Catalyst Acceleration for Gradient-Based Non-Convex Optimization

TL;DR

This paper presents a universal scheme that adapts gradient-based algorithms for non-convex optimization, ensuring convergence to stationary points and automatic acceleration when the problem is convex, with promising experimental results.

Contribution

It introduces a generic, adaptive scheme that extends convex optimization algorithms to weakly convex non-convex functions without prior knowledge of convexity.

Findings

Guarantees convergence to stationary points with first-order efficiency.

Automatically accelerates for convex objectives, achieving near-optimal rates.

Demonstrates effectiveness on neural network training and matrix factorization.

Abstract

We introduce a generic scheme to solve nonconvex optimization problems using gradient-based algorithms originally designed for minimizing convex functions. Even though these methods may originally require convexity to operate, the proposed approach allows one to use them on weakly convex objectives, which covers a large class of non-convex functions typically appearing in machine learning and signal processing. In general, the scheme is guaranteed to produce a stationary point with a worst-case efficiency typical of first-order methods, and when the objective turns out to be convex, it automatically accelerates in the sense of Nesterov and achieves near-optimal convergence rate in function values. These properties are achieved without assuming any knowledge about the convexity of the objective, by automatically adapting to the unknown weak convexity constant. We conclude the paper by showing promising experimental results obtained by applying our approach to incremental algorithms such as SVRG and SAGA for sparse matrix factorization and for learning neural networks.