Nonasymptotic Theory of Gain-Dependent Error Dynamics in Behavior Cloning
This research paper introduces a nonasymptotic theory addressing how controller gains influence error dynamics in Behavior Cloning (BC) for position-controlled robots. The study reveals that independent sub-Gaussian action errors propagate through gain-dependent closed-loop dynamics, resulting in sub-Gaussian position errors governed by a proxy matrix. A key finding is that the probability of task failure over a finite horizon factorizes into a gain-dependent amplification index and validation loss, indicating that training loss alone is insufficient for predicting closed-loop performance. The authors decompose the proxy bound into label difficulty, injection strength, and contraction, ranking four canonical control regimes. Specifically, compliant-overdamped systems exhibit the tightest bounds, while stiff-underdamped systems are the loosest. For scalar second-order PD systems, the analysis provides a closed-form stationary variance that is strictly monotone in stiffness and damping. This work extends previous explanations of gain-dependent error attenuation, offering rigorous theoretical insights for improving robustness in robotic learning policies.
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Nonasymptotic Theory of Gain-Dependent Error Dynamics in Behavior Cloning
This research paper introduces a nonasymptotic theory addressing how controller gains influence error dynamics in Behavior Cloning (BC) for position-controlled robots. The study reveals that independent sub-Gaussian action errors propagate through gain-dependent closed-loop dynamics, resulting in sub-Gaussian position errors governed by a proxy matrix. A key finding is that the probability of task failure over a finite horizon factorizes into a gain-dependent amplification index and validation loss, indicating that training loss alone is insufficient for predicting closed-loop performance. The authors decompose the proxy bound into label difficulty, injection strength, and contraction, ranking four canonical control regimes. Specifically, compliant-overdamped systems exhibit the tightest bounds, while stiff-underdamped systems are the loosest. For scalar second-order PD systems, the analysis provides a closed-form stationary variance that is strictly monotone in stiffness and damping. This work extends previous explanations of gain-dependent error attenuation, offering rigorous theoretical insights for improving robustness in robotic learning policies.
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