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

arXiv:2609.15679v1 Announce Type: cross 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.

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.

Optimization and Control (math.OC); Machine Learning (cs.LG)

arXiv:2609.15679 [math.OC]

(or arXiv:2609.15679v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2609.15679 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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

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