Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization

TL;DR

This paper develops new projection-free variance reduction algorithms for stochastic constrained multi-level optimization, achieving optimal sample complexities and extending analysis to convex and strongly convex objectives.

Contribution

Introduction of novel projection-free variance reduction algorithms with improved complexity analysis for various convexity settings in multi-level stochastic optimization.

Findings

Algorithms match optimal sample complexities for unconstrained problems.

Theoretical guarantees extend to convex and strongly convex functions.

Numerical experiments demonstrate the effectiveness of the proposed methods.

Abstract

This paper investigates projection-free algorithms for stochastic constrained multi-level optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Existing projection-free algorithms for solving this problem suffer from two limitations: 1) they solely focus on the gradient mapping criterion and fail to match the optimal sample complexities in unconstrained settings; 2) their analysis is exclusively applicable to non-convex functions, without considering convex and strongly convex objectives. To address these issues, we introduce novel projection-free variance reduction algorithms and analyze their complexities under different criteria. For gradient mapping, our complexities improve existing results and match the optimal rates for unconstrained problems. For the widely-used Frank-Wolfe gap criterion, we provide theoretical guarantees that align with those for single-level problems. Additionally, by using a stage-wise adaptation, we further obtain complexities for convex and strongly convex functions. Finally, numerical experiments on different tasks demonstrate the effectiveness of our methods.