Beyond Online Balanced Descent: An Optimal Algorithm for Smoothed Online Optimization

TL;DR

This paper establishes fundamental limits and proposes new algorithms for smoothed online convex optimization with strong convexity and squared Euclidean movement costs, achieving near-optimal competitive ratios.

Contribution

It proves a lower bound on the competitive ratio and introduces two algorithms, G-OBD and R-OBD, that attain competitive ratios matching this bound under various conditions.

Findings

Lower bound on competitive ratio as m approaches zero.

G-OBD achieves O(m^{-1/2}) ratio for quasiconvex costs.

R-OBD achieves O(m^{-1/2}) ratio for strongly convex costs and Bregman divergences.

Abstract

We study online convex optimization in a setting where the learner seeks to minimize the sum of a per-round hitting cost and a movement cost which is incurred when changing decisions between rounds. We prove a new lower bound on the competitive ratio of any online algorithm in the setting where the costs are $m$-strongly convex and the movement costs are the squared $\ell_2$ norm. This lower bound shows that no algorithm can achieve a competitive ratio that is $o(m^{-1/2})$ as $m$ tends to zero. No existing algorithms have competitive ratios matching this bound, and we show that the state-of-the-art algorithm, Online Balanced Decent (OBD), has a competitive ratio that is $\Omega(m^{-2/3})$. We additionally propose two new algorithms, Greedy OBD (G-OBD) and Regularized OBD (R-OBD) and prove that both algorithms have an $O(m^{-1/2})$ competitive ratio. The result for G-OBD holds when the hitting costs are quasiconvex and the movement costs are the squared $\ell_2$ norm, while the result for R-OBD holds when the hitting costs are $m$-strongly convex and the movement costs are Bregman Divergences. Further, we show that R-OBD simultaneously achieves constant, dimension-free competitive ratio and sublinear regret when hitting costs are strongly convex.