Machine learning

CS229 Lecture Notes

Andrew Ng and Tengyu Ma

June 11, 2023

,Contents

I Supervised learning 5
1 Linear regression 8
1.1 LMS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 The normal equations . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Matrix derivatives . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Least squares revisited . . . . . . . . . . . . . . . . . . 14
1.3 Probabilistic interpretation . . . . . . . . . . . . . . . . . . . . 15
1.4 Locally weighted linear regression (optional reading) . . . . . . 17

2 Classification and logistic regression 20
2.1 Logistic regression . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Digression: the perceptron learning algorithm . . . . . . . . . 23
2.3 Multi-class classification . . . . . . . . . . . . . . . . . . . . . 24
2.4 Another algorithm for maximizing `(θ) . . . . . . . . . . . . . 27

3 Generalized linear models 29
3.1 The exponential family . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Constructing GLMs . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Ordinary least squares . . . . . . . . . . . . . . . . . . 32
3.2.2 Logistic regression . . . . . . . . . . . . . . . . . . . . 33

4 Generative learning algorithms 34
4.1 Gaussian discriminant analysis . . . . . . . . . . . . . . . . . . 35
4.1.1 The multivariate normal distribution . . . . . . . . . . 35
4.1.2 The Gaussian discriminant analysis model . . . . . . . 38
4.1.3 Discussion: GDA and logistic regression . . . . . . . . 40
4.2 Naive bayes (Option Reading) . . . . . . . . . . . . . . . . . . 41
4.2.1 Laplace smoothing . . . . . . . . . . . . . . . . . . . . 44
4.2.2 Event models for text classification . . . . . . . . . . . 46



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5 Kernel methods 48
5.1 Feature maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 LMS (least mean squares) with features . . . . . . . . . . . . . 49
5.3 LMS with the kernel trick . . . . . . . . . . . . . . . . . . . . 49
5.4 Properties of kernels . . . . . . . . . . . . . . . . . . . . . . . 53

6 Support vector machines 59
6.1 Margins: intuition . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Notation (option reading) . . . . . . . . . . . . . . . . . . . . 61
6.3 Functional and geometric margins (option reading) . . . . . . 61
6.4 The optimal margin classifier (option reading) . . . . . . . . . 63
6.5 Lagrange duality (optional reading) . . . . . . . . . . . . . . . 65
6.6 Optimal margin classifiers: the dual form (option reading) . . 68
6.7 Regularization and the non-separable case (optional reading) . 72
6.8 The SMO algorithm (optional reading) . . . . . . . . . . . . . 73
6.8.1 Coordinate ascent . . . . . . . . . . . . . . . . . . . . . 74
6.8.2 SMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


II Deep learning 79
7 Deep learning 80
7.1 Supervised learning with non-linear models . . . . . . . . . . . 80
7.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3 Modules in Modern Neural Networks . . . . . . . . . . . . . . 92
7.4 Backpropagation . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4.1 Preliminaries on partial derivatives . . . . . . . . . . . 99
7.4.2 General strategy of backpropagation . . . . . . . . . . 102
7.4.3 Backward functions for basic modules . . . . . . . . . . 105
7.4.4 Back-propagation for MLPs . . . . . . . . . . . . . . . 107
7.5 Vectorization over training examples . . . . . . . . . . . . . . 109


III Generalization and regularization 112
8 Generalization 113
8.1 Bias-variance tradeoff . . . . . . . . . . . . . . . . . . . . . . . 115
8.1.1 A mathematical decomposition (for regression) . . . . . 120
8.2 The double descent phenomenon . . . . . . . . . . . . . . . . . 121
8.3 Sample complexity bounds (optional readings) . . . . . . . . . 126

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8.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 126
8.3.2 The case of finite H . . . . . . . . . . . . . . . . . . . . 128
8.3.3 The case of infinite H . . . . . . . . . . . . . . . . . . 131

9 Regularization and model selection 135
9.1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.2 Implicit regularization effect (optional reading) . . . . . . . . . 137
9.3 Model selection via cross validation . . . . . . . . . . . . . . . 139
9.4 Bayesian statistics and regularization . . . . . . . . . . . . . . 142


IV Unsupervised learning 144
10 Clustering and the k-means algorithm 145

11 EM algorithms 148
11.1 EM for mixture of Gaussians . . . . . . . . . . . . . . . . . . . 148
11.2 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . 151
11.3 General EM algorithms . . . . . . . . . . . . . . . . . . . . . . 152
11.3.1 Other interpretation of ELBO . . . . . . . . . . . . . . 158
11.4 Mixture of Gaussians revisited . . . . . . . . . . . . . . . . . . 158
11.5 Variational inference and variational auto-encoder (optional
reading) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

12 Principal components analysis 165

13 Independent components analysis 171
13.1 ICA ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . 172
13.2 Densities and linear transformations . . . . . . . . . . . . . . . 173
13.3 ICA algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

14 Self-supervised learning and foundation models 177
14.1 Pretraining and adaptation . . . . . . . . . . . . . . . . . . . . 177
14.2 Pretraining methods in computer vision . . . . . . . . . . . . . 179
14.3 Pretrained large language models . . . . . . . . . . . . . . . . 181
14.3.1 Open up the blackbox of Transformers . . . . . . . . . 183
14.3.2 Zero-shot learning and in-context learning . . . . . . . 186