Online Optimization of Difference-of-Convex Compositions with Smooth Mappings

Abstract:We study online optimization for a broad class of structured non-convex non-smooth problems where each loss is a composition of a difference-of-convex function with a smooth mapping, and the feasible region is defined by constraint functions of the same kind. We propose a time-smoothed proximal linear algorithm and a local-regret measure based on a proximal residual mapping. We show that this residual is a proper stationarity measure for the original problem: its fixed-point condition implies first-order stationarity. Our analysis relies on a tangent-cone characterization for a feasible region described by composite difference-of-convex constraints, which is of independent interest and allows each update to be computed via a convex optimization oracle, despite the non-convexity of the problem. We establish a local-regret bound and a bound on the total number of inner convex subproblems. We also derive an error bound connecting the proximal residual to the distance to stationarity, providing a quantitative certificate of approximate stationarity.

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

arXiv:2607.19553 [math.OC]

(or arXiv:2607.19553v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2607.19553 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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