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This paper provides a nonasymptotic analysis of error propagation in Behavior Cloning (BC) for position-controlled robots, explicitly considering the influence of PD controller gains. They demonstrate that action errors propagate through gain-dependent closed-loop dynamics, resulting in position errors governed by a proxy matrix that dictates failure probability. Their key result is a factorization of task failure probability into a gain-dependent amplification index and the validation loss, revealing that training loss alone is insufficient for predicting closed-loop performance and that compliant-overdamped controllers yield tighter error bounds.
Compliant, overdamped controllers improve Behavior Cloning success rates not by magic, but because they minimize error amplification, a fact now rigorously proven.
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 $\Gamma_T(K)$ 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 $X_\infty(K)\preceq\Psi(K)\bar X$ with $\Psi(K)$ 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 $X_\infty^{\mathrm{c}}(\alpha,\beta)=\sigma^2\alpha/(2\beta)$ 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.