Deep Learning
Ian Goodfellow
Yoshua Bengio
Aaron Courville
Contents
Website vii
Acknowledgments viii
Notation xi
1 Introduction 1
1.1 Who Should Read This Book? . . . . . . . . . . . . . . . . . . . . 8
1.2 Historical Trends in Deep Learning . . . . . . . . . . . . . . . . . 11
I Applied Math and Machine Learning Basics 29
2 Linear Algebra 31
2.1 Scalars, Vectors, Matrices and Tensors . . . . . . . . . . . . . . . 31
2.2 Multiplying Matrices and Vectors . . . . . . . . . . . . . . . . . . 34
2.3 Identity and Inverse Matrices . . . . . . . . . . . . . . . . . . . . 36
2.4 Linear Dependence and Span . . . . . . . . . . . . . . . . . . . . 37
2.5 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Special Kinds of Matrices and Vectors . . . . . . . . . . . . . . . 40
2.7 Eigendecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.8 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . 44
2.9 The Moore-Penrose Pseudoinverse . . . . . . . . . . . . . . . . . . 45
2.10 The Trace Operator . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.11 The Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.12 Example: Principal Components Analysis . . . . . . . . . . . . . 48
3 Probability and Information Theory 53
3.1 Why Probability? . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
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3.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . 56
3.4 Marginal Probability . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . 59
3.6 The Chain Rule of Conditional Probabilities . . . . . . . . . . . . 59
3.7 Independence and Conditional Independence . . . . . . . . . . . . 60
3.8 Expectation, Variance and Covariance . . . . . . . . . . . . . . . 60
3.9 Common Probability Distributions . . . . . . . . . . . . . . . . . 62
3.10 Useful Properties of Common Functions . . . . . . . . . . . . . . 67
3.11 Bayes’ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.12 Technical Details of Continuous Variables . . . . . . . . . . . . . 71
3.13 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.14 Structured Probabilistic Models . . . . . . . . . . . . . . . . . . . 75
4 Numerical Computation 80
4.1 Overflow and Underflow . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Poor Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Gradient-Based Optimization . . . . . . . . . . . . . . . . . . . . 82
4.4 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . 93
4.5 Example: Linear Least Squares . . . . . . . . . . . . . . . . . . . 96
5 Machine Learning Basics 98
5.1 Learning Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Capacity, Overfitting and Underfitting . . . . . . . . . . . . . . . 110
5.3 Hyperparameters and Validation Sets . . . . . . . . . . . . . . . . 120
5.4 Estimators, Bias and Variance . . . . . . . . . . . . . . . . . . . . 122
5.5 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . 131
5.6 Bayesian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.7 Supervised Learning Algorithms . . . . . . . . . . . . . . . . . . . 139
5.8 Unsupervised Learning Algorithms . . . . . . . . . . . . . . . . . 145
5.9 Stochastic Gradient Descent . . . . . . . . . . . . . . . . . . . . . 150
5.10 Building a Machine Learning Algorithm . . . . . . . . . . . . . . 152
5.11 Challenges Motivating Deep Learning . . . . . . . . . . . . . . . . 154
II Deep Networks: Modern Practices 165
6 Deep Feedforward Networks 167
6.1 Example: Learning XOR . . . . . . . . . . . . . . . . . . . . . . . 170
6.2 Gradient-Based Learning . . . . . . . . . . . . . . . . . . . . . . . 176
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6.3 Hidden Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.4 Architecture Design . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.5 Back-Propagation and Other Differentiation Algorithms . . . . . 203
6.6 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7 Regularization for Deep Learning 228
7.1 Parameter Norm Penalties . . . . . . . . . . . . . . . . . . . . . . 230
7.2 Norm Penalties as Constrained Optimization . . . . . . . . . . . . 237
7.3 Regularization and Under-Constrained Problems . . . . . . . . . 239
7.4 Dataset Augmentation . . . . . . . . . . . . . . . . . . . . . . . . 240
7.5 Noise Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.6 Semi-Supervised Learning . . . . . . . . . . . . . . . . . . . . . . 244
7.7 Multi-Task Learning . . . . . . . . . . . . . . . . . . . . . . . . . 245
7.8 Early Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.9 Parameter Tying and Parameter Sharing . . . . . . . . . . . . . . 251
7.10 Sparse Representations . . . . . . . . . . . . . . . . . . . . . . . . 253
7.11 Bagging and Other Ensemble Methods . . . . . . . . . . . . . . . 255
7.12 Dropout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
7.13 Adversarial Training . . . . . . . . . . . . . . . . . . . . . . . . . 267
7.14 Tangent Distance, Tangent Prop, and Manifold Tangent Classifier 268
8 Optimization for Training Deep Models 274
8.1 How Learning Differs from Pure Optimization . . . . . . . . . . . 275
8.2 Challenges in Neural Network Optimization . . . . . . . . . . . . 282
8.3 Basic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
8.4 Parameter Initialization Strategies . . . . . . . . . . . . . . . . . 301
8.5 Algorithms with Adaptive Learning Rates . . . . . . . . . . . . . 306
8.6 Approximate Second-Order Methods . . . . . . . . . . . . . . . . 310
8.7 Optimization Strategies and Meta-Algorithms . . . . . . . . . . . 318
9 Convolutional Networks 331
9.1 The Convolution Operation . . . . . . . . . . . . . . . . . . . . . 332
9.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
9.3 Pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
9.4 Convolution and Pooling as an Infinitely Strong Prior . . . . . . . 346
9.5 Variants of the Basic Convolution Function . . . . . . . . . . . . 348
9.6 Structured Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . 359
9.7 Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
9.8 Efficient Convolution Algorithms . . . . . . . . . . . . . . . . . . 363
9.9 Random or Unsupervised Features . . . . . . . . . . . . . . . . . 364
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9.10 The Neuroscientific Basis for Convolutional Networks . . . . . . . 365
9.11 Convolutional Networks and the History of Deep Learning . . . . 372
10 Sequence Modeling: Recurrent and Recursive Nets 374
10.1 Unfolding Computational Graphs . . . . . . . . . . . . . . . . . . 376
10.2 Recurrent Neural Networks . . . . . . . . . . . . . . . . . . . . . 379
10.3 Bidirectional RNNs . . . . . . . . . . . . . . . . . . . . . . . . . . 396
10.4 Encoder-Decoder Sequence-to-Sequence Architectures . . . . . . . 397
10.5 Deep Recurrent Networks . . . . . . . . . . . . . . . . . . . . . . 399
10.6 Recursive Neural Networks . . . . . . . . . . . . . . . . . . . . . . 401
10.7 The Challenge of Long-Term Dependencies . . . . . . . . . . . . . 403
10.8 Echo State Networks . . . . . . . . . . . . . . . . . . . . . . . . . 406
10.9 Leaky Units and Other Strategies for Multiple Time Scales . . . . 409
10.10 The Long Short-Term Memory and Other Gated RNNs . . . . . . 411
10.11 Optimization for Long-Term Dependencies . . . . . . . . . . . . . 415
10.12 Explicit Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
11 Practical methodology 424
11.1 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 425
11.2 Default Baseline Models . . . . . . . . . . . . . . . . . . . . . . . 428
11.3 Determining Whether to Gather More Data . . . . . . . . . . . . 429
11.4 Selecting Hyperparameters . . . . . . . . . . . . . . . . . . . . . . 430
11.5 Debugging Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 439
11.6 Example: Multi-Digit Number Recognition . . . . . . . . . . . . . 443
12 Applications 446
12.1 Large Scale Deep Learning . . . . . . . . . . . . . . . . . . . . . . 446
12.2 Computer Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
12.3 Speech Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . 461
12.4 Natural Language Processing . . . . . . . . . . . . . . . . . . . . 464
12.5 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 480
III Deep Learning Research 489
13 Linear Factor Models 492
13.1 Probabilistic PCA and Factor Analysis . . . . . . . . . . . . . . . 493
13.2 Independent Component Analysis (ICA) . . . . . . . . . . . . . . 494
13.3 Slow Feature Analysis . . . . . . . . . . . . . . . . . . . . . . . . 496
13.4 Sparse Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
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13.5 Manifold Interpretation of PCA . . . . . . . . . . . . . . . . . . . 502
14 Autoencoders 505
14.1 Undercomplete Autoencoders . . . . . . . . . . . . . . . . . . . . 506
14.2 Regularized Autoencoders . . . . . . . . . . . . . . . . . . . . . . 507
14.3 Representational Power, Layer Size and Depth . . . . . . . . . . . 511
14.4 Stochastic Encoders and Decoders . . . . . . . . . . . . . . . . . . 512
14.5 Denoising Autoencoders . . . . . . . . . . . . . . . . . . . . . . . 513
14.6 Learning Manifolds with Autoencoders . . . . . . . . . . . . . . . 518
14.7 Contractive Autoencoders . . . . . . . . . . . . . . . . . . . . . . 524
14.8 Predictive Sparse Decomposition . . . . . . . . . . . . . . . . . . 526
14.9 Applications of Autoencoders . . . . . . . . . . . . . . . . . . . . 527
15 Representation Learning 529
15.1 Greedy Layer-Wise Unsupervised Pretraining . . . . . . . . . . . 531
15.2 Transfer Learning and Domain Adaptation . . . . . . . . . . . . . 539
15.3 Semi-Supervised Disentangling of Causal Factors . . . . . . . . . 544
15.4 Distributed Representation . . . . . . . . . . . . . . . . . . . . . . 549
15.5 Exponential Gains from Depth . . . . . . . . . . . . . . . . . . . 556
15.6 Providing Clues to Discover Underlying Causes . . . . . . . . . . 557
16 Structured Probabilistic Models for Deep Learning 561
16.1 The Challenge of Unstructured Modeling . . . . . . . . . . . . . . 562
16.2 Using Graphs to Describe Model Structure . . . . . . . . . . . . . 566
16.3 Sampling from Graphical Models . . . . . . . . . . . . . . . . . . 583
16.4 Advantages of Structured Modeling . . . . . . . . . . . . . . . . . 584
16.5 Learning about Dependencies . . . . . . . . . . . . . . . . . . . . 585
16.6 Inference and Approximate Inference . . . . . . . . . . . . . . . . 586
16.7 The Deep Learning Approach to Structured Probabilistic Models 587
17 Monte Carlo Methods 593
17.1 Sampling and Monte Carlo Methods . . . . . . . . . . . . . . . . 593
17.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 595
17.3 Markov Chain Monte Carlo Methods . . . . . . . . . . . . . . . . 598
17.4 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
17.5 The Challenge of Mixing between Separated Modes . . . . . . . . 602
18 Confronting the Partition Function 608
18.1 The Log-Likelihood Gradient . . . . . . . . . . . . . . . . . . . . 609
18.2 Stochastic Maximum Likelihood and Contrastive Divergence . . . 610
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18.3 Pseudolikelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
18.4 Score Matching and Ratio Matching . . . . . . . . . . . . . . . . 620
18.5 Denoising Score Matching . . . . . . . . . . . . . . . . . . . . . . 622
18.6 Noise-Contrastive Estimation . . . . . . . . . . . . . . . . . . . . 623
18.7 Estimating the Partition Function . . . . . . . . . . . . . . . . . . 626
19 Approximate inference 634
19.1 Inference as Optimization . . . . . . . . . . . . . . . . . . . . . . 636
19.2 Expectation Maximization . . . . . . . . . . . . . . . . . . . . . . 637
19.3 MAP Inference and Sparse Coding . . . . . . . . . . . . . . . . . 638
19.4 Variational Inference and Learning . . . . . . . . . . . . . . . . . 641
19.5 Learned Approximate Inference . . . . . . . . . . . . . . . . . . . 653
20 Deep Generative Models 656
20.1 Boltzmann Machines . . . . . . . . . . . . . . . . . . . . . . . . . 656
20.2 Restricted Boltzmann Machines . . . . . . . . . . . . . . . . . . . 658
20.3 Deep Belief Networks . . . . . . . . . . . . . . . . . . . . . . . . . 662
20.4 Deep Boltzmann Machines . . . . . . . . . . . . . . . . . . . . . . 665
20.5 Boltzmann Machines for Real-Valued Data . . . . . . . . . . . . . 678
20.6 Convolutional Boltzmann Machines . . . . . . . . . . . . . . . . . 685
20.7 Boltzmann Machines for Structured or Sequential Outputs . . . . 687
20.8 Other Boltzmann Machines . . . . . . . . . . . . . . . . . . . . . 688
20.9 Back-Propagation through Random Operations . . . . . . . . . . 689
20.10 Directed Generative Nets . . . . . . . . . . . . . . . . . . . . . . . 694
20.11 Drawing Samples from Autoencoders . . . . . . . . . . . . . . . . 712
20.12 Generative Stochastic Networks . . . . . . . . . . . . . . . . . . . 716
20.13 Other Generation Schemes . . . . . . . . . . . . . . . . . . . . . . 717
20.14 Evaluating Generative Models . . . . . . . . . . . . . . . . . . . . 719
20.15 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721
Bibliography 723
Index 780
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Website
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variety of supplementary material, including exercises, lecture slides, corrections of
mistakes, and other resources that should be useful to both readers and instructors.
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