Online Optimization with Predictions and Non-convex Losses

TL;DR

This paper introduces a general framework for online optimization with predictions and non-convex costs, providing near-optimal algorithms under broad conditions and establishing fundamental limits in certain cases.

Contribution

It presents two broad sufficient conditions for leveraging predictions in non-convex online optimization with movement costs, and introduces the SFHC algorithm with a near-optimal competitive ratio.

Findings

SFHC achieves a 1+O(1/w) competitive ratio under the conditions.

First constant, dimension-free competitive ratio for non-convex online optimization with movement costs.

Identifies cases where no online algorithm can approach optimality, such as Convex Body Chasing.

Abstract

We study online optimization in a setting where an online learner seeks to optimize a per-round hitting cost, which may be non-convex, while incurring a movement cost when changing actions between rounds. We ask: \textit{under what general conditions is it possible for an online learner to leverage predictions of future cost functions in order to achieve near-optimal costs?} Prior work has provided near-optimal online algorithms for specific combinations of assumptions about hitting and switching costs, but no general results are known. In this work, we give two general sufficient conditions that specify a relationship between the hitting and movement costs which guarantees that a new algorithm, Synchronized Fixed Horizon Control (SFHC), provides a $1+O(1/w)$ competitive ratio, where $w$ is the number of predictions available to the learner. Our conditions do not require the cost functions to be convex, and we also derive competitive ratio results for non-convex hitting and movement costs. Our results provide the first constant, dimension-free competitive ratio for online non-convex optimization with movement costs. Further, we give an example of a natural instance, Convex Body Chasing (CBC), where the sufficient conditions are not satisfied and we can prove that no online algorithm can have a competitive ratio that converges to 1.