Smoothed Online Learning for Prediction in Piecewise Affine Systems

TL;DR

This paper introduces the first efficient algorithms for prediction and simulation in piecewise affine systems using smoothed online learning, achieving polynomial regret bounds under weak smoothness assumptions.

Contribution

It develops novel algorithms for PWA systems with polynomial regret bounds, leveraging smoothed online learning and optimization oracles, addressing discontinuities in sequential learning.

Findings

Algorithms achieve polynomial regret bounds.

Efficient in calls to optimization oracle.

Applicable to trajectory simulation with Wasserstein distance.

Abstract

The problem of piecewise affine (PWA) regression and planning is of foundational importance to the study of online learning, control, and robotics, where it provides a theoretically and empirically tractable setting to study systems undergoing sharp changes in the dynamics. Unfortunately, due to the discontinuities that arise when crossing into different ``pieces,'' learning in general sequential settings is impossible and practical algorithms are forced to resort to heuristic approaches. This paper builds on the recently developed smoothed online learning framework and provides the first algorithms for prediction and simulation in PWA systems whose regret is polynomial in all relevant problem parameters under a weak smoothness assumption; moreover, our algorithms are efficient in the number of calls to an optimization oracle. We further apply our results to the problems of one-step prediction and multi-step simulation regret in piecewise affine dynamical systems, where the learner is tasked with simulating trajectories and regret is measured in terms of the Wasserstein distance between simulated and true data. Along the way, we develop several technical tools of more general interest.