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A Nonasymptotic Theory of Gain-Dependent Error Dynamics in Behavior Cloning
Junghoon Seo
Intelligence
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Summary
This paper provides a nonasymptotic theoretical framework for understanding how controller gains (proportional and derivative) in position-controlled robots affect the failure rates of behavior cloning (BC) policies. It demonstrates that action errors propagate through gain-dependent closed-loop dynamics, and that task failure probability is governed by a gain-dependent amplification index rather than just validation loss. The analysis explains why compliant, overdamped controllers empirically improve BC success rates by minimizing this amplification.
Entities (5)
Relation Signals (3)
Amplification Index → determines → Task Failure
confidence 95% · The probability of horizon-T task failure factorizes into a gain-dependent amplification index
PD Controller → influences → Behavior Cloning
confidence 95% · BC policies on position-controlled robots inherit the closed-loop response of the underlying PD controller
Compliant-Overdamped Regime → improves → Behavior Cloning
confidence 90% · compliant, overdamped controllers improve BC success rates
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Abstract
Abstract: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.
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- Source: https://arxiv.org/abs/2604.14484v1
- Canonical: https://arxiv.org/abs/2604.14484v1
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A Nonasymptotic Theory of Gain-Dependent Error Dynamics in Behavior Cloning Junghoon Seo Junghoon Seo is with PIT IN Corp., South Korea (e-mail: junghoon.seo@pitin.co.kr). Abstract 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∞(K)X_∞(K) governs the failure tail. The probability of horizon-T task failure factorizes into a gain-dependent amplification index ΓT(K) _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¯X_∞(K) (K) X with Ψ(K) (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∞c(α,β)=σ2α/(2β)X_∞^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 [7]. I Introduction Position-controlled action spaces have become the standard interface for learned manipulation policies [8, 9, 28]. At each step a policy π^K π_K outputs a position target at=π^K(st)∈ℝna_t= π_K(s_t) ^n that a PD controller [15] converts to motor torques via τt=Kp(at−qt)−Kdq˙t+g(qt), _t=K_p(a_t-q_t)-K_d\, q_t+g(q_t), (1) where qt,q˙t∈ℝnq_t, q_t ^n are the measured joint positions and velocities, Kp,Kd∈ℝn×nK_p,K_d ^n× n are the diagonal positive-definite proportional and derivative gain matrices, and g(qt)∈ℝng(q_t) ^n is gravity compensation. Despite the ubiquity of this interface, the choice of gains (Kp,Kd)(K_p,K_d) for learning pipelines remains largely heuristic. Recent empirical work by Bronars et al. [7] systematically tunes controller gains across BC, reinforcement learning, and sim-to-real transfer. The salient finding is that BC benefits from compliant (low KpK_p) and overdamped (high KdK_d) gains. Policies trained under these settings achieve higher closed-loop success rates despite exhibiting higher validation loss, an apparent contradiction between open-loop prediction accuracy and closed-loop task performance that has no existing theoretical explanation. Parallel theoretical work identifies fundamental challenges in continuous-action imitation learning, including sensitivity to function approximation error [24] and noise-driven amplification during autoregressive rollout [3], but neither line of work models the controller through which actions are physically executed. The classical compounding-error analysis pioneered by Ross and Bagnell [22] and extended in DAgger [23] bounds the policy-versus-expert performance gap by the horizon and the per-step error, treating the dynamics as a black box. Spencer et al. [25] characterize three regimes of covariate shift, again without reference to the gain-dependent structure of the closed loop. In control theory, the propagation of stochastic disturbances through linear systems is well understood via Lyapunov equations and ℋ2/ℋ∞H_2/H_∞ norms [21], and recent work on learning-based control [5, 26] connects sample complexity and regret to system-theoretic quantities. These analyses target reinforcement learning or adaptive control, however, not the offline BC pipeline where a fixed policy is deployed open-loop with respect to the expert. We bridge these two lines of research. The key distinction from classical compounding-error analysis [22, 23] is that our bounds depend explicitly on the controller gains through the closed-loop matrices AK,BKA_K,B_K, and reveal that the relevant quantity for closed-loop performance is not the prediction loss alone but its product with a gain-dependent amplification. Our contributions are fourfold as follows: 1. Sub-Gaussian propagation (Theorem 1). Independent sub-Gaussian action errors yield a sub-Gaussian position error whose proxy matrix Xt(K)X_t(K) is the gain-dependent Lyapunov sum. 2. Failure bound via validation loss (Theorem 3). The probability of horizon-T task failure factorizes into a gain-dependent amplification index ΓT(K) _T(K) and the validation loss plus generalization slack, showing that higher validation loss can coexist with a lower failure bound. 3. Scalar ordering and regime comparison (Theorems 5 and 7). Under shape-preserving upper-bound assumptions, the proxy obeys X∞(K)⪯Ψ(K)X¯X_∞(K) (K) X with Ψ(K)=b(K)l(K)/(1−ρ∗(K)2) (K)=b(K)l(K)/(1- _*(K)^2), and monotonicity of Ψ ranks the four canonical regimes. 4. Global canonical monotonicity (Theorem 9). For the canonical scalar second-order PD system, X∞c(α,β)=σ2α/(2β)X_∞^c(α,β)=σ^2α/(2β) is strictly monotone in stiffness and damping over the entire stable orthant, and the exact ZOH discretization inherits this monotonicity. I Related Works Behavior cloning theory Behavior cloning [20, 2] learns a policy by supervised regression on expert state-action pairs. The foundational analysis of Ross and Bagnell [22] shows that the expected cost of the learned policy grows quadratically with horizon T from compounding errors, motivating interactive approaches such as DAgger [23]. Subsequent work refines these bounds under structural assumptions [25, 11], sharpens the horizon dependence [12], and links low-level dynamical stability to task-level guarantees [4]. Our analysis introduces the controller gain as a structural variable that modulates error propagation, a dimension absent from prior theory. Impedance control The interaction between compliance and task performance has a long history in robotics [14, 13, 16], with variable impedance learning [6, 27, 18] adapting compliance during execution. Our contribution is orthogonal. We analyze how fixed gains modulate the propagation of prediction errors during BC rollout. Controller design for robot learning The choice of action space and controller gains has long been recognized as a critical design axis for robot learning [1, 10, 19]. Kim et al. [17] show that torque-level control improves task-agnostic transfer, and Bronars et al. [7] provide the first systematic empirical study of how PD gains affect BC, RL, and sim-to-real transfer, identifying compliant-overdamped settings as the BC optimum. The theoretical analysis below supplies the missing foundation for that observation. I Problem Formulation I-A Gain-Dependent Closed-Loop Error Dynamics Consider a robot with n joints and inertia M∈ℝn×nM ^n× n operating under the PD controller (1) with gain setting K=(Kp,Kd)K=(K_p,K_d). Let qt⋆q_t denote the expert joint trajectory and at⋆a_t the expert action. Define the position error et:=qt−qt⋆e_t:=q_t-q_t and the prediction error ξt:=π^K(st)−at⋆ _t:= π_K(s_t)-a_t . Substituting at=at⋆+ξta_t=a_t + _t into (1) and linearizing about the expert trajectory yields Me¨t=−Kpet−Kde˙t+Kpξt,M\, e_t=-K_p\,e_t-K_d\, e_t+K_p\, _t, (2) so the action error enters scaled by KpK_p and stiffer gains amplify both the restoring force on ete_t and the injection of action errors. Setting xt:=[et⊤,e˙t⊤]⊤∈ℝ2nx_t:=[e_t ,\, e_t ] ^2n gives the continuous state-space form x˙t=AKcxt+BKcξt,et=Cxt, x_t=A_K^cx_t+B_K^c _t, e_t=Cx_t, (3) with AKc=[0In−M−1Kp−M−1Kd]A_K^c= bmatrix0&I_n\\ -M^-1K_p&-M^-1K_d bmatrix, BKc=[0M−1Kp]B_K^c= bmatrix0\\ M^-1K_p bmatrix, and C=[In, 0]C=[I_n,\,0]. The policy emits a new target every Δt t seconds and the controller holds it constant between updates, so the sampled state obeys xt+1=AKxt+BKξt,et=Cxt,x_t+1=A_Kx_t+B_K _t, e_t=Cx_t, (4) with matrices given by the exact zero-order-hold (ZOH) sampling of (3), AK=eAKcΔt,BK=(∫0ΔteAKcss)BKc.A_K=e^A_K^c t, B_K= ( _0 t\!e^A_K^cs\,ds )B_K^c. (5) We consider gains for which the sampled closed loop is asymptotically stable, ρ(AK)<1ρ(A_K)<1. I-B Action Prediction Error Model We model the prediction errors ξt\ _t\ as independent, mean-zero, sub-Gaussian random vectors. There exists a PSD matrix ΣKroll⪰0 _K^roll 0 such that [exp(λ⊤ξt)]≤exp(12λ⊤ΣKrollλ),∀λ∈ℝm.E\! [ (λ _t) ]≤ \! ( 12λ _K^rollλ ), ∀\,λ ^m. (6) The proxy covariance ΣKroll _K^roll governs the tail behavior, with the subscript K reflecting that different gains induce different action labels and hence different residual statistics. Independence is justified in the early-rollout regime before significant covariate shift, and sub-Gaussianity accommodates bounded errors and light-tailed neural network residuals. The rollout-local population action MSE is Lroll(K):=[‖ξt‖22]=tr(ΣKroll)L_roll(K):=E[ \| _t \|_2^2]=tr( _K^roll). I-C Task Success, Validation Loss, and Amplification A rollout is successful if the position error stays within a success tube of radius r>0r>0 over horizon T. The complementary failure event is FailT:=∃t≤T:‖et‖2≥r.Fail_T:= \∃\,t≤ T: \|e_t \|_2≥ r \. (7) After training, the empirical validation loss over NvaN_va held-out pairs (s~j,a~j)\( s_j, a_j)\ is L^va(K):=Nva−1∑j‖π^K(s~j)−a~j‖22 L_va(K):=N_va^-1 _j \| π_K( s_j)- a_j \|_2^2, and the training loss L^tr(K) L_tr(K) is defined analogously. A standard generalization argument gives, with probability at least 1−δ1-δ, Lroll(K)≤L^va(K)+εgen(K,δ).L_roll(K)≤ L_va(K)+ _gen(K,δ). (8) We define the finite-horizon amplification index ΓT(K):=max0≤t≤T∑s=0t−1‖CAKsBK‖op2, _T(K):= _0≤ t≤ T _s=0^t-1 \|CA_K^sB_K \|_op^2, (9) which measures the worst-case cumulative operator-norm gain from action errors to position errors and depends on the gains through AKA_K and BKB_K. IV Main Results We present four results in a logical chain: a sub-Gaussian propagation theorem, a closed-loop failure bound, a scalar ordering reduction in the upper-bound direction, and a regime comparison. IV-A Sub-Gaussian Error Propagation Theorem 1 lifts the per-step sub-Gaussian assumption on the action error to a horizon-wide statement about the position error. The lift is a direct consequence of linearity: a weighted sum of independent sub-Gaussians is sub-Gaussian, and the weights are exactly the closed-loop impulse response at lag s. The Lyapunov limit S∞(K)S_∞(K) is the standard quantity for disturbance-to-state propagation in discrete LTI systems, and the novelty here is that every ingredient depends on the controller gain, so tuning (Kp,Kd)(K_p,K_d) reshapes the entire tail of ete_t. Theorem 1 (Sub-Gaussian propagation). Consider (4) with x0=0x_0=0 and action errors satisfying (6). Define the proxy matrix St(K):=∑s=0t−1AKsBKΣKrollBK⊤(AK⊤)s,S_t(K):= _s=0^t-1A_K^sB_K _K^rollB_K (A_K )^s, (10) and its position-error projection Xt(K):=CSt(K)C⊤X_t(K):=C\,S_t(K)\,C . Then ete_t is mean-zero sub-Gaussian with proxy Xt(K)X_t(K). For all u∈ℝnu ^n, [exp(u⊤et)]≤exp(12u⊤Xt(K)u),E\! [ (u e_t) ]≤ \! ( 12u X_t(K)\,u ), (11) ℙ(|u⊤et|≥r)≤2exp(−r22u⊤Xt(K)u).P\! (|u e_t|≥ r )≤ 2 \! (- r^22\,u X_t(K)\,u ). (12) If ρ(AK)<1ρ(A_K)<1, then St(K)→S∞(K)S_t(K)→ S_∞(K), the unique PSD solution of the discrete Lyapunov equation S∞(K)=AKS∞(K)AK⊤+BKΣKrollBK⊤,S_∞(K)=A_KS_∞(K)A_K +B_K _K^rollB_K , (13) and the stationary proxy is X∞(K)=CS∞(K)C⊤X_∞(K)=C\,S_∞(K)\,C . Proof. Unrolling (4) from x0=0x_0=0 gives et=∑s=0t−1CAKsBKξt−1−se_t= _s=0^t-1CA_K^sB_K _t-1-s. For u∈ℝnu ^n and hs:=BK⊤(AK⊤)sC⊤uh_s:=B_K (A_K )^sC u, we have u⊤et=∑shs⊤ξt−1−su e_t= _sh_s _t-1-s. By independence and the sub-Gaussian MGF bound, [eu⊤et] \! [e^u e_t ] ≤∏sexp(12hs⊤ΣKrollhs) ≤ _s \! ( 12h_s _K^rollh_s ) =exp(12u⊤Xt(K)u). = \! ( 12u X_t(K)u ). (14) The tail bound follows from the Chernoff method, and Schur stability with the discrete Lyapunov theorem yields convergence to X∞(K)X_∞(K). ∎ Corollary 2 (Euclidean tail bound). Under Theorem 1, ℙ(‖et‖2≥r)≤2nexp(−r22nλmax(Xt(K))).P\! ( \|e_t \|_2≥ r )≤ 2n\, \! (- r^22n\, _ (X_t(K)) ). (15) Proof. Diagonalize Xt=QΛQ⊤X_t=Q Q and set y=Q⊤ety=Q e_t. If ‖y‖2≥r \|y \|_2≥ r then some |yi|≥r/n|y_i|≥ r/ n, and each yiy_i is sub-Gaussian with proxy λi≤λmax(Xt) _i≤ _ (X_t). A union bound over coordinates gives the result. ∎ The dimensional factor n enters twice, once as a linear prefactor and once inside the denominator of the exponent. In the scalar case n=1n=1 both factors disappear and the bound reduces to the one-dimensional Chernoff tail. For robot manipulators n is typically between six and twelve, and the corresponding slack is mild on a logarithmic scale. IV-B Closed-Loop Failure Bound Theorem 3 (Failure bound via validation loss). Under the framework of Section I, with probability at least 1−δ1-δ over training randomness, ℙ(FailT∣π^K)≤2n(T+1)exp(−r22nΓT(K)(L^va(K)+εgen(K,δ))).P\! (Fail_T π_K )≤\\ 2n(T+1) \! (- r^22n\, _T(K)\! ( L_va(K)+ _gen(K,δ) ) ). (16) Proof. From Theorem 1, λmax(Xt(K))≤∑s‖CAKsBK‖op2λmax(ΣKroll)≤ΓT(K)Lroll(K) _ (X_t(K))≤ _s \|CA_K^sB_K \|_op^2\, _ ( _K^roll)≤ _T(K)\,L_roll(K), where the second inequality uses λmax(ΣKroll)≤tr(ΣKroll)=Lroll(K) _ ( _K^roll) ( _K^roll)=L_roll(K). The generalization bound (8) replaces Lroll(K)L_roll(K) by L^va(K)+εgen(K,δ) L_va(K)+ _gen(K,δ) with probability 1−δ1-δ. Substituting into (15) and union-bounding over t=0,1,…,Tt=0,1,…,T yields (16). ∎ The exponent of (16) is governed by the product ΓT(K)(L^va(K)+εgen(K,δ)) _T(K)\,( L_va(K)+ _gen(K,δ)) rather than by the loss alone, so a gain K1K_1 with lower training or validation loss than K2K_2 can incur a strictly worse failure bound once its amplification is correspondingly larger. This is precisely the empirical pattern of [7], in which compliant-overdamped gains attain higher closed-loop success despite higher validation MSE because their amplification is small enough to dominate the product. A related root-mean-square quantity Kp/(2Kd)ϵπ(K) K_p/(2K_d)\, _π(K) appears in the State-Space Impact of Policy Errors corollary of [7], which Theorem 3 strengthens to a nonasymptotic finite-horizon tail bound carrying the explicit generalization slack εgen(K,δ) _gen(K,δ). IV-C Scalar Ordering Reduction To compare gain regimes through a single scalar, we impose structural conditions on the error dynamics that yield an upper bound on the proxy matrix. Every inequality below is an upper bound, and reversing any one invalidates the chain. Definition 4 (Shape-preserving upper-bound structure). The error dynamics (4) have shape-preserving upper-bound structure if there exist fixed PSD matrices W¯,Σ¯⪰0 W, 0 and scalar functions l(K),b(K)>0l(K),b(K)>0, ρ∗(K)∈[0,1) _*(K)∈[0,1) such that (i) label difficulty: ΣKroll _K^roll ⪯l(K)Σ¯, l(K)\, , (i) injection: BKΣ¯BK⊤ B_K B_K ⪯b(K)W¯, b(K)\, W, (i) Lyapunov contraction: AKW¯AK⊤ A_K WA_K ⪯ρ∗(K)2W¯. _*(K)^2\, W. Condition (i) bounds the rollout-error proxy at gain K, condition (i) bounds how strongly action errors are injected into the state space, and condition (i) certifies W¯ W as a Lyapunov function for AKA_K with rate ρ∗(K)2 _*(K)^2. Theorem 5 (Scalar ordering reduction). Under Definition 4, define the ordering index Ψ(K):=b(K)l(K)1−ρ∗(K)2. (K):= b(K)\,l(K)1- _*(K)^2. (17) Then X∞(K)⪯Ψ(K)X¯,X¯:=CW¯C⊤.X_∞(K) (K)\, X, X:=C\, W\,C . (18) Consequently, λmax(X∞(K))≤Ψ(K)λmax(X¯) _ (X_∞(K))≤ (K)\, _ ( X). If Ψ(K1)≤Ψ(K2) (K_1)≤ (K_2), then K1K_1’s proxy upper bound is no larger than K2K_2’s, and substituting into (16) yields a failure-probability upper bound for K1K_1 that is no larger than the corresponding upper bound for K2K_2. Proof. By (i), left- and right-multiplying ΣKroll⪯l(K)Σ¯ _K^roll l(K) by BKB_K and its transpose and applying (i) gives BKΣKrollBK⊤⪯l(K)b(K)W¯B_K _K^rollB_K l(K)b(K)\, W. The map X↦AKsX(AK⊤)sX A_K^sX(A_K )^s preserves ⪯ for s≥0s≥ 0, so AKsBKΣKrollBK⊤(AK⊤)s⪯l(K)b(K)AKsW¯(AK⊤)sA_K^sB_K _K^rollB_K (A_K )^s l(K)b(K)A_K^s W(A_K )^s. Iterating (i) yields AKsW¯(AK⊤)s⪯ρ∗(K)2sW¯A_K^s W(A_K )^s _*(K)^2s W, and the geometric sum gives S∞(K)⪯l(K)b(K)∑s≥0ρ∗(K)2sW¯=Ψ(K)W¯S_∞(K) l(K)b(K) _s≥ 0 _*(K)^2s W= (K) W. Projecting through C yields (18), and operator monotonicity of λmax _ followed by substitution into (16) gives the failure-bound monotonicity. ∎ The index Ψ(K) (K) assembles three independent penalties: a large label difficulty l(K)l(K), a large injection b(K)b(K), and slow contraction (large ρ∗(K) _*(K)). A gain can remain optimal even when its label difficulty is the highest of the four candidates, provided the injection and contraction terms are correspondingly smaller. This three-way competition is precisely what makes the BC gain-tuning problem nontrivial, and Theorem 5 reduces the resolution of that competition to a single scalar comparison. Importantly, the three penalties are not symmetric in the gains: the injection b(K)b(K) is typically a strictly increasing function of KpK_p (stiffer gains inject action errors more aggressively), while the contraction ρ∗(K) _*(K) is typically a strictly decreasing function of KdK_d (stronger damping tightens the Lyapunov rate). The label-difficulty term l(K)l(K) is the only component with a qualitatively data-driven behavior: harder labels are induced by compliant settings because the policy must compensate for richer closed-loop trajectories. Corollary 6 (Finite-horizon convergence). Under Definition 4, the truncation error of the proxy admits the upper bound X∞(K)−Xt(K)⪯ρ∗(K)2t1−ρ∗(K)2b(K)l(K)X¯,X_∞(K)-X_t(K) _*(K)^2t1- _*(K)^2\,b(K)\,l(K)\, X, (19) and consequently ‖X∞(K)−Xt(K)‖op≤ρ∗(K)2tΨ(K)‖X¯‖op \|X_∞(K)-X_t(K) \|_op≤ _*(K)^2t\, (K)\, \| X \|_op, so the approximation error decays geometrically at rate ρ∗(K)2 _*(K)^2 per step. Proof. S∞(K)−St(K)=∑s=t∞AKsWK(AK⊤)sS_∞(K)-S_t(K)= _s=t^∞A_K^sW_K(A_K )^s with WK=BKΣKrollBK⊤W_K=B_K _K^rollB_K . Repeating the upper-bound chain from the proof of Theorem 5 on the tail sum gives X∞(K)−Xt(K)⪯l(K)b(K)ρ∗(K)2t/(1−ρ∗(K)2)X¯X_∞(K)-X_t(K) l(K)b(K) _*(K)^2t/(1- _*(K)^2) X. The operator-norm bound follows since ∥⋅∥op \|· \|_op is monotone under ⪯ for PSD matrices. ∎ IV-D Regime Ordering We parameterize gains as K=(α,β)K=(α,β) where α denotes stiffness (KpK_p scale) and β denotes damping (KdK_d scale), and define four canonical regimes CO =(αL,βH), =( _L, _H), SO =(αH,βH), =( _H, _H), (20) CU =(αL,βL), =( _L, _L), SU =(αH,βL), =( _H, _L), with αH>αL _H> _L and βH>βL _H> _L. Theorem 7 (Regime ordering). Suppose Ψ(α,β) (α,β) is non-decreasing in α and non-increasing in β. Then Ψ(CO) (CO) ≤minΨ(SO),Ψ(CU), ≤ \! \ (SO),\, (CU) \, (21) Ψ(SU) (SU) ≥maxΨ(SO),Ψ(CU), ≥ \! \ (SO),\, (CU) \, (22) so CO admits the tightest failure-bound upper bound and SU the loosest. The SO vs. CU comparison reduces to Ψ(αH,βH)≶Ψ(αL,βL) ( _H, _H) ( _L, _L) and is therefore system-dependent. Proof. Direct from monotonicity. Ψ(αL,βH)≤Ψ(αH,βH) ( _L, _H)≤ ( _H, _H) by stiffness-monotonicity, and Ψ(αL,βH)≤Ψ(αL,βL) ( _L, _H)≤ ( _L, _L) by damping-antimonotonicity. Similarly Ψ(αH,βH)≤Ψ(αH,βL) ( _H, _H)≤ ( _H, _L) and Ψ(αL,βL)≤Ψ(αH,βL) ( _L, _L)≤ ( _H, _L). The SO-CU comparison involves opposing effects and cannot be resolved without additional information. ∎ Figure 1: Hasse diagram of the regime ordering established by Theorem 7. Solid arrows indicate Ψ(K1)≤Ψ(K2) (K_1)≤ (K_2). CO is the unique minimum and SU the unique maximum. The dashed line between SO and CU indicates that their ordering is system-dependent. IV-E Verification for Canonical Second-Order Systems Theorem 7 reduces the regime ordering to monotonicity of Ψ(K) (K). We verify this monotonicity directly for the canonical second-order PD system without invoking the shape-preserving abstraction. The continuous-time analysis yields a closed-form expression that is monotone over the entire stable gain space, covering both underdamped and overdamped regimes, and the exact ZOH sampling used in Section V inherits the same monotonicity. Lemma 8 (Continuous-time stationary covariance). For the scalar (n=1n=1) continuous-time error dynamics (3) with mass m and gains Kp=αK_p=α, Kd=βK_d=β driven by a white action-error process of intensity σ2σ^2, the stationary covariance of xtx_t is the unique PSD solution of AKcP+P(AKc)⊤+BKcσ2(BKc)⊤=0A_K^cP+P(A_K^c) +B_K^cσ^2(B_K^c) =0, given by Pc(α,β)=[σ2α2β00σ2α22βm].P^c(α,β)= bmatrix σ^2α2β&0\\ 0& σ^2α^22β\,m bmatrix. (23) The position-error stationary variance is therefore X∞c(α,β)=CPcC⊤=σ2α2β.X_∞^c(α,β)=CP^cC = σ^2α2β. (24) Proof. Write Ac=[01−α/m−β/m]A_c= bmatrix0&1\\ -α/m&-β/m bmatrix, Bc=(α/m)e2B_c=(α/m)e_2, and P=[p11p12p12p22]P= bmatrixp_11&p_12\\ p_12&p_22 bmatrix. Expanding AcP+PAc⊤+Bcσ2Bc⊤=0A_cP+PA_c +B_cσ^2B_c =0 entry-wise gives the three independent equations 2p12=02p_12=0, p22=(α/m)p11p_22=(α/m)p_11, and −2(β/m)p22+(α/m)2σ2=0-2(β/m)p_22+(α/m)^2σ^2=0, solved in order to obtain p12=0p_12=0, p22=σ2α2/(2βm)p_22=σ^2α^2/(2β m), and p11=σ2α/(2β)p_11=σ^2α/(2β). Hurwitz stability of AcA_c gives uniqueness of the PSD solution. ∎ Theorem 9 (Global monotonicity in canonical systems). The scalar canonical stationary variance X∞c(α,β)=σ2α/(2β)X_∞^c(α,β)=σ^2α/(2β) is strictly increasing in stiffness α and strictly decreasing in damping β over the entire stable region (α,β)∈ℝ2:α,β>0\(α,β) ^2:α,β>0\, covering both underdamped (β2<4mαβ^2<4mα) and overdamped (β2≥4mαβ^2≥ 4mα) regimes. The canonical regime ordering of Theorem 7 therefore holds with Ψc(α,β):=X∞c(α,β)/X¯c ^c(α,β):=X_∞^c(α,β)/ X^c for any positive scalar reference X¯c X^c. Proof. From (24), ∂αX∞c=σ2/(2β)>0 _αX_∞^c=σ^2/(2β)>0 and ∂βX∞c=−σ2α/(2β2)<0 _βX_∞^c=-σ^2α/(2β^2)<0 on the positive orthant. The Hurwitz condition for the canonical scalar system is exactly α,β>0α,β>0, so the closed form covers the entire stable region without an underdamped/overdamped split, and the four regimes inherit the ordering of Theorem 7. ∎ Working in continuous time keeps the Lyapunov solution in closed form and avoids any branching on the underdamped/overdamped split, while inheritance through exact ZOH discretization matches the input matrix used throughout Section V. The same scalar σ2Kp/(2Kd)σ^2K_p/(2K_d) appears in the analytical appendix of [7] under a one-degree-of-freedom Gaussian white-noise model. Lemma 8 instead recovers it from the continuous-time Lyapunov equation, and Theorem 9 together with Proposition 10 extends the identity to sub-Gaussian forcing and to strict monotonicity on the entire stable orthant under exact ZOH. Proposition 10 (Discrete-time inheritance under ZOH). For the canonical scalar system sampled by (5) with control period Δt>0 t>0, the discrete stationary proxy X∞d(α,β,Δt)X_∞^d(α,β, t) is smooth in (α,β)(α,β) on the open Schur-stable region, and X∞d(α,β,Δt)/Δt→X∞c(α,β)X_∞^d(α,β, t)/ t→ X_∞^c(α,β) as Δt→0+ t→ 0^+. For control periods small enough that the leading-order continuous-time term dominates, X∞dX_∞^d inherits the global strict monotonicity of X∞cX_∞^c in (α,β)(α,β), covering the gain ranges used at typical BC control rates of 5050–10001000 Hz. Proof. The ZOH matrices AK(Δt)=eAKcΔtA_K( t)=e^A_K^c t and BK(Δt)=(∫0ΔteAKcss)BKcB_K( t)=( _0 te^A_K^csds)B_K^c are smooth in (α,β,Δt)(α,β, t) on the stable region, with AK=I+AKcΔt+O(Δt2)A_K=I+A_K^c t+O( t^2) and BK=ΔtBKc+O(Δt2)B_K= t\,B_K^c+O( t^2). Substituting into (13) and expanding to leading order yields AKcS∞d+S∞d(AKc)⊤+ΔtBKcσ2(BKc)⊤+O(Δt2)=0A_K^cS_∞^d+S_∞^d(A_K^c) + t\,B_K^cσ^2(B_K^c) +O( t^2)=0, so S∞d/Δt→PcS_∞^d/ t→ P^c. Strict inequalities are open under continuous perturbation, so the strict monotonicity of X∞cX_∞^c from Theorem 9 carries over, and the numerical sweep of Fig. 3 confirms it on the experimental gain range at Δt=0.02 t=0.02 s. ∎ Corollary 11 (Explicit failure bound, canonical case). For the canonical scalar second-order system, the failure-bound exponent of Theorem 3 admits the explicit form ℙ(FailT)≤2(T+1)exp(−r2βαΔtLroll(K))+o(Δt),P(Fail_T)≤ 2(T+1) \! (- r^2\,βα\, t\,L_roll(K) )+o( t), (25) where the remainder collects Δt2 t^2 and higher discretization corrections via Proposition 10. The exponent therefore scales as β/αβ/α for fixed validation loss, so increasing damping or decreasing stiffness sharpens the bound. Proof. Proposition 10 gives X∞d/Δt=X∞c(α,β)+O(Δt)X_∞^d/ t=X_∞^c(α,β)+O( t), and Lemma 8 identifies the leading term as σ2α/(2β)σ^2α/(2β). Substituting X∞d=Δtσ2α/(2β)+o(Δt)X_∞^d= t\,σ^2α/(2β)+o( t) into (15) with n=1n=1 and σ2=Lroll(K)σ^2=L_roll(K) yields (25). ∎ For a fixed prediction-error budget, equation (25) depends only on the ratio β/αβ/α. Doubling both gains leaves the bound unchanged, which decouples the gain-tuning question from the absolute scale of the controller. Corollary 12 (Connection to the squared ℋ2H_2 norm). For the canonical second-order system, the closed-form continuous-time stationary variance equals the squared ℋ2H_2 norm of the disturbance-to-position transfer function HK(s)=(α/m)/(s2+(β/m)s+α/m)H_K(s)=(α/m)/(s^2+(β/m)s+α/m) scaled by σ2σ^2, and the amplification index of Theorem 3 converges in the long-horizon limit to limT→∞ΓT(K)=‖K‖ℋ22 _T→∞ _T(K)=\|G_K\|_H_2^2. Minimizing Ψ(K) (K) recovers the classical ℋ2H_2-optimal gain selection rule as the failure-bound minimizer. Proof. The transfer function HK(s)H_K(s) has natural frequency ωn=α/m _n= α/m and damping ratio ζ=β/(2mα)ζ=β/(2 mα). The standard ℋ2H_2 formula gives ‖HK‖ℋ22=(α/m)2/(4ζωn3)=α/(2β)\|H_K\|_H_2^2=(α/m)^2/(4ζ _n^3)=α/(2β), matching Lemma 8 after the noise scaling. The asymptotic identification of ΓT _T is standard for Schur-stable LTI systems [21]. ∎ V Numerical Illustration Figure 2: Monte Carlo position-error envelopes (N=50,000N=50,000 rollouts) for the four gain regimes under exact ZOH discretization. Shaded bands show the 95th and 99th percentiles. The dashed line is the steady-state 95th-percentile bound from Theorem 1. All panels share the same vertical scale. CO yields the tightest envelopes, confirming the predicted ordering. Figure 3: Stationary proxy X∞(K)X_∞(K) (normalized by X∞COX_∞^CO, log scale) over the gain parameter space, computed via exact ZOH discretization at Δt=0.02 t=0.02 s. Markers locate the four canonical regimes. The landscape is monotone increasing in KpK_p and decreasing in KdK_d over the entire stable region, consistent with Theorem 9 and Proposition 10. Figure 4: Empirical failure rate P^(FailT) P(Fail_T) (solid) versus the Theorem 3 upper bound (dashed) as a function of the success-tube radius r for the four canonical regimes. The bound dominates the Monte Carlo curve in every regime, and the regime ordering CO ≺ SO ≈ CU ≺ SU is preserved at all r. We illustrate the framework on the canonical scalar second-order system, verifying sub-Gaussian propagation, the failure bound, and the regime ordering. The closed-loop matrices are obtained by exact ZOH sampling (5), matching Proposition 10. V-A Setup Consider unit mass (m=1m=1 kg) under PD control sampled at Δt=0.02 t=0.02 s (5050 Hz). We evaluate the four canonical gain regimes of (20) with αL=50 _L=50, αH=100 _H=100, βL=20 _L=20, βH=40 _H=40. Action prediction errors are drawn i.i.d. from (0,1)N(0,1), satisfying (6) with ΣKroll=1 _K^roll=1. For each regime we simulate N=50,000N=50,000 independent rollouts of T=50T=50 steps (11 s task horizon), and compute the stationary proxy X∞(K)X_∞(K) from the discrete Lyapunov equation (13). TABLE I: System quantities and empirical failure rates for the scalar canonical system (m=1m=1 kg, Δt=0.02 t=0.02 s, T=50T=50, r=0.3r=0.3). X∞cX_∞^c from (24), X∞dX_∞^d from the ZOH discrete Lyapunov. CO SO CU SU KpK_p 50 100 50 100 KdK_d 40 40 20 20 ρ(AK)ρ(A_K) 0.974 0.948 0.943 0.819 X∞c=σ2Kp/(2Kd)X_∞^c=σ^2K_p/(2K_d) 0.625 1.250 1.250 2.500 X∞dX_∞^d (ZOH discrete) 0.012 0.025 0.025 0.050 X∞d/X∞d,COX_∞^d/X_∞^d,CO 1.0 2.0 2.0 4.0 P^(FailT) P(Fail_T) 0.01 0.26 0.22 0.75 V-B Sub-Gaussian Error Propagation Fig. 2 shows the position-error magnitude ‖et‖\|e_t\| across regimes, with median, 9595th-percentile, and 9999th-percentile envelopes from the Monte Carlo ensemble. The dashed line is the steady-state 9595th-percentile threshold predicted by Theorem 1 via r95=2X∞(K)ln40r_95= 2X_∞(K) 40, derived from (12) with n=1n=1. The empirical 9595th percentile stays inside the predicted bound in every regime, confirming sub-Gaussian propagation. The error magnitudes differ systematically: CO yields steady-state errors roughly half those of SO or CU and four times smaller than SU, despite identical action-error distributions across the four regimes. V-C Failure Bound and Regime Ordering Table I reports both the closed-form continuous-time stationary variance X∞c=σ2α/(2β)X_∞^c=σ^2α/(2β) and the exact discrete ZOH proxy X∞dX_∞^d. Both quantities respect the predicted ordering CO≺SO≈CU≺SUCO , with the near-equality of SO and CU exemplifying the system-dependent SO/CU comparison. The Monte Carlo failure rates corroborate the ordering: at r=0.3r=0.3, CO fails on 1%1\% of rollouts, SO and CU on 26%26\% and 22%22\%, and SU on 75%75\%. Notably CO has a larger spectral radius (0.9740.974) than SU (0.8190.819) yet achieves a fourfold smaller proxy because its injection is proportionally weaker, reflecting the three-way competition between label difficulty, injection strength, and contraction inside Ψ(K) (K). Fig. 3 sweeps X∞(K)X_∞(K) over the (Kp,Kd)(K_p,K_d) parameter space. The landscape is monotone increasing in stiffness and decreasing in damping over the entire sampled region, covering both underdamped and overdamped regimes, with CO at the global minimum and SU at the maximum, consistent with Theorem 9. V-D Test of the Failure Bound Fig. 4 compares the empirical failure rate P^(FailT) P(Fail_T) with the upper bound of Theorem 3 as the success-tube radius r varies. For each regime, the dashed curve plots min1, 2(T+1)exp(−r2/(2ΓT(K)Lroll(K))) \1,\,2(T+1) (-r^2/(2 _T(K)L_roll(K)))\ from Corollary 2 with Lroll=1L_roll=1 and the empirical ΓT(K) _T(K) from (9). The bound dominates the Monte Carlo curve uniformly across regimes and tube radii, with CO retaining the largest gap (well-conditioned tail) and SU sitting closest to its bound, and the relative ordering matches Theorem 7 at every r. The failure-bound exponent is therefore qualitatively correct as a comparator and quantitatively conservative for absolute prediction. The sub-Gaussian assumption is the dominant source of looseness, since LrollL_roll replaces the per-coordinate proxy by its trace and the union over T+1T+1 steps adds an unconditional log-factor. VI Discussion VI-A Gain-Selection Criterion Theorem 3 converts the counterintuitive empirical finding into a decision rule. Rather than minimizing the training or validation loss, practitioners should minimize the product ΓT(K)⋅(L^va(K)+εgen(K,δ)) _T(K)·( L_va(K)+ _gen(K,δ)), equivalently Ψ(K) (K) when shape-preserving structure holds or X∞c(K)X_∞^c(K) for canonical second-order systems. Theorems 7 and 9 recommend the compliant-overdamped regime in agreement with [7]. The paradox that CO gains exhibit higher validation MSE yet better closed-loop performance is resolved inside the product: CO incurs larger label difficulty l(K)l(K) but a proportionally smaller amplification, and the exponent of (16) rewards that trade. A practical workflow follows directly. Start from compliant-overdamped gains, monitor ΓT(K)⋅L^va(K) _T(K)· L_va(K) rather than the training loss alone, and resolve the SO versus CU choice empirically because the ordering is fundamentally system-dependent. VI-B Relation to Classical Analysis Classical compounding-error analysis [22] bounds the performance gap by O(εT2)O( T^2). Our amplification index refines the T2T^2 factor. For Schur-stable systems, ΓT(K)=O(1/(1−ρ(AK))2) _T(K)=O(1/(1-ρ(A_K))^2) remains bounded as T→∞T→∞ and converges to the squared ℋ2H_2 norm limT→∞ΓT(K)=‖K‖ℋ22 _T→∞ _T(K)=\|G_K\|_H_2^2 [21], connecting the analysis to classical robust control. Block et al. [3] quantify how training noise compounds during autoregressive rollout, while our amplification index complements that line by exposing how the controller gains modulate the same pathway through AKA_K and BKB_K. The indeterminacy of the SO versus CU comparison matches the system-dependent observations reported in [7]. VI-C Extensions and Limitations The linearization about the expert trajectory is the main simplification. Nonlinear, contact-rich settings are a natural next step via contraction theory [5]. The independence assumption on ξt _t weakens once covariate shift accumulates, and in that regime the bounds remain valid after replacing ΣKroll _K^roll by an upper bound on the conditional sub-Gaussian proxy. Multi-step action chunks [8, 28] enter the framework unchanged by treating a length-H chunk as a single HmHm-dimensional prediction error and evaluating ΓT(K) _T(K) at the chunk-execution rate. For multi-joint systems with non-uniform gains the matrix-level comparison through X∞(K)X_∞(K) replaces the scalar reduction whenever the shape-preserving structure of Definition 4 fails to hold uniformly across joints. VII Conclusion We have presented a theoretical framework that quantifies how controller gains shape behavior cloning failure through four results: sub-Gaussian error propagation, a closed-loop failure bound that factorizes into a gain-dependent amplification index and the validation loss, a scalar ordering reduction via shape-preserving upper-bound structure, and a global monotonicity theorem for canonical second-order systems covering both underdamped and overdamped regimes. The closed-form continuous-time stationary variance X∞c=σ2Kp/(2Kd)X_∞^c=σ^2K_p/(2K_d) supplies the first rigorous justification for the empirical observation [7] that compliant-overdamped controllers are preferred for BC. Natural extensions include nonlinear contact dynamics, co-optimization of gains with policy architecture, and tightening the discrete-time inheritance bound beyond the leading-order continuous-time limit. Appendix A Asymptotic Form of ΓT _T for the Canonical System For the canonical scalar second-order system under exact ZOH discretization with sampling period Δt t, the closed-loop matrix AKA_K has eigenvalues λ1,2(K)=exp(−β±β2−4mα2mΔt), _1,2(K)= \! ( -β± β^2-4mα2m\, t ), (26) which lie strictly inside the unit disk on the open Hurwitz region α,β>0α,β>0. Bounding the geometric series of operator norms in (9) yields ΓT(K)≤‖BK‖op21−ρ(AK)2(1−ρ(AK)2T), _T(K)≤ \|B_K \|_op^21-ρ(A_K)^2 (1-ρ(A_K)^2T ), (27) where ‖BK‖op2=O(Δt2) \|B_K \|_op^2=O( t^2) and ρ(AK)2=1−βΔt/m+O(Δt2)ρ(A_K)^2=1-β t/m+O( t^2) as Δt→0+ t→ 0^+. Substituting these expansions into (27) and using 1−ρ2T→11-ρ^2T→ 1 in the long-horizon limit gives ΓT(K)∼αΔt/(2β) _T(K) α t/(2β), which reproduces the leading-order failure exponent of Corollary 11 and identifies X∞cX_∞^c as the rate constant of the long-horizon failure tail. Appendix B Multi-Joint Reduction via Definition 4 For an n-joint robot with diagonal mass M=diag(mi)M=diag(m_i) and diagonal gains Kp=diag(αi)K_p=diag( _i), Kd=diag(βi)K_d=diag( _i), the closed-loop dynamics decouple per joint. Choosing W¯:=I2n W:=I_2n and Σ¯:=In :=I_n, the conditions of Definition 4 hold with l(K) l(K) =λmax(ΣKroll), = _ ( _K^roll), b(K) b(K) =maxiαi2Δt24mi2, = _i _i^2\, t^24m_i^2, ρ∗(K) _*(K) =maxie−βiΔt/(2mi). = _i\,e^- _i t/(2m_i). 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