Projection-Free Multi-level Algorithms for Stochastic Constrained Compositional Optimization

2026-09-14

AI Summary

研究无投影算法解决随机约束多级复合优化问题，通过线性最小化oracle和动量方法减少方差，对非凸、凸及强凸目标函数提供复杂度保证。

Abstract

This paper studies projection-free algorithms for stochastic constrained multi-level compositional optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Since projection onto the constraint set can be computationally expensive, we develop projection-free methods that rely on linear minimization oracles. For non-convex objectives, we propose variance-reduced projection-free algorithms and establish complexity guarantees under both the Frank-Wolfe gap and the gradient mapping criteria. We also develop momentum-based methods that achieve convergence guarantees under weaker smoothness assumptions. Additionally, by using a stage-wise design, we derive a parameter-free variant that preserves the same complexities for the Frank-Wolfe gap. Such a design can be further used to develop algorithms for convex and strongly convex functions whose rates match those of single-level projection-free counterparts. Finally, we consider finite-sum problems and derive complexities for non-convex, convex, and strongly convex objectives. Numerical experiments across multiple tasks demonstrate the effectiveness of the proposed methods.

Problem

• stochastic constrained optimization
• projection-free algorithms
• multi-level compositional optimization
• non-convex objectives
• linear minimization oracles

Innovation

• projection-free methods
• stochastic constrained compositional optimization
• variance-reduced algorithms
• momentum-based methods
• parameter-free variant