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

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.