Stochastic Compositional Optimization via Hybrid Momentum Frank--Wolfe

Abstract:Stochastic compositional optimization minimizes $F(f(x), x)$ over a compact convex set when the inner map $f$ is available only through noisy samples. We propose Hybrid Momentum Stochastic Frank--Wolfe, a projection-free method for outer functions $F$ that are jointly convex and Lipschitz but not necessarily differentiable, such as maxima, CVaR, and norms. The method tracks the Jacobian and the value of $f$ with momentum, using a Taylor correction for the value, and passes the tracked linearization to a generalized linear minimization oracle. Under finite-variance noise, it attains an expected generalized Frank--Wolfe gap of order $K^{-1/4}$ for non-convex objectives and, when the linearized models minorize the objective, a last-iterate primal error of order $K^{-1/3}$. The analysis extends to heavy-tailed noise without clipping, with separate moment orders for values and Jacobians. With an exact oracle, the method reduces to the deterministic method of Vladarean et al. (2023).

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

arXiv:2605.15350 [math.OC]

(or arXiv:2605.15350v2 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2605.15350 Focus to learn more arXiv-issued DOI via DataCite

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