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Smoothed Online Learning for Prediction in Piecewise Affine Systems

Adam Block, Max Simchowitz, Russ Tedrake

TL;DR

The paper tackles learning and prediction in piecewise-affine systems where discontinuities hinder online learning. It proposes a smoothed online learning framework using directional smoothness to suppress boundary pathology, paired with an oracle-efficient ERM-based algorithm that operates in epochs to recover region boundaries and affine parameters. The main results show sublinear regret that scales polynomially in key problem parameters, along with parameter recovery for frequently visited modes and sublinear misclassification of modes; this is extended to multi-step trajectory prediction with Wasserstein-based simulation regret, under a Lyapunov-type stability condition. The approach introduces new analytical tools, including disagreement covers and martingale-based concentration arguments, to achieve these guarantees without strong realizability assumptions. This work advances theory for learning in non-smooth dynamical systems and has implications for online control and model-based planning in contact-rich robotics.

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.

Smoothed Online Learning for Prediction in Piecewise Affine Systems

TL;DR

The paper tackles learning and prediction in piecewise-affine systems where discontinuities hinder online learning. It proposes a smoothed online learning framework using directional smoothness to suppress boundary pathology, paired with an oracle-efficient ERM-based algorithm that operates in epochs to recover region boundaries and affine parameters. The main results show sublinear regret that scales polynomially in key problem parameters, along with parameter recovery for frequently visited modes and sublinear misclassification of modes; this is extended to multi-step trajectory prediction with Wasserstein-based simulation regret, under a Lyapunov-type stability condition. The approach introduces new analytical tools, including disagreement covers and martingale-based concentration arguments, to achieve these guarantees without strong realizability assumptions. This work advances theory for learning in non-smooth dynamical systems and has implications for online control and model-based planning in contact-rich robotics.

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.
Paper Structure (48 sections, 40 theorems, 251 equations, 4 algorithms)

This paper contains 48 sections, 40 theorems, 251 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Key challenges.
  4. Our techniques.
  5. Setting
  6. Directional Smoothness.
  7. Further Assumptions.
  8. Algorithm and Guarantees
  9. Regret for One-Step Prediction in PWA Systems
  10. Guarantees for Simulation Regret
  11. Analysis
  12. Parameter Recovery
  13. Mode Prediction
  14. Discussion
  15. General Tools
  16. ...and 33 more sections

Key Result

Proposition 1

In the above setting, there exists an adversary with $m = d = 1$, K=2, that chooses $\mathbf{\Theta}^\star$ and $g_{\star}$, as well as $\mathbf{x}_1, \dots, \mathbf{x}_T$ such that any learner experiences

Theorems & Definitions (90)

  • Proposition 1
  • Definition 2: Definition 52 from block2022efficient
  • Theorem 3: Regret Bound
  • Remark 1
  • Remark 2
  • Theorem 4: Parameter Recovery
  • Lemma 3.1
  • Theorem 5: One-Step Regret in PWA Systems
  • proof
  • Definition 6: Simulation Regret
  • ...and 80 more