Convex Optimization: Algorithms and Complexity

TL;DR

This monograph comprehensively reviews convex optimization algorithms, complexity theorems, and recent advances, emphasizing black-box, structural, and stochastic optimization techniques relevant to machine learning.

Contribution

It provides a unified presentation of classical and modern convex optimization methods, including new insights into non-Euclidean settings and stochastic algorithms.

Findings

Analysis of cutting plane and gradient descent methods

Discussion of non-Euclidean optimization algorithms like Frank-Wolfe and mirror descent

Overview of stochastic optimization techniques such as stochastic gradient descent

Abstract

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.