Oracle-Efficient Smoothed Online Learning for Piecewise Continuous Decision Making

TL;DR

This paper introduces a new complexity measure called generalized bracketing numbers to develop oracle-efficient algorithms with optimal regret bounds for smoothed online learning, especially in piecewise continuous decision spaces.

Contribution

It proposes a novel complexity measure and an efficient algorithm for smoothed online learning that achieves optimal regret with minimal oracle calls, applicable to piecewise continuous functions.

Findings

Low regret algorithms with optimal oracle call complexity.

Effective in online prediction and planning of piecewise continuous functions.

Applicable to diverse fields like econometrics and robotics.

Abstract

Smoothed online learning has emerged as a popular framework to mitigate the substantial loss in statistical and computational complexity that arises when one moves from classical to adversarial learning. Unfortunately, for some spaces, it has been shown that efficient algorithms suffer an exponentially worse regret than that which is minimax optimal, even when the learner has access to an optimization oracle over the space. To mitigate that exponential dependence, this work introduces a new notion of complexity, the generalized bracketing numbers, which marries constraints on the adversary to the size of the space, and shows that an instantiation of Follow-the-Perturbed-Leader can attain low regret with the number of calls to the optimization oracle scaling optimally with respect to average regret. We then instantiate our bounds in several problems of interest, including online prediction and planning of piecewise continuous functions, which has many applications in fields as diverse as econometrics and robotics.