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

• URL: http://arxiv.org/abs/2609.15679v1
• Date: Mon, 14 Sep 2026 14:53:18 GMT
• Title: Projection-Free Multi-level Algorithms for Stochastic Constrained Compositional Optimization
• Abstract summary: This paper is about algorithms for computationally constrained multi-level compositional optimization.<n>The objective function is a composition of several smooth nested functions, and the decision is convex and closed.<n>By using a stage-wise design, we derive a parameter-free variant that preserves the same complexities for the Frank-Wolf gap.
• Score: 57.66069228055418
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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