Smoothed Online Learning for Prediction in Piecewise Affine Systems
This addresses a foundational problem in online learning, control, and robotics by enabling practical algorithms for systems with sharp dynamic changes, though it builds on existing smoothed online learning frameworks.
The paper tackles the problem of learning in piecewise affine systems, which is challenging due to discontinuities, by providing the first algorithms for prediction and simulation with polynomial regret under a weak smoothness assumption and efficient optimization oracle calls.
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.