Smoothed Online Convex Optimization in High Dimensions via Online Balanced Descent

TL;DR

This paper introduces the Online Balanced Descent (OBD) algorithm for smoothed online convex optimization, achieving dimension-free competitive ratios and sublinear regret by balancing switching and hitting costs through iterative projections.

Contribution

The paper presents a novel OBD framework that improves performance guarantees in high-dimensional settings and achieves the first dimension-free competitive ratio for certain cost functions.

Findings

OBD achieves a dimension-free competitive ratio of 3 + O(1/α) for locally polyhedral costs.

OBD provides dimension-free bounds on dynamic regret.

OBD outperforms previous algorithms in high-dimensional smoothed online convex optimization.

Abstract

We study Smoothed Online Convex Optimization, a version of online convex optimization where the learner incurs a penalty for changing her actions between rounds. Given a $\Omega(\sqrt{d})$ lower bound on the competitive ratio of any online algorithm, where $d$ is the dimension of the action space, we ask under what conditions this bound can be beaten. We introduce a novel algorithmic framework for this problem, Online Balanced Descent (OBD), which works by iteratively projecting the previous point onto a carefully chosen level set of the current cost function so as to balance the switching costs and hitting costs. We demonstrate the generality of the OBD framework by showing how, with different choices of "balance," OBD can improve upon state-of-the-art performance guarantees for both competitive ratio and regret, in particular, OBD is the first algorithm to achieve a dimension-free competitive ratio, $3 + O(1/\alpha)$, for locally polyhedral costs, where $\alpha$ measures the "steepness" of the costs. We also prove bounds on the dynamic regret of OBD when the balance is performed in the dual space that are dimension-free and imply that OBD has sublinear static regret.