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、控制及机器人领域的研究具有基础性意义，它为研究系统动态发生急剧变化的情况提供了理论和经验上易于处理的分析框架。然而，由于在不同"片段"间切换时产生的不连续性，在一般序列化场景中学习是不可能的，实际算法被迫采用启发式方法。本文基于新近发展的平滑在线学习框架，首次提出在弱平滑性假设下针对PWA系统的预测与仿真算法，其遗憾值在所有相关问题参数上呈多项式增长；此外，我们的算法在调用优化预言机的次数上具有高效性。我们进一步将结果应用于分段仿射动力系统中的单步预测与多步仿真遗憾问题——其中学习器需要模拟轨迹，并且通过模拟数据与真实数据之间的Wasserstein距离衡量遗憾。在此过程中，我们发展了若干具有更广泛通用价值的技术工具。