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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.14484 |
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| _version_ | 1866909032656142336 |
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| author | Seo, Junghoon |
| author_facet | Seo, Junghoon |
| contents | 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 $X_\infty(K)$ 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. 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 $X_\infty^{\mathrm{c}}(α,β)=σ^2α/(2β)$ 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 gives a nonasymptotic finite-horizon extension of the gain-dependent error-attenuation explanation of Bronars et al. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14484 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Nonasymptotic Theory of Gain-Dependent Error Dynamics in Behavior Cloning Seo, Junghoon Robotics Artificial Intelligence Optimization and Control 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 $X_\infty(K)$ 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. 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 $X_\infty^{\mathrm{c}}(α,β)=σ^2α/(2β)$ 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 gives a nonasymptotic finite-horizon extension of the gain-dependent error-attenuation explanation of Bronars et al. |
| title | A Nonasymptotic Theory of Gain-Dependent Error Dynamics in Behavior Cloning |
| topic | Robotics Artificial Intelligence Optimization and Control |
| url | https://arxiv.org/abs/2604.14484 |