Abstract
Behavior cloning (BC) policies on position-controlled robots inherit the closed-loop response of the underlying PD controller, yet the nonasymptotic finite-horizon consequences of controller gains for BC failure remain open. 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_∞(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_∞(K)Ψ(K)X, with Ψ(K) decomposed into label difficulty, injection strength, and contraction. This ranks 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_∞^c(α,β)=σ²α/(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 gives a nonasymptotic finite-horizon extension of the gain-dependent error-attenuation explanation of Bronars et al.