  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.
