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

2026-07-21

AI Summary

This work addresses a class of structured nonconvex and nonsmooth online optimization problems where the loss functions are compositions of difference-of-convex (DC) functions with smooth mappings, and the constraint sets share the same structural form. The authors propose a time-smoothed proximal linear algorithm that, at each iteration, requires only a convex optimization oracle to compute the update step. The key innovation lies in introducing a tangent cone characterization tailored to composite DC constraints, constructing a first-order stationarity measure based on the proximal residual, and establishing an error bound linking this residual to the distance to stationary points. The theoretical analysis provides a local regret bound and an upper bound on the total number of inner convex subproblems, while also quantifying the relationship between the proximal residual and approximate stationarity.

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.

Problem

• online optimization
• difference-of-convex
• non-convex non-smooth
• composite functions
• constraint satisfaction

Innovation

• difference-of-convex
• online optimization
• proximal residual
• tangent cone
• stationarity certificate