Online Optimization of Smoothed Piecewise Constant Functions

TL;DR

This paper develops algorithms for online optimization of smoothed piecewise constant functions, achieving sublinear regret in both full information and bandit settings, addressing challenges posed by non-Lipschitz functions.

Contribution

It introduces novel algorithms for online optimization of smoothed piecewise constant functions with sublinear regret guarantees, expanding beyond traditional Lipschitz assumptions.

Findings

Achieves sublinear regret in full information setting

Achieves sublinear regret in bandit setting

Addresses non-Lipschitz, piecewise constant functions in online learning

Abstract

We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipschitz-continuous functions or functions with bounded gradients. 1 This is with good reason---any learning algorithm suffers linear regret even against piecewise constant functions that are chosen adversarially, arguably the simplest of non-Lipschitz continuous functions. The smoothed setting we consider is inspired by the seminal work of Spielman and Teng (2004) and the recent work of Gupta and Roughgarden---in this setting, the sequence of functions may be chosen by an adversary, however, with some uncertainty in the location of discontinuities. We give algorithms that achieve sublinear regret in the full information and bandit settings.