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.

翻译：分段仿射（PWA）回归与规划问题对于在线学习、控制及机器人学研究具有基础重要性，它为研究系统动力学突变提供了理论和经验上易于处理的框架。然而，由于跨越不同"片段"时产生的非连续性，在一般序贯场景中学习是不可行的，实际算法被迫采用启发式方法。本文基于最新发展的平滑在线学习框架，提出了首个在弱平滑性假设下所有相关问题参数的遗憾函数呈多项式形式的分段仿射系统预测与仿真算法；此外，我们的算法在优化预言机调用次数方面具有高效性。我们进一步将结果应用于分段仿射动态系统中的单步预测与多步仿真遗憾问题——在该问题中，学习者需完成轨迹仿真任务，并通过仿真数据与真实数据之间的Wasserstein距离度量遗憾。在此过程中，我们开发了若干具有更广泛适用性的技术工具。