Learning Theory

Learning theory provides the mathematical foundations for how machines can generalize from data. This page covers key concepts from statistical learning to modern deep learning.

Statistical Learning Theory

Generalization and Overfitting

• Training vs Test Error: The generalization gap
• Bias-Variance Tradeoff: Model complexity and performance
• Cross-Validation: Estimating generalization error
• Regularization: L1, L2, dropout, early stopping

PAC Learning

Probably Approximately Correct (PAC) Learning:

• Sample Complexity: How much data is needed?
• VC Dimension: Measuring model expressiveness
• Realizability: When the true function is in the hypothesis class
• Agnostic Learning: Learning without realizability assumptions

Risk Minimization

• Empirical Risk Minimization (ERM): Minimizing training loss
• Structural Risk Minimization (SRM): Trading fit for complexity
• Consistency: Convergence as data grows

Supervised Learning

Classification

• Linear Classifiers: Perceptrons, logistic regression, SVM
• Decision Trees: Splitting criteria, pruning
• Ensemble Methods: Bagging, boosting, random forests
• Neural Networks: Universal approximation theorem

Regression

• Linear Regression: Least squares, gradient descent
• Non-linear Regression: Polynomial features, kernel methods
• Gaussian Processes: Probabilistic regression

Unsupervised Learning

Clustering

• K-means: Centroid-based clustering
• Hierarchical Clustering: Dendrograms, agglomerative vs divisive
• DBSCAN: Density-based clustering
• Gaussian Mixture Models: Probabilistic clustering

Dimensionality Reduction

• PCA: Principal component analysis
• t-SNE: Visualization of high-dimensional data
• Autoencoders: Neural network-based compression
• Manifold Learning: Discovering low-dimensional structure

Deep Learning Fundamentals

Neural Network Basics

• Feedforward Networks: Fully connected layers
• Activation Functions: ReLU, sigmoid, tanh, softmax
• Backpropagation: Chain rule for gradient computation
• Optimization: SGD, momentum, Adam

Convolutional Neural Networks

• Convolutional Layers: Local connectivity, weight sharing
• Pooling: Max pooling, average pooling
• Architectures: LeNet, AlexNet, VGG, ResNet
• Applications: Computer vision, pattern recognition

Recurrent Neural Networks

• Sequential Data: Time series, language
• LSTM and GRU: Gating mechanisms for long-term dependencies
• Bidirectional RNNs: Forward and backward context
• Applications: Language modeling, sequence-to-sequence

Transformers

• Self-Attention: Relating positions in sequences
• Multi-Head Attention: Parallel attention mechanisms
• Positional Encoding: Sequence order information
• Applications: BERT, GPT, language understanding

Reinforcement Learning Basics

Markov Decision Processes

• States, Actions, Rewards: MDP formulation
• Policy: Mapping from states to actions
• Value Functions: Expected return from states/actions
• Bellman Equations: Recursive value relationships

Learning Methods

• Q-Learning: Off-policy TD learning
• Policy Gradient: Directly optimizing policies
• Actor-Critic: Combining value and policy methods
• Deep RL: DQN, A3C, PPO

Meta-Learning

Learning to Learn

• Few-Shot Learning: Generalizing from few examples
• Transfer Learning: Using knowledge from related tasks
• Neural Architecture Search: Automated model design
• Hyperparameter Optimization: Bayesian optimization, grid search

Information-Theoretic Learning

Mutual Information

• Maximizing I(X;Y): Learning useful representations
• InfoMax Principle: Self-supervised learning
• Variational Information Maximization

Compression and Generalization

• Minimum Description Length: Model selection via compression
• Rate-Distortion Theory: Information and reconstruction tradeoff

Online Learning

Sequential Decision Making

• Regret Bounds: Performance relative to best fixed strategy
• Multi-Armed Bandits: Exploration vs exploitation
• Contextual Bandits: Incorporating feature information

Curriculum Learning

Structured Learning Paths

• Easy-to-Hard: Training on increasingly difficult examples
• Self-Paced Learning: Letting the model choose examples
• Teacher-Student: Distillation and knowledge transfer

Practical Considerations

Data Augmentation

• Image Augmentation: Rotation, cropping, color jittering
• Text Augmentation: Back-translation, paraphrasing
• Synthetic Data: Simulation, generative models

Batch Normalization

• Internal Covariate Shift: Stabilizing layer inputs
• Training Acceleration: Faster convergence
• Regularization Effect: Implicit regularization

Recommended Resources

• Understanding Machine Learning - Shalev-Shwartz and Ben-David
• Deep Learning - Goodfellow, Bengio, and Courville
• Pattern Recognition and Machine Learning - Christopher Bishop
• Reinforcement Learning: An Introduction - Sutton and Barto

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