Stochastic Optimization for DC Functions and Non-smooth Non-convex Regularizers with Non-asymptotic Convergence

TL;DR

This paper introduces new stochastic algorithms for optimizing difference of convex (DC) functions, achieving non-asymptotic convergence without requiring smoothness, and extends to non-convex regularizers with efficient proximal mappings.

Contribution

It proposes practical, efficient stochastic algorithms for broad DC functions with non-asymptotic convergence analysis that adapts to gradient Hölder continuity, and extends to non-convex regularizers.

Findings

Algorithms are more user-friendly, requiring smaller mini-batches.

Convergence analysis does not need smoothness or Lipschitz gradient assumptions.

First non-asymptotic convergence results for non-convex problems with general non-convex non-differentiable regularizers.

Abstract

Difference of convex (DC) functions cover a broad family of non-convex and possibly non-smooth and non-differentiable functions, and have wide applications in machine learning and statistics. Although deterministic algorithms for DC functions have been extensively studied, stochastic optimization that is more suitable for learning with big data remains under-explored. In this paper, we propose new stochastic optimization algorithms and study their first-order convergence theories for solving a broad family of DC functions. We improve the existing algorithms and theories of stochastic optimization for DC functions from both practical and theoretical perspectives. On the practical side, our algorithm is more user-friendly without requiring a large mini-batch size and more efficient by saving unnecessary computations. On the theoretical side, our convergence analysis does not necessarily require the involved functions to be smooth with Lipschitz continuous gradient. Instead, the convergence rate of the proposed stochastic algorithm is automatically adaptive to the H\"{o}lder continuity of the gradient of one component function. Moreover, we extend the proposed stochastic algorithms for DC functions to solve problems with a general non-convex non-differentiable regularizer, which does not necessarily have a DC decomposition but enjoys an efficient proximal mapping. To the best of our knowledge, this is the first work that gives the first non-asymptotic convergence for solving non-convex optimization whose objective has a general non-convex non-differentiable regularizer.