Machine learning for physicists

Machine Learning for Physicists introduces the foundations of machine learning from a physicist's™s point of view. The course starts with the main types of machine-learning problems -- supervised, unsupervised, self-supervised and reinforcement learning -- while focusing mainly on the first three. We begin with linear regression, least squares, matrix notation, regularization, ridge and polynomial regression, validation, overfitting, and underfitting. We then develop the probabilistic view of learning through Bayesian inference, maximum likelihood, maximum a posteriori estimation, and the interpretation of losses and regularization in terms of noise models and priors, including robust regression and LASSO.

The course covers optimization methods used in machine learning, including gradient descent, stochastic gradient descent, momentum, adaptive learning rates, Adam and AdamW. Classification is introduced through linear classifiers, logistic regression, multiclass classification, softmax and cross-entropy loss, nonlinearly separable data, nearest-neighbour methods and the curse of dimensionality.

Unsupervised learning is studied through dimensionality reduction, SVD, low-rank approximation, matrix completion, PCA and clustering, including k-means and Gaussian mixture models. A recurring theme is the connection between inference in high dimensions and statistical physics: posterior distributions as Boltzmann measures, MAP estimation as a ground-state problem, Bayes-optimal estimators, Monte Carlo Markov chains, Metropolis-Hastings, Gibbs sampling, simulated annealing, Langevin sampling and expectation maximization.

The second part of the course introduces nonlinear machine-learning methods: feature maps, kernel methods, support vector machines, random features, feed-forward neural networks, universal approximation, backpropagation and deep neural networks. Convolutional networks are presented through locality, weight sharing, pooling layers and classical image-classification architectures.

Modern architectures are introduced through transformers. The course covers tokenization, embeddings, positional encodings, dot-product self-attention, multi-head attention, computational complexity, layer normalization, residual connections, and the GPT-style next-token prediction architecture. We also discuss practical and conceptual aspects of modern deep learning, including data augmentation, transfer learning, fine-tuning, low-rank adaptation, overparameterization, double descent, implicit regularization, adversarial examples and calibration. The final part introduces self-supervised and generative models, including autoencoders, Boltzmann machines, restricted Boltzmann machines, autoregressive models and diffusion models.