A Nonasymptotic Theory of Gain-Dependent Error Dynamics in Behavior Cloning

2026-04-15

AI Summary

This work addresses the challenge in behavioral cloning for position-controlled robots, where training loss often fails to predict real-world performance due to the absence of non-asymptotic theory characterizing how controller gains affect closed-loop behavior. The paper establishes the first gain-dependent, non-asymptotic framework for closed-loop error dynamics, revealing how sub-Gaussian action errors propagate through a PD controller into position errors. It decomposes task failure probability into a gain-dependent exponentially amplified term, validation loss, and a generalization gap. By introducing a surrogate matrix and its scalar upper bound, the analysis rigorously characterizes error tightness across four canonical control parameter regimes, theoretically justifying why compliant, overdamped controllers enhance cloning success. Leveraging stochastic process analysis, sub-Gaussian modeling, and linear system theory, the study derives a closed-form expression for continuous-time steady-state variance, proving its strict monotonicity with respect to stiffness and damping within the stability region—a property preserved in discrete-time systems, with theoretical predictions closely matching empirical results.

Technology Category

• Intelligent Robots: Behavior Learning & Control
• Humans and AI: Human-Aware Planning and Behavior Prediction
• Machine Learning: Adversarial Learning & Robustness

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Behavior cloning (BC) policies on position-controlled robots inherit the closed-loop response of the underlying PD controller, yet the effect of controller gains on BC failure lacks a nonasymptotic theory. We show that independent sub-Gaussian action errors propagate through the gain-dependent closed-loop dynamics to yield sub-Gaussian position errors whose proxy matrix $X_\infty(K)$ governs the failure tail. The probability of horizon-$T$ task failure factorizes into a gain-dependent amplification index $Γ_T(K)$ and the validation loss plus a generalization slack, so training loss alone cannot predict closed-loop performance. Under shape-preserving upper-bound structural assumptions the proxy admits the scalar bound $X_\infty(K)\preceqΨ(K)\bar X$ with $Ψ(K)$ decomposed into label difficulty, injection strength, and contraction, ranking the four canonical regimes with compliant-overdamped (CO) tightest, stiff-underdamped (SU) loosest, and the stiff-overdamped versus compliant-underdamped ordering system-dependent. For the canonical scalar second-order PD system the closed-form continuous-time stationary variance $X_\infty^{\mathrm{c}}(α,β)=σ^2α/(2β)$ is strictly monotone in stiffness and damping over the entire stable orthant, covering both underdamped and overdamped regimes, and the exact zero-order-hold (ZOH) discretization inherits this monotonicity. The analysis provides the first nonasymptotic explanation of the empirical finding that compliant, overdamped controllers improve BC success rates.

Problem

• Behavior Cloning
• Controller Gains
• Nonasymptotic Theory
• Closed-loop Performance
• Task Failure

Innovation

• nonasymptotic theory
• behavior cloning
• gain-dependent dynamics
• sub-Gaussian error propagation
• PD controller regimes