regression/predictions_lasso.csv
@@ -0,0 +1,101 @@
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diff --git a/doc/Programs/Regression/predictions_linear.csv b/doc/Programs/Regression/predictions_linear.csv
new file mode 100644
index 000000000..2e12b8a27
--- /dev/null
+++ b/doc/Programs/Regression/predictions_linear.csv
@@ -0,0 +1,101 @@
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diff --git a/doc/Programs/Regression/predictions_ridge.csv b/doc/Programs/Regression/predictions_ridge.csv
new file mode 100644
index 000000000..1fdf1125d
--- /dev/null
+++ b/doc/Programs/Regression/predictions_ridge.csv
@@ -0,0 +1,101 @@
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diff --git a/doc/pub/week36/ipynb/week36.ipynb b/doc/pub/week36/ipynb/week36.ipynb
index 326b2fa5d..468fda883 100644
--- a/doc/pub/week36/ipynb/week36.ipynb
+++ b/doc/pub/week36/ipynb/week36.ipynb
@@ -3296,10 +3296,11 @@
"cell_type": "markdown",
"id": "d6b1ae12",
"metadata": {
- "editable": true
+ "editable": true,
+ "jp-MarkdownHeadingCollapsed": true
},
"source": [
- "## Material for lab sessions sessions Tuesday and Wednesday\n",
+ "## Material for lab sessions Tuesday and Wednesday\n",
"\n",
"The material here contains a summary of the lecture on Monday and discussion of SVD, Ridge and Lasso regression with examples"
]
@@ -5129,7 +5130,7 @@
},
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": 3,
"id": "92ca291e",
"metadata": {
"collapsed": false,
diff --git a/doc/src/week35/programs/lasso.py b/doc/src/week35/programs/lasso.py
new file mode 100644
index 000000000..23e03f53a
--- /dev/null
+++ b/doc/src/week35/programs/lasso.py
@@ -0,0 +1,139 @@
+import numpy as np
+
+# Seed for reproducibility
+np.random.seed(0)
+
+# Dimensions of the synthetic dataset
+N = 100 # number of samples (observations)
+p = 10 # number of features
+
+# True sparse coefficients (only a few non-zero)
+w_true = np.array([5, -3, 0, 0, 2, 0, 0, 0, 0, 0], dtype=float)
+# For example, feature 0 has coefficient 5, feature 1 has -3, feature 4 has 2, rest are 0.
+
+# Generate feature matrix X from a normal distribution
+X = np.random.randn(N, p)
+
+# Generate target values: linear combination of X with w_true + noise
+noise = np.random.randn(N) * 1.0 # noise with standard deviation 1.0
+y = X.dot(w_true) + noise
+
+
+# Standardize features (zero mean, unit variance for each column)
+X_mean = X.mean(axis=0)
+X_std = X.std(axis=0)
+X_std[X_std == 0] = 1.0 # avoid division by zero if any constant feature
+X_norm = (X - X_mean) / X_std
+
+# Center the target to zero mean
+y_mean = y.mean()
+y_centered = y - y_mean
+
+
+def soft_threshold(rho, lam):
+ """Soft thresholding operator: S(rho, lam) = sign(rho)*max(|rho|-lam, 0)."""
+ if rho < -lam:
+ return rho + lam
+ elif rho > lam:
+ return rho - lam
+ else:
+ return 0.0
+
+def lasso_coordinate_descent(X, y, alpha, max_iter=1000, tol=1e-6):
+ """
+ Perform LASSO regression using coordinate descent.
+ X : array of shape (n_samples, n_features), assumed to be standardized.
+ y : array of shape (n_samples,), assumed centered.
+ alpha : regularization strength (L1 penalty coefficient).
+ max_iter : maximum number of coordinate descent iterations (full cycles).
+ tol : tolerance for convergence (stop if max coef change < tol).
+ """
+ n_samples, n_features = X.shape
+ w = np.zeros(n_features) # initialize weights to zero
+ for it in range(max_iter):
+ w_old = w.copy()
+ # Loop over each feature coordinate
+ for j in range(n_features):
+ # Compute rho_j = x_j^T (y - X w + w_j * x_j)
+ # (This is the contribution of feature j to the residual)
+ X_j = X[:, j]
+ # temporarily exclude feature j's effect
+ residual = y - X.dot(w) + w[j] * X_j
+ rho_j = X_j.dot(residual)
+ # Soft thresholding update for w_j
+ w[j] = soft_threshold(rho_j, alpha) / (X_j.dot(X_j))
+ # Check convergence: if all updates are very small, break
+ if np.max(np.abs(w - w_old)) < tol:
+ break
+ return w
+
+
+alpha = 50.0 # regularization strength
+w_learned = lasso_coordinate_descent(X_norm, y_centered, alpha)
+
+print("True coefficients:", w_true)
+print("Learned coefficients:", w_learned)
+
+
+# Plot y vs a relevant feature (0) and an irrelevant feature (2)
+fig, axes = plt.subplots(1, 2, figsize=(10, 4))
+axes[0].scatter(X[:, 0], y, color='blue', alpha=0.6)
+axes[0].set_title("Feature 0 (Relevant) vs Target")
+axes[0].set_xlabel("Feature 0 values")
+axes[0].set_ylabel("Target (y)")
+axes[1].scatter(X[:, 2], y, color='red', alpha=0.6)
+axes[1].set_title("Feature 2 (Irrelevant) vs Target")
+axes[1].set_xlabel("Feature 2 values")
+axes[1].set_ylabel("Target (y)")
+plt.tight_layout()
+plt.show()
+
+# Track cost history during coordinate descent for plotting
+def lasso_with_cost_history(X, y, alpha, max_iter=1000):
+ n_samples, n_features = X.shape
+ w = np.zeros(n_features)
+ cost_history = []
+ # initial cost
+ cost_history.append(0.5 * np.sum((y - X.dot(w))**2) + alpha * np.sum(np.abs(w)))
+ for it in range(max_iter):
+ w_old = w.copy()
+ for j in range(n_features):
+ X_j = X[:, j]
+ residual = y - X.dot(w) + w[j] * X_j
+ rho_j = X_j.dot(residual)
+ w[j] = soft_threshold(rho_j, alpha) / (X_j.dot(X_j))
+ # compute cost after this iteration
+ cost = 0.5 * np.sum((y - X.dot(w))**2) + alpha * np.sum(np.abs(w))
+ cost_history.append(cost)
+ if np.max(np.abs(w - w_old)) < 1e-6:
+ break
+ return w, cost_history
+
+# Run coordinate descent and get cost history
+w_fit, cost_history = lasso_with_cost_history(X_norm, y_centered, alpha=50.0)
+
+# Plot cost vs iteration
+plt.figure(figsize=(6,4))
+plt.plot(cost_history, marker='o', color='purple')
+plt.title("LASSO Cost Decrease over Iterations")
+plt.xlabel("Iteration")
+plt.ylabel("Cost function value")
+plt.grid(True)
+plt.show()
+
+
+# Compare true vs learned coefficients
+import numpy as np
+import matplotlib.pyplot as plt
+
+indices = np.arange(p)
+width = 0.4
+plt.figure(figsize=(6,4))
+plt.bar(indices - width/2, w_true, width=width, label='True Coefficient')
+plt.bar(indices + width/2, w_fit, width=width, label='Learned Coefficient')
+plt.xlabel("Feature index")
+plt.ylabel("Coefficient value")
+plt.title("True vs Learned Coefficients")
+plt.legend()
+plt.show()
+
diff --git a/doc/src/week37/exercisesweek37.do.txt b/doc/src/week37/exercisesweek37.do.txt
index e86f55d0c..94c1d068a 100644
--- a/doc/src/week37/exercisesweek37.do.txt
+++ b/doc/src/week37/exercisesweek37.do.txt
@@ -7,7 +7,7 @@ DATE: September 8-12, 2025
After having completed these exercises you will have:
o Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression
-o Be able to compare the analytical expressions for OLS and Rudge regression with the gradient descent approach
+o Be able to compare the analytical expressions for OLS and Ridge regression with the gradient descent approach
o Explore the role of the learning rate in the gradient descent approach and the hyperparameter $\lambda$ in Ridge regression
o Scale the data properly
@@ -72,7 +72,7 @@ same scale).
Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.
-===== Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\bm{\theta}$ =====
+===== Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters $\bm{\theta}$ =====
!bc pycod
# Set regularization parameter, either a single value or a vector of values
@@ -124,13 +124,14 @@ theta = np.zeros(n_features)
cost_history = np.zeros(num_iters)
# Gradient descent loop
-m = n_samples # number of examples
+m = n_samples # number of data points
for t in range(num_iters):
# Compute prediction error
error = X_norm.dot(theta) - y_centered
# Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring
cost_OLS = ?
cost_Ridge = ?
+ # You could add a history for both methods (optional)
cost_history[t] = ?
# Compute gradients for OSL and Ridge
grad_OLS = ?
@@ -150,7 +151,7 @@ print("Gradient Descent Ridge coefficients:", theta_gdRidge)
Discuss the results as function of the learning rate parameters and the number of iterations.
=== 4b) ===
-Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?
+Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\theta$ values. How would you define a stopping criterion?
===== Exercise 5, Ridge regression and a new Synthetic Dataset =====