Recent Theoretical Advances in Non-Convex Optimization

TL;DR

This paper reviews recent theoretical advances in non-convex optimization, focusing on convergence guarantees of various algorithms and problem classes relevant to deep learning and data analysis.

Contribution

It provides a comprehensive overview of recent theoretical results, including convergence rates and problem structures that enable efficient solutions in non-convex optimization.

Findings

Efficient algorithms exist for structured non-convex problems.

Convergence rates are established for first-order and stochastic methods.

Certain classes like Polyak--Lojasiewicz functions allow guaranteed convergence.

Abstract

Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical results on global performance guarantees of optimization algorithms for non-convex optimization. We start with classical arguments showing that general non-convex problems could not be solved efficiently in a reasonable time. Then we give a list of problems that can be solved efficiently to find the global minimizer by exploiting the structure of the problem as much as it is possible. Another way to deal with non-convexity is to relax the goal from finding the global minimum to finding a stationary point or a local minimum. For this setting, we first present known results for the convergence rates of deterministic first-order methods, which are then followed by a general theoretical analysis of optimal stochastic and randomized gradient schemes, and an overview of the stochastic first-order methods. After that, we discuss quite general classes of non-convex problems, such as minimization of $\alpha$-weakly-quasi-convex functions and functions that satisfy Polyak--Lojasiewicz condition, which still allow obtaining theoretical convergence guarantees of first-order methods. Then we consider higher-order and zeroth-order/derivative-free methods and their convergence rates for non-convex optimization problems.