Smoothed Online Optimization for Regression and Control

TL;DR

This paper analyzes online convex optimization with strong convexity and switching costs, demonstrating that the Online Balanced Descent algorithm achieves constant competitiveness and near-optimal regret, applicable to various learning and control tasks.

Contribution

The paper introduces the use of Online Balanced Descent in strongly convex online optimization with switching costs, achieving constant competitiveness and broad applicability.

Findings

OBD is constant competitive with ratio 3 + O(1/m)

OBD attains near-optimal dynamic regret for smooth cost sequences

Framework applies to online regression, classification, and control problems

Abstract

We consider Online Convex Optimization (OCO) in the setting where the costs are $m$-strongly convex and the online learner pays a switching cost for changing decisions between rounds. We show that the recently proposed Online Balanced Descent (OBD) algorithm is constant competitive in this setting, with competitive ratio $3 + O(1/m)$, irrespective of the ambient dimension. Additionally, we show that when the sequence of cost functions is $\epsilon$-smooth, OBD has near-optimal dynamic regret and maintains strong per-round accuracy. We demonstrate the generality of our approach by showing that the OBD framework can be used to construct competitive algorithms for a variety of online problems across learning and control, including online variants of ridge regression, logistic regression, maximum likelihood estimation, and LQR control.