# Mathematics of data: from theory to computation

## Summary

This course provides an overview of key advances in continuous optimization and statistical analysis for machine learning. We review recent learning formulations and models as well as their guarantees, describe scalable solution techniques and algorithms, and illustrate the trade-offs involved.

## Content

The course consists of the following lectures (2h each)

Lecture 1: Introduction. The role of models and data. Maximum-likelihood formulations. Error decomposition for estimation and prediction.

Lecture 2: Generalized linear models. Logistic regression.

Lecture 3: Linear algebra reminders. Computation of Gradients. Reading convergence plots.

Lecture 4: The role of computation. Challenges to optimization algorithms. Optimality measures. Structures in optimization. Gradient descent. Convergence rate of gradient descent for smooth functions.

Lecture 5: Optimality of convergence rates. Lower bounds. Accelerated gradient descent. Concept of total complexity. Stochastic gradient descent.

Lecture 6: Concise signal models. Compressive sensing. Sample complexity bounds for estimation and prediction. Challenges to optimization algorithms for non-smooth optimization. Subgradient method.

Lecture 7: Introduction to proximal-operators. Proximal gradient methods. Linear minimization oracles. Conditional gradient method for constrained optimization.

Lecture 8: Time-data trade-offs. Variance reduction for improving trade-offs.

Lecture 9: Introduction to deep learning. Generalization through uniform convergence bounds. Rademacher complexity.

Lecture 10: Double descent curves and over-parameterization. Implicit regularization. Generalization bounds using stability.

Lecture 11: Escaping saddle points. Adaptive gradient methods.

Lecture 12: Adversarial machine learning and generative adversarial networks (GANs). Wasserstein GAN. Difficulty of minimax optimization. Pitfalls of gradient descent-ascent approach.

Lecture 13: Primal-dual optimization-I: Fundamentals of minimax problems. Fenchel conjugates. Duality.

Lecture 14: Primal-dual optimization-II: Extra gradient method. Chambolle-Pock algorithm. Stochastic primal-dual methods.

Lecture 15: Primal-dual III: Lagrangian gradient methods. Lagrangian conditional gradient methods.

## Learning Prerequisites

## Required courses

Previous coursework in calculus, linear algebra, and probability is required. Familiarity with optimization is useful.

Familiarity with python, and basic knowledge of one deep learning framework (Pytorch, TensorFlow, JAX) is needed.

## In the programs

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematics of data: from theory to computation**Lecture:**3 Hour(s) per week x 14 weeks**Project:**3 Hour(s) per week x 14 weeks**Type:**optional