Regrets of Proximal Method of Multipliers for Online Non-convex Optimization with Long Term Constraints

TL;DR

This paper introduces the OPMM algorithm for online non-convex optimization with long-term constraints, providing regret bounds for constraint violations and objective reduction, applicable even with non-convex feasible sets.

Contribution

It proposes a proximal method of multipliers with quadratic approximations for online non-convex problems, analyzing its regret bounds and demonstrating its effectiveness under various conditions.

Findings

Regret bounds of ${\\cO}(T^{-1/8})$ for constraint violations.

Regret bounds of ${\\cO}(T^{-1/4})$ for objective reduction.

Applicability to non-convex feasible sets and dual-based subproblem solutions.

Abstract

The online optimization problem with non-convex loss functions over a closed convex set, coupled with a set of inequality (possibly non-convex) constraints is a challenging online learning problem. A proximal method of multipliers with quadratic approximations (named as OPMM) is presented to solve this online non-convex optimization with long term constraints. Regrets of the violation of Karush-Kuhn-Tucker conditions of OPMM for solving online non-convex optimization problems are analyzed. Under mild conditions, it is shown that this algorithm exhibits ${\cO}(T^{-1/8})$ Lagrangian gradient violation regret, ${\cO}(T^{-1/8})$ constraint violation regret and ${\cO}(T^{-1/4})$ complementarity residual regret if parameters in the algorithm are properly chosen, where $T$ denotes the number of time periods. For the case that the objective is a convex quadratic function, we demonstrate that the regret of the objective reduction can be established even the feasible set is non-convex. For the case when the constraint functions are convex, if the solution of the subproblem in OPMM is obtained by solving its dual, OPMM is proved to be an implementable projection method for solving the online non-convex optimization problem.