Action Chunking and Data Augmentation Yield Exponential Improvements in Behavior Cloning for Continuous Spaces
Authors: Thomas TCK Zhang
Daniel Pfrommer
Chaoyi Pan
Nikolai Matni
Max Simchowitz
Venue: ICLR 2026 Poster (2026)
Keywords: Imitation learning
compounding errors
distribution shift
control theory
dynamical systems
robotics
action chunking
noise injection
Primary area: learning theory
TL;DR: Imitation learning in continuous spaces is more nuanced than over discrete spaces, and can suffer exponential-in-horizon errors. We distill core mechanisms from popular practices to provably and demonstrably mitigate these errors.

Abstract: This paper presents a theoretical analysis of two of the most impactful interventions in modern learning from demonstration in robotics and continuous control: the practice of *action-chunking* (predicting sequences of actions in open-loop) and *exploratory augmentation* of expert demonstrations. Though recent results show that learning from demonstration, also known as imitation learning (IL), can suffer errors that compound *exponentially* with task horizon in continuous settings, we demonstrate that action chunking and exploratory data collection circumvent exponential compounding errors in different regimes. Our results identify control-theoretic stability as the key mechanism underlying the benefits of these interventions. On the empirical side, we validate our predictions and the role of control-theoretic stability through experimentation on popular robot learning benchmarks. On the theoretical side, we demonstrate that the control-theoretic lens provides fine-grained insights into how compounding error arises, leading to tighter statistical guarantees on imitation learning error when these interventions are applied than previous techniques based on information-theoretic considerations alone.

Published as a conference paper at ICLR 2026
ACTION CHUNKING AND EXPLORATORY DATA COL- LECTION YIELD EXPONENTIAL IMPROVEMENTS IN BEHAVIOR CLONING FOR CONTINUOUS CONTROL
Thomas T. Zhang" Daniel Pfrommer Chaoyi Pan Nikolai Matni" Max Simchowitz
University of Pennsylvania
Massachusetts Institute of Technology Carnegie Mellon University
ABSTRACT
This paper presents theoretical analysis of two of the most impactful interven- tions in modern learning from demonstration in robotics and continuous control: the practice of cation-chunking (predicting sequences of actions open-loop) and exploratory augmentation of expert demonstrations. Though recent results show that learning from demonstration, also known as imitation learning (IL), can suf- fer errors that compound exponentially with task horizon in continuous settings, we demonstrate that action chunking and exploratory data collection circumvent exponential compounding errors in different regimes. Our results identify control- theoretic stability as the key mechanism underlying the benefits of these interven- tions. On the empirical side, we validate predictions and the role of control- theoretic stability through experimentation on popular rohot learning benchmarks. On the theoretical side, we demonstrate that the tohoroc- lens provides fine-grained insights into how compounding error arises, leading to tighter statis- tical guarantees on imitation learning error when these interventions are applied than previous techniques based on information-theomm theeoretic considerations alone.
INTRODUCTION
Imitation learning (IL) is the problem of learning complex behaviors from data labeled with tions from an expert demonstrator policy. This methodology encompasses both some of the earliest examples and most recent state-of-the-arr in control for autonomous robotic systems (Pomerleau, 1988; Ross & Bagnell, 2010; Bojarski et al., 2016; Teng et al., 2023; Zhao et al., 2023). Following the rise of large language models (LLMs), IL has also become increasingly prevalent in settings where an agent predicts discrete tokens, such as words in a sentence, lines in proof, positions on chessboard (Chen et al., 2021). Such methods have also seen adoption in the context of both continuous and discretized-action control of continuous state-space dynamical systems in an autore- gressive fashion.
The recent and dramatic successes of imitation learning in continuous control applications has coin- cided with a range of algorithmic interventions which appear essential to ensure strong performance: 1. the prediction of open-loop sequences, or "chunks" of actions by the control policy, called action- chunking (AC), 2. the careful curation of expert data to be imitated and 3. the adoption of generative neural architectures (c.g. conditional diffusion models (Chi et al., 2023)) learned policies. While the benefits of 3. have been studied broadly, a precise understanding of how action-chunking and curated expert data improve behavior cloning performance remains elusive. Current hypotheses around AC foreground partial observability as the underlying mechanism, de- spite its clear benefits in fully-observable, state-based control (see e.g. Figure 2). Moreover, studies on active data collection, recent and classical, focus on multi-round interactive data collection to witness expert corrections, but do not isolate core mechanisms of how exploratory data can cover susceptibilities of behavior cloning, especially for single or few-shot dataset generation.
parameterizations of
In this work, we provide the first theoretical guarantees justifying the practices of AC and exploratory data augmentation during expert data collection (defined formally below) in the minimal setting of imitation of an expert in state-based continuous-contro problem. Our point of departure is the finding in recent work (Simchowitz et al., 2025) that imitation learning in continuous settings- even those whose dynamics and expert demonstrator appear benign-can be considerably more
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Danaber Trahiing Cinjeteroo
Number T'Tabinng "onjeterko
Figure 1: Visualization of the benefits of action-chunking (Practice 1) and noise-injection (Practice 2). Left: even on synthetic globally EISS (Definition 2.1) dynamics f. frequent feedhack can cause exponential com- pounding error, which action-chunking mitigates. Center: HalfChectah-v5 environment. We see sufficiently large white noise injection yields significant performance improvement. on par with more advanced iterative methods. Right: Humanoid-v5 environment. lterative methods like DAGGER and DART can be suboptimal due to poor learned policy rollouts or aggressive noise-covariance shaping, while naive noise-injection reliably pro- vides the necessary local exploration; error bars omitted for clarity. Experiment details in $F and Appendix G.
challenging than imitation in discrete settings, such as those encountered in language modeling, demonstrating compounding errors can grow exponentially with horizon, as opposed to polynomi- ally (or none) (Foster et al., 2024). As Simchowitz et al. (2025) eliminates the possibility of simple "fix" to the learning procedure, we instead consider how changes to either 1. the policy parameter- ization 2. the data-collection process
circumvent this negative result. We thereby elucidate both the design space of "sound" offline learning methodologies and better understand the success of widely-deployed practices such as action-chunking (Zhao al., 2023) and data-augmennatiinn (Ross et al., 2011; Laskey et al., 2017; Ke et al., 2021; Hu et al., 2025).
Contributions. We provide the first theoretical guarantees in continuous state-action IL for inter- ventions that provably prevent compounding error without iterative expert feedback. Whereas previous work (Ross al., 2011; Laskey al., 2017; Pfrommer al., 2022) require either itera- tive interaction with the expert or knowledge of the underlying system, we establish our results without access such oracles, using near-" "vanilvaalla" behavior cloning. We study two key practices: Practice 1: Setion-Chunking. When the environment is benign, we show the algorithmic modi- fication of action- settionchuuking, i.e., predicting and playing open-loop sequences of actions, mitigates compounding errors without requiring any modification to the expert data (Theorem 1).
Practice 2: Exploratory Data Collection via Expert Injeetinn When the environment is less benign, some alteration of the expert data distribution is necessary. We demonstrate noise- injection, ie., adding noise while executing expert actions, is a simple and practical tool for avoiding compounding errors (Theorem 2).
Surprising Takeaways. While Practice and 2 are reflective of popular practices at the intersection of (reinforcement) learning and control, analysis additionally uncovers phenomena that contrast with the common perspectives of both literatures. In particular:
Rationale for action-chunking, Action chhtiinn has been motivated by both enabling larger policy latency, and encouraging
planning and stronger policy representations (Chi et al., 2023; Liu et al., 2025). We illuminate an orthogonal rationale: action-chunking encourages control- theoretic stability of policies learned, mitigating possibly exponential compounding errors from unstable closed-loop interactions.
Moreover, analysis of expert-noise injection reveals that, from theoretical perspective, adaptive iterative interaction or queries with an expert is not needed.
Sufficiency of Non-Interactive Exploration. Various prior works have approached the compound- ing errors problem by adaptively querying the expert policy at possibly suboptimal states (Ross et al., 2011; Laskey et al., 2017), with the overarching goal of wifnessing how the expert policy recovers from errors. We expose the key directions of (additional) supervision around expert trajectories re- quired to achieve stable imitation, and we attain this supervision through noising expert actions at a fixed, isotropic scale. In doing so, we establish that-unlike what online learning per- spective may suggest- iterative, adaptive interaction with an expert is unnecessary to achieve near-optimal expert imitation.
Finally, our results reveal the inadequacy of existing theoretical tools for describing or mitigating compounding emror in continuous state spaces.
Towards New Notions of "Coverage" Continuous State Space. The Simchowitz et al. (2025) lower bounds shows that standard notions of "coverage" in theoretical reinforcement learning (Jiang
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& Xie, 2024) (which is maximal when learning from expert data) is insufficient for mitigating com- pounding error in continuous behavior cloning. Our work reveals that that novel, stronger notions of coverage realized via noise injection do suffice in continuous state spaces, and lead to guar- antees that are sharper than coverage-based arguments which leverage exploratory data.
Towards New Notions of "Excitation" in Continuous State Space. Despite requiring stronger coverage, our bounds do not require the injected noise to produce "persistent excitation" from con- trol theory (Bai & Sastry, 1985), i.e. uniform variation across all state directions. Rather naive exploration (Simchowitz& & Foster, 2020) via white noise suffices, even when the underlying system is not controllable (Kailath, 1980). This is because the error-directions which are susceptible to compounding error are those along which noise injection (Practice 2) provides supervision.
Related Work. Imitation learning from expert demonstrations has emerged as a dominant tech- nique for learning performant models across many sequential decision-making applications. As such, the compounding error phenomenon is well-documented, dating back even to the introduc- tion of IL (Pomerleau, 1988). In discrete state-action settings, compounding errors appears more benign (Ross& & Bagnell, 2010; Ross et al., 2011), where recent work by Foster et al. (2024) demon- strates that just modifying the loss may result in performance that has no adverse dependence on horizon. However, these settings are ill-suited for continuous control, where the expert policy must be estimated tnformation-theoretiic distances that are not feasible, e.g., even for deterministic policies. A complementary line of work has attempted to understand the theoretical foundations of imitating in continuous settings. Tu et al. (2022) parameterize a scale of "incremental stabil- ity" (see Definition 2.1) and study its impact on the statistical generalization of IL. Pfrommer et al. (2022) proposes sufficient conditions for benign compounding errors in a similar setting. However, the resulting algorithms have exceedingly strong requirements, e.g., stability oracles or Simpt/assate derivative sketching, respectively. Rounding off this line of work, Simchowitz et al. (2025) offers definitive evidence that exponential compounding errors cannot be avoided by altering the learning procedure, motivating the interventions we study. We restate the relevant lower bounds in Theo- rem A. In addition to the works discussed above, we provide extended background in Appendix B.
2 PRELIMINARIES
We consider a discrete-time, continuous state-action control system with states E X Rd and inputs' u, E = Rdd, where dynamics deterministically evolve according to X++1 = [(x,, uz). A deterministic policy π maps histories of states, inputs, and the current time step to a control input u, = TX1x,, 11 t). We assume the initial state is drawn X1 ~ for some distribution D fixed throughout. We say is Markovian and time-invariant invariant if we simply express u, TT Txx)). In this case, define the closed-looy loop dynamics fx((x, u) & f(x, (x) + u), and f'((x) = f" 0). We let E,, (resp. PRx) denote expectation (resp. law) under X1 ~ D, the dynamics f, and inputs selected by the policy म. Given two deterministic policies Ii, T2, let Ez,, denote the expectation of sequences (x.", u (1²) under the dynamics f. coupled so that x; x"" ~ D. We consider estimation of deterministic, Markovian expert policies n² X' l given a problem horizon T. Our aim is to learn some policy ă which accumulates low gquared-trajecton
error:
JTRAIT E.,.*
min (1, ||x - x7 112
1 u, 1P]
(2.1)
Above, the practice of taking a minimum with 1 accounts for the possibility of unbounded trajec- tories rare events; 1 can be replaced by an arbitrary constant. Upper bounds on JTRNLT imply upper bounds on the difference in expected Lipschitz costs, see e.g., Appendix D. Let Pdemo denote probability distribution over demonstrations (x,, Uz) u))SSITT We set P demmo = PT. in Practice 1, but consider modifications beyond the expert distribution for Practice 2. For a given candidate imitator policy म, we may define the population-level risk:
JDEMO, ( Pdeeo)) Er Eyaeano
Σ (xa-11) 1
(2.2)
We broadly term J the on-expert error. i.c. the generalization error of मे imitating over PP Pdlmo- This can be interchanged with an error scaling law from standard supervised learning, e.g. given n independent training trajectories from IP Peemoo JDEMO, T (ft; IP PPdmmo) S n a E (0. 1). Such
We refer inputs and actions interchangeably.
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bounds can realized by standard learning algorithms such empirical risk minimization (ERM), ie. behavior cloning, but are not restricted to it; any algorithm that seeks to perfectly match the expert will drive low on-expert eπror Uft P. ). We defer discussions of supervised learning formalisms to Appendix B. Gauging how well various algorithmic interventions (or lack thereof) mitigate compounding errors therefore translates to bounding the trajectory error JTRAIT in terms of the on-expert error JDEMO,T.
The Compounding Errors problem. We now formally describe the compounding errors problem.
Let alg be (possibly randomized) mapping from a sample of n trajectories S, iid Pdemo to an imitator policy î ~ alg(S.). The problem instance suffers exponential compounding errors if:
Ea,s, [JTRAI,((F)] Z CT Es,,,, [JDEMO, P Pdemo)l:
(2.3)
for some C> 1. In other words, imitating via empirical risk minimization distribution Pdemo leads to learned policies Ωt that suffer exponentially more trajectory error rolled out in closed-loop compared their on-expert regression error. As proposed in prior work (Tu al., 2022; Pfrommer et al., 2022), compounding error can be understood through the lens of control- theoretic stability, which describes the sensitivity of the dynamics input. By the primer key control-theorete concepts in Appendix C. We consider notion of incremental stability (Angeli, 2002; Tran et al., 2016).
given demonstration
perturbations of the state or
nature of the ensuing definitions and analysis, we provide a concise
Definition 2.1 (EISS, Figure 5). A system X = [xx, uz) is (Ciss, p)- expponeentlllly incremen- tally input-to- to-uttto-state stable (EISS) if for all pairs of initial conditions (X1.X) x1) and input sequences (furhe2, (u/),,), there exist constants Cass 2 1, p (0, such that for any 2 1:
x x S Cissp
-|x] -
x11l + Ciss
Σ pp--e
Epl
u/ell.
t 2
k=1
We say a policy-dynamics pair (T,, f) is (Ciss: P)-EISS if the induced closed- looed-loop dynamics f is (Ciss P)SSS We also denote the shorthands Cstah A CLss I-p" Cstab
A
Cytab
In other words, incremental stability ensures that bounded input perturbations lead to bounded future state deviations, with their effect decaying in time. Thus, incremental stability can be viewed as a continuous-comtrol analog to notions of "recoverabilityy (Ross et al., 2011; Foster et al., 2024). Henceforth, we will refer to "stability" and EISS interchangeably. Particularly relevant to Section 3, we note that "open-loop stable" dynamics (i.e., satisfying EISS even in the absence of feedback policies) are salient in various robotic applications via low-level controlles..
Incremental Stability as a Natural Abstraction for End-Effector Control. End-effector con- trol outputs desired motion relative to position, which are then tracked by low-level high-frrquency controllers, such as proportional-derin -errvvattvee (PD) controllers. Hence, the presence of low-level tracking controller renders the closed-loop between the desired position and system state in- crementally stable (c.f. Block et al. (2024)). In other words, end-effector control renders imitation of, e.g. position commands as taking place in an open-loop stable dynamical system.
Our base assumption across both Section 3 and Section is that the expert-induced closed-loop system fz is incrementally stable. This formalizes a notion of expert robustness by implying the expert eventually recover from bounded input perturbations. As strong this may seem, EISS of the expert does not assume away the compounding errors issue (see Theorem A); for example, the candidate policy a destabilizes the system, the resulting "input perturbations" to f are exponentially growing.
Theorem A (Motivating lower bounds, Informal vers. of Simehowitz et al. (2025, Theorems 1 & 4)). There exists families Petab and Punst of policies and dynamics such that:
(i) For every (T,9) E Pstab is open-loop EISS and (z,g) is closed-loop EISS, and n, 9. are Lips- chitz and smooth. However, any learning algorithm which returns smooth, Lipschitz, Markovian policies with state inde-independent stochasticity must suffer exponential-in-T compounding error (2.3) when learning from n expert trajectories from some (T,, g) E Pstab-
2 We note traditional definitions of nonlinear stability may track separate 3, for the transient bound [[]|x
||, f) and the input gain >(Ilu- - u1') (l) It suffices for our purposes to combine these under Ciss for clarity.
'The stability definition and ensuing results can be loosened to polynomial decay or local variants with appropriate modifications, though we note the lower-bound constructions in Theorem A are EISS systems.
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(ii) For every (T.g) Punst. (T.g) is closed-loop EISS but g need not be open-loop EISS, and n. are Lipschitz and smooth. However, any learning algorithm, without restriction, suffers exponential-in-T" compounding error (2.3) when learning from n expert trajectories on some (T,g) E Pstab-
The bounds ensure that for at least one instance (T,g) in Pstab (resp. Punst), the learner suffers exponential-in-7 compounding error if that instance (T, is the ground truth, and the learner re- ceives n expert demonstrations from that instance. Our results constitute positive converses:
When g is open-loop EISS, the chunked policies prescribed by Practice are smooth and Lips- chitz but non-Markovian (by virtue of being chunked), and thus bypass Theorem A.(i)
If g is not necessarily EISS, Practice 2 alters the distribution over expert demonstrations to circumvent Theorem (i)). Note that Theorem (iii precludes any purely algorithmic changes that do not change the distribution ,,, over expert demonstrations.
We note Simchowitz et al. (2025) also provide stylized policies which bypass the particular con- struction used in Theorem A.(i); Practice serves natural, generally applicable solution, and Practice 2 circumvents the more challenging setting that even these stylized policies cannot address.
Additional Notation. Blue (e.g. πT" indicates expert-induced quantities, and red indicates quan- tities induced by learned policy (e.g. f). Positive semi-definite matrices are indicated by Q > O, and the corresponding partial order = Q = (P (P-Q) Q) Z O. We use S, = omit universal con- stants. In the main body, we also use (-) to omit polynomial dependence on instance-dependen constants, but not
constants or horizon T', e.g. T-p
This o² = O, (To.).
3
SUFFICES IN OPEN-LOOP STABLE SYSTEMS
Action-chunking is popular practice in modern sequential modeling pipelines, where policy predicts a sequence of actions, of which some number are played in open-loop (Chen et al., 2021; Chi et al., 2023; Shafiullah al., 2022). There are various intuitions of the practical benefits of action-chunking, ranging from: 1. robustness to non-Markovian partial observability quirks in the data (Liu et al., 2025), 2. amenability learning via multi-step prediction, and 4. simulating receding-horte horizon control. Yet, we show that even in control settings with unimodal, Markovian, ftete-fedbucck experts, action-chunking serves a critical role in subverting exponential compounding errors. All proofs and extended details in this section are contained in Appendix D. We may conveniently describe chunking as follows.
prediction, improved representation
Definition 3.1 (Chunking Policy). A chunking policy specified by a chunk-length L, and mappings chunk, [T] X - it i E [C] such that, for k E Zza and E [e] and = kÉ+i for, π (X1:, u1:t-1,t) = chunke(x)).
chunk, ( (x&e+1, i). We also write chunk[z]
(chunk, (s)) For simplicity, we
always assume e divides T - 1.
Practice (Learning over Chunked Policies). We sample Sn as denote i.i.d. trajectories drawn from the expert distribution ,.... We aim to find सं from a class of length-é chunked policies, llehunk és defined formally in Definition 3.2 that attains low on-expert error, e.g., by empirical risk minimization. We note that for chunked policies.
1/c
JEMOo T (#;Px.) Ez,,
Σ
I| [1i+(k1))eNt - chunk (xx)
(3.1)
We now formally define the policies induced by chunking with dynamics model.
Definition 3.2 (Induced Chunking Policy). Let g : X the true dynamics f), and क X' - U a Markovian, deterministic policy. Given chunk length E E N, we define the induced chunked policy म chunked g. f). मे X' (2) as returning
U be a dynamics map (possibly not
chunk(F](x) = (x(x), 7 (0 (x)). ((())) ... (()))((((x))
(3.2)
where above (x) A g(x, (x)), and ( is understood repeated composition.
In other words, chunked(T, (T. g, E) returns policy that, conditioning on the current state, outputs the next actions given by simulating on dynamics y in closed-loop. We state the core assumptions.
'In the sense of distribution having multiple modes.
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Thil:: tool hang
Pod befon Iectens
Fedetox
04
Zont
Pad befion 26
02
02
64
Length
Leugh
Figure 2: Success rates as function of evaluated action-chunk lengths the challenging robomimic "tool hang" environment with full-state observations. Left:: Each line corresoonds to a model trained given prediction horizon on 100 expert trajectories. Each point corresponds to the model evaluating given chunk length ranging from receding-horiaz (transient) effect, evaluating slightly longer chunks improves success drastically. Right: We repeat similar set-up with 50 expert training trajectories. We see that noise-injection (Practice can also synergize in this open-loop stable setting (see Appendix F), though requires modifying the data-collecting procedure rather than simply adjusting policy parameterization and evaluation as in AC.
(E 1) to the full chunk. While prediction horizon has some
Assumption 3.1 (Regularity and Stability). We make the following assumptions: 1. the true dynam- ies f are (Ciss P)-EISS in open-loop. without loss of generality with p 2 1/e, all base policies E U {Tr) } in consideration are L, -Liischcii:: [|/T(x) (x')ll s La ||x x
All ensuing results stem from the following key result.
Proposition 3.1. Let Assumption 3.I hold. Let (ft, f) be a policy-dynamics pair that is (Ciss, P)- EISS, and consider the corresponding chunked policy में = chunked f f.l). Then the closed- loop system the chunked policy induces on the true dynamics (ā, is (C, P)-ESS, where Ö log(1/p) poly(Lo (L,, Ciss) and = p2/22, as long the chunk length is sufficiently long: 8>
log(1 (1p)) -1. log(poly(Ler Ciss)).
Therefore, combining Proposition 3.1 and Proposition 3.2 leads to the following compounding error guarantee on any sufficiently chunked policy. This result states that: as long as a policy "believes" stabilizes the simulated dynamics at hand, then it is guaranteed to be stable on the actual dynamics chunked accordingly. For notational simplicity, we set the prediction horizon and the ex- ecuted chunk length (i.e., how many predicted actions played before re-predicting) the same f. However, we note that the requirement > log(1/p) (1 logg(poly Ciss)) is on the executed chunk length: playing a chunked policy म = chunked ft, f, e) in Markovian (i.e., grreding-horizon fashion does not subvert Thcorem (i).. Crucially, without action-chunking open-loop stability of the nominal dynamics f and closed-loop stability of the expert (T* f) does not imply closed-loop stability of (f, f) for the learned policy without action chunking é = 1. Contrast this to Propo- sition 3.1, which depends only on the stability properties of the true system f and the closed-loop simulated system (#, f), and requires assumption on the closeness of f. or â to any reference policy. Define the class of possible policy-dynamics pairs 2 {(TT( g) (π, g) is (Ciss: p)-EISS), and the induced length-f chunked policy class: Iehunnk - { chunked (TT, g, (T.g) E P). A key consequence of chunked policies inducing stable closed-loop dynamics is that they induce limited compounding error.
Proposition 3.2. Let Assumption 3.1 hold. Let ã chunked ( f, E Ilehunn,6, and assume ().f). (Fr, f) are EEESS. Then, the following bound holds:
JIRAST (+) S poly (I (Ix, - T
Theorem Let Assumption 3.1 hold. For sufficiently long chunk-length > log(1/p) log(poly(Ln, Cras)), let गे chunked [f.g. g, H.hhnn..e We have the trajectory-error bound:
JTRALT (7) < O+(1) JDEMO, ((ū; P..).
Theorem implies that when the ambient dynamics f are EISS, then a sufficiently chunked imitator policy will accrue limited compounding emoors horizon-free- errrioon-rree-rllative relative to the on-expert error it sees. particular, given से attaining regression generalization error (f;; implies (+) & O+)))) (1))n=5 To summarize the key takeaways of the theoretical results:
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Key Findings
Theorem I implies that executing chunks of actions in open-loop is key: multi-step prediction
alone cannot subvert Theorem A.(i) if the policy is only played in greeding-horizon fashion. Requisite chunk lengths are small, scaling logarithmically with system-depender constants, and longer chunk lengths beyond that point provide marginal benefit (and clashes with practical concerns of prolonged open-loop execution).
Action-chunking is demonstrably erucial in state-based, deterministic control, revealing its key role in IL independent of non-Markovianity or partial observability of demonstrations,
.
However, action chunking is not silver bullet: it relies crucially on stability of the open-loop system, which typically requires end-effector control to ensure in robot manipulation setups.
4 NOISE INJECTION MITIGATES COMPOUNDING ERROR UNDER SMOOTH, UNSTABLE DYNAMICS
We now consider the difficult setting where the ambient dynamics f may not be open-loop stable. In this case, purely algorithmic interventions like action-chunking are generally insufficient, as er- roneous actions quickly lead to unstable behavior. In fact, recall Theorem A states that no algorithm, even permitting stochastic and non-Markovian policies, can circumvent exponential compounding errors in the worst-case, provided only data from the expert-induceed law Pa.. This necessitates altering the demonstration distribution IP Pdemoo beyond the expert's xpx., i.e., some form of additional exploratory data collection is required. GER (Ross et al., 2011) and DART (Laskey et al., 20 can be summarized as attempting to witness how the expert recovers from errors, where the former queries the expert along learned-policy roll- similar to the approach we propose. Exploratory Data Collection. However, beyond the motivating intuition, still lack fine- grained insights into what kinds of recovery or policy errors need to be witnessed to circumvent compound- ing errors, if even possible. Furthermore, these works require iterative rounds of expert data collec- tion based on learned policy statistics. Our point of departure is the following: if we are tracking the expert sufficiently closely, we should only need witness how the expert policy recovers near the expert distribution. To this end, we consider arguably the simplest approach to inducing local ex- ploration in the expert dataset: noise injection. In the discussion below, we fix noise level ou >> O, which controls the magnitude of the noise added, and mixture fraction E [0, 1], that controls the proportion of trajectories collected without noise injection.
particular, prior approaches such as DAG-
outs, and the latter injects policy-shaped noise into the expert-similar
Definition 4.1. We define the expert distribution under noise injection as the distribution P Fu over trajectories (R,, ú,)(21 with x1 D, and ú, " (X,,; &(+1 = f(x,, ūn + ouZi) for 1 2 1, where 2,
L.d (Bd (1)) is drawn uniformly over the unit ball.5
In other words, noise injection collects trajectories induced when the expert's 's commanded actions are executed with additive noise duZ,. We then consider fitting a policy ι by augmenting standard (un-noised) expert trajectories with noise-injected ones.
Practice (Exploratory Data Collection via Expert Noise Injection). For the noise-injected dis- tribution Prr.... defined above, provide a sample Sn,a.a of (xx") ( where for 1 S i < |an] the trajectories are i.i.d. from P,,,, and the remaining trajectories are drawn i...d from Par.o.- Define the corresponding mixture distribution Px* ,0 ou.a then find स that attains low J (f; Pa a. c.g., by empirical risk minimization.
2 a pr. (1-aa P sx.. We
Notably, Practice 2 only collects data once before fitting the policy म. and thus does not depend on learned policy rollouts. We now lay out the core regularity assumptions this section.
Assumption 4.1 (Regularity and Stability). Recall that function h Rd + IR C-smooth if for all x, E Rdd VVz((x) V=h(x')lz < C||x x'Il. We make the following assumptions: 1. the expert policy and true dynamics (m*,f) are (Cz, Creg)-smooth respectively. 2. all policies E IT {x²* are Lipschitt 3. the closed-loop system induced by (r*, is (Ciss, EISS.
To understand the exploratory role of noise-injection, we gather intuition through linearizations.
*Our results hold for generic bounded noise, but it suffices to consider z Unif(B (1)) or Unif($ (1)).

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Analysis via Linearizations. Our analysis of Practice uses smoothness of the dynamics and pol- icy to reason about its local linear approximation to the dynamical system along a given trajectory, called the Jacobian linearization.
Definition 4.2 (Jacobian Linearization). For a fixed initial condition xi" ~ D, we define the Jaco-
bian Linearization of the expert trajectory by setting x = ff (x] ), uy = x (x,"" ), and define
a linear varying system determined by the transition matrices: A, [x] .u, ) B.
Vn/(x !u?" ), as well as the local linearization of the controller KY" = Vx (x,")
For smooth dynamical system, consider perturbed trajectory x, given by k = f(x,, ū.), X1
xf + 8x1, and u, n" (xe) + ōu,- Then, the linearization is such that the trajectory differences 5x, = x x satisfy up to first-order: åx(+1 R (A, + B,K," )ox, + B, Therefore, for sufficiently small perturbations ~u, the evolution of the trajectory difference is primarily determined by the linear transition matrices derived from linearizations along the clean expert trajectory. We now introduce a measure of how "sensitive" the closed-loop dynamics is around the expert trajectory.
Definition 4.3 (Linearized Controllability Gramian). The 1-step controllability Gramian is defined 1:z (x7) _ - Agl 14 B,B, TA s+ where we define closed-loop transfer matrix as A s2t = (A,-+ B, K( + B, B,-2K(-2) ... + B,K,")
as: WH
The linear controllability Gramian can be interpreted as capturing the sensitive directions of the closed-loop dynamics to perturbations (see Appendix C). In particular, Wi:r (xT") is the (linearized) covariance matrix of the trajectory difference ox, under uniform stochastic perturbations (e.g., ou (0, I)). Therefore, directions corresponding large eigenvalues of WW. (xT") correspond to axes of 8x; that are most magnified under perturbation, and small (or zero) eigendirections correspond to those that naturally dissipate (or are unreachable). Therefore, under mean-zero, -o-ovaiia noise- injection ou, the local excitation (i.e. exploration) around the expert state x is approximated by: E[ōx(6x] xx'' x o² o' (xf ). Though the Gramian provides notion of local exploration, fully realizing its benefits requires certain crucial subtleties not captured in prior literature.
Suboptimal Approaches. We now remark on subtle but important features of Practice 2.
Actions under P,, .o, are executed noisily, but recorded action labels are noiseless ii x=' (X,), preventing additional regression error. This may run counter to RL theory, where noising the policy, e.g. ü, ~ (x== III may be desirable to induce coverage (Jiang & Xie, 2024).
Only a proportion (1 ax) of trajectories are noise-injected; the rest are clean expert trajectories. We relegate a detailed description of standard RL and control-theoretic perspectives (and their de- ficiencies) to Appendix 1. In either case, i.e. if noisy policy is adopted, or only noise-injected trajectories are collected, we encounter fundamental problem. Due to the non- vineaiiy of the dy- namics, the noised actions induce a trajectory drift compared to the nominal noiseless expert. This drift means policies fitted on the noisy trajectories, even with clean action labels P.. .ou necessarily acerue an additive trajectory error scaling with ou. regardless of the on-expert regression error.
Proposition 4.1 (Drift lower bound, informal). For any given ou > O and Ca > O, there ex- ists a pair of two -mosthh policies Ti, 72 such that one trajectory from the rollout distribu- tion under each can distinguish them perfectly, but given trajectories with o² noino injection, any learning algorithm on n. trajectories sampled under either Tı, n2 will yield a policy x that incurs ( O(C2ot) trajectory error with probability 2 1- n
The formal statement and set-up of Proposition 4.1 is found in Appendix E.7. We notice that this bound scales with Cr, indicating that smoothness is key quantity in any argument based on noising. A consequence of an additive drift is that it suggests an "optimal" choice of ou > 0 is miniscule: J (f) Z poly(a,, " (f;; IP ) + poly(au) implies of z poly(J (ft; P Peemo)) which we will see a suboptimal scaling in theory and empirically.
As for a corresponding upper bound, let us first entertain the implications of the too-strong as- sumption of one-step controllability. where WT., (xT') M AwIL, Aw > 0, for all t 2 2, such that under an appropriate input sequence, the (linearized) expert system fa can reach any state at any time. Therefore, noise-injection will excite all modes of the linearized system, translating to persistency-of-exciaatii (PE) (Bai & Sastry, 1985) as traditional control theory would desire (see Appendix E. This yields the following (suboptimal) bound when imitating over Par..au.
Suboptimal Proposition 4.2. Let Assumption 4.1 hold, and let WIL,( (xF')) ł Aw t 2 2 w.p. 1 over xi" D for some Aw > 0. Let से be a Cm-smooth candidate policy. For oî that satisfies
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Naxber
Figure 3: Features of Practice 2 exhibited on HalfCheetah--S. Left: we compare collecting clean action labels as in Practice 2, versus the noised ones as in coverage-based approach. We note ou = corresponds to sizable entry-wise input perturbations z 0.4 an action space of []1,1]" Imitating with noisy lahels is therefore catastrophic, yet using clean labels achieves improved performance. Center: Fixing du 0.5, vary the proportion of clean expert trajectories a E [0. 1]. The performance difference marginal past a sufficient number of noised trajectories; see Eq. (4.1). Right: naively action-chunking (Practice 1) is disastrous due to
open-loop instability; compare to Figure 2. Experiment details in Appendix G.
à S (poly(1//C 1/Creg)) Aw: have:
JTRALT((F) $ (T) Aw
JDEMO (f; (f; + C2 Ctaaba
The full statement and proof can be found in Appendix E.3. Though this bound avoids exponential- in-7' compounding trajectory error, it has several shortcomings. Besides the strictness of one-step controllability controllability at all (see Appendix C) the bound suffers: drift term that scales as o. which is even worse than Proposition 4.1 suggests, 2. the requirement on ou and result- ing bound scaling with Aw. which is miniscule for Gramians with fast-decaying spectra. As such, combination of algorithmic (e.g., Pat,on + Fataa and analytical innovations are required to advance the result.
4.1 A SHARP ANALYSIS OF EXPLORATORY DATA: EXCITING THE UNSTABLE DIRECTIONS
In light of Suboptimal Proposition 4.2, we make few key observations. Firstly, compounding errors are not arbitrary state perturbations: they result from policy errors, and thus enter the state via the input channels. For smooth systems, this implies the trajectory error is primarily contained in the controllable subspace range( W (::)) However, nonlinearity in the dynamics f and policies म, n means error will leak outside of range( (W:)), which would seem to require PE, i.e. full-dimensional coverage. to detect. Our first key insight that as long subspace, the nonlinear error automatically regulates itself.
enforce low error the controllable
Proposition 4.3. Let Assumption 4.1 hold, and assume the candidate policy ft is Cx-smooth. Fix xf x X1, and define R.¹' & range W.:)). Thenfor any given [0, 11, T N. as long
11 - n (xx +ew + + S Catabe,
max 131T
sup WIWE
we are guaranteed maxigisT |x]
x S.
This result shows that if we ensure the "generic" error v term scales as e², Lipschitzness of ît, n n² is o(Cstab)) for small enough For smooth systems
automatically ensures its contribution the nonlinear error is indeed higher-order. However, it remains to control the first-order error term ew lying in R. where the bound in Suboptimal Proposition 4.2 incurs dependence on the smallest (positive) eigenvalues of W l:t This is unintuitive: small eigendirections of Wu those that hard to excite, such that emrors should compound slowly on them. Thus, there should exist a certain O, (Catab) threshold below which we may ignore them. Restricting our attention to excitable directions means we only pay the statistical cost for the level of excitation we need.
Proposition 4.4. Let Assumption 4.1 hold. For x ~ D, let {(Xi,, Vi be the eigenvalues and vectors of Wu l: & 2 2. Define R[((A) A span(Vi, : dE, 2 A} and P PRo"(X) the corresponding orthogonal projection. Recall Pat Jaua and set 0.5. Then, for on S Ox)),, we have:
du Tu |l - (x,))
doo? A
Ep,. 1PR7 Vx xx² 11,p S
C204
This is precisely where our algorithmic prescription arises: Proposition 4.4 suggests that certify- ing the learned policy â matches x² up first-order Ry ({(()) requires data both at x" and

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around it (e.g. via noise-injection). This translates to imitating on the mixture distribution Px aa- Therefore, combining Proposition 4.3, which translates imitating т well in a neighborhood to low trajectory error, with Proposition 4.4, which guarantees imitating I x matches x x²) up to a flexible excitation level, leads to our main guarantee of Practice 2.
Theorem 2. Let Assumption 4.1 hold. Let ft be L -ipischht,, C=-smooth policy. Then, for ou S
O. (poly(1/Cr: 1 //Cre)) = O,(1), we have:
JTRNIT S (T) ( JDEMO, (f;P (f;Px*,...0.5). ....0.)).
In particular, sefting ou O+(1), we have: (R) & (+) JDEMO, (f;; PPx x .m....s)
Notably, by regressing the mixture distribution au ou,a. we are able set ou large as smooth- ness permits, rather than trading off with the regression eπrrr J in Suboptimal Proposition 4.2. We note that a detailed analysis in fact reveals:
( (1) JDEMO, (f; P.. T
Σ PP au [l[( =*) -
(4.1)
In other words, the trajectory error can be bounded un-moised on-expert error and a sum over "error events" on the mixture expert distribution. 6 On the other hand, if mild moment-equiivalence conditions such hypercontractivity (Wain- wright, 2019; Ziemann & Tu, 2022) hold on the estimation eгror, then the dependence on both T and ou can be attached to higher-order factors, e.g., (T S (1) JDEMO,T(f;;PP In particular, this would imply the impact of T and on vanish when J (ft P.. is sufficiently small (i.e. when is large), c.g.. J JDEM (ft; P .a..)) (ft & O. (o /T) implies J x (1) JDEMo, (ft: .o..a) As consequence, this reveals that we may only need "sufficiently many" noise- injected trajectories to ensure stable closed-loop behav- ior (see e.g. Figure center) that scales horizon-free. We direct detailed derivations and discussion
term scaling horizon-free with the
+
O. (T/oa) J IP Pr
to Appendix E. To summarize the key takeaways of this section:
Key Findings
Proposition 4.3 and Proposition 4.4 isolate that the key role of noise-injection is ensuring the first-order policy error on the controllable subspace R is detectable.
Proposition 4.4 further shows we only require supervision on the excitable subspace Rे (A) therein. In particular, we bypass stringent requirements of RL-theoretic coverage or control- theoretic PE.
Simple white noise suffices to explore (A), as the most excitable (fastest compounding error) directions receive the most supervision.
Imitating mixture of clean and noise-injected trajectories bypasses additive error scaling with ou (see Proposition 4.1). This further larger noise- implics du > 0 can be beneficial.
5 DISCUSSION AND LIMITATIONS
Our action-chunking guarantees rely structural assumption of (ft, f) P being an EISS pair. We believe either explicitly enforcing this, e.g., via regularization (Sindhwani et al., 2018; Mehta et al., 2025) or hierarchy (Matni et al., 2024), attaining it indirectly via implicit biases (Chi al., 2023), are interesting directions of inquiry. We assume smoothness in Section 4, which is not strictly satisfied in some applications, such as in model-predictive control (Garcia et al., 1989). We remark our lower bound Proposition 4.1 depends on smoothness in C., which implies it is in some sense fundamental aspect of noise-injection. However, we believe our results should extend to piece- wise notions (Block et al., 2023), and note ongoing research exploring smoothing for learning in dynamical systems (Suh et al., 2022: Pang et al., 2023; Pfrommer et al., 2024). In general, we leave a sharp characterization of the role of smoothness and control-theoretic quantities in IL as an open problem. We also note though our theory suggests isotropic noise injection suffices, this may not be desirable in some practical contexts, such highly dexterous robotics. In light of our findings elucidating the precise role of local exploration, we leave designing robust practical recipes for perturbative data collection as future inquiry. Lastly, we leave investigating the marginal benefit of iterative interaction (Ross et al., 2011; Laskey et al., 2017: Kelly et al., 2019; 9; Hu et al., 2025) as future work.
Note applying Markov's inequality to the second recovers Theorem 2.
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A
SUPPLEMENTARY FIGURES
Open Loop
Closed Loop
Controller x(x,)
Dynamies Rx 4)
Ax+1
14
4
Dynamics fi, 1b)
Controller x
Figure 4: A comparison of open-loop control, where the policy generates actions without accessing the system state, and closed-loop control, where the policy's 's generated actions condition on the system state. While action- chunks are generated closed-loop, the actions within chunk exccuted "open-loop."
Initialization
difference decays with rate
Xio
A1o
Without persistent error, trajectories converge pairwise p
x
Any 15
With persistent input error magnitude r, trajectories remain most 1-p apart
Figure S: A visualization of EISS (Definition 2.1), which guarantees pairwise contraction of trajectories.
Action Chunks
Trajectory
Open-Loop Stability Modulus
Figure We visualize the stabilizing effect of using multiple action chunks (shown in different colors) when evaluating a "chunked" policy (with corresponding trajectory shown in black). As the open-loop dynamics on each chunk is stabilizing, this ensures closed- looo--oooss------- FISS of the resulting learned policy over multiple chunks.
B ADDITIONAL DISCUSSION
Extended Related Work. Imitation learning from expert demonstrations has emerged as a dom- inant technique for learning performant models across applications such as: self-driving vehicles (Hussein et al., 2017: Bojarski et al., 2016; Bansal al., 2018), visuomotor policies (Finn et al., 2017: Zhang et al., 2018), and large-scale robotic decision-making models (Zitkovich et al., 2023; Black et al., 2024). As such, the compounding error phenomenon is well-documented,, dating back the introduction of IL (Pomerleau, 1988).
In discrete state-action settings, the seminal work in Ross & Bagnell (2010); Ross et al. (2011) propose an iterative, interactive procedure to collect examples of corrective data, seeing widespread adoption (Kelly et al., 2019; Sun et al., 2023). On the theoretical side, compounding errors appears more benign in discrete settings, with naive behavior cloning (BC) attaining discrepancy between
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Noise Injection on the Excitable Manifold
Linearization errors and off manifold excitation naturally dissipates
EISS-stability prevents compounding
Noise injection explores constant-radius
around training data
Figure 7: The effect of noise injection for controllable versus uncontrollable subspaces. We illustrate the key advantage of Proposition 4.3, namely, that noise injection occurs primarily in more excitable directions. By leveraging this mechanism, tion 4.3)
able derive better rates (Suboptimal Proposition 4.2 Proposi-
training and execution enror at most quadratic in the horizon. Recent work by Foster et al. (2024) even demonstrates that modifying the loss may result in performance that has no adverse dependence on horizon. However, these works operate in settings ill-suited for continuous control, where the expert policy must be estimated information thcooretic distances that are not feasible, c.g., even for deterministic policies in continuous action spaces.
Accordingly, prior work which applies to continuous control settings has involved more elabo- rate set-ups enable stable performance. For example, recent advances in generative policies typically paired with action-chunked execution (Practice 1), see e.g., (Chen et al., 2021; Shafiullah al., 2022; Chi al., 2023; Zhao & Grover, 2023; Liu al., 2025). Other works have considered tools from robust control (Hertneck et al., 2018; Yin et al., 2021) and stability regularization (Sind- hwani al., 2018; Mehta al., 2025) promote stability around observed data. Lastly, various works have proposed different forms of data augmentation as way to promote robustness to distri- bution shift, including iteratively shaped noise injection during expert demonstrations (Laskey al., 2017), and noising observed states/actions (Ke et al., 2021; 2024; Block et al., 2024). Our proposed Practice can be viewed as a distilled, non-iterative version DART (Laskey al., 2017).
A complementary line of work has attempted to understand the theoretical foundations of imitat- (2022) parameterize a scale of "incremental stability" (see
ing in continuous settings. Definition 2.1) and study its impact the statistical generalization of IL. Pfrommer al. (2022) proposes sufficient conditions for benign compounding errors in a similar setting. However, the re- sulting algorithms have exceedingly strong requirements, e.g., stability oracles Oinput /aaate deriva- tive sketching, respectively. Rounding off this line work, Simchowitz et al. (2025) offers definitive evidence that exponential compounding errors cannot be avoided by altering the learning procedure, motivating the interventions we study.
Supervised Learning Preliminaries. Given demonstration distribution over trajectories Pdemos consider a sample S, of n i.i.d. trajectories (x() (+) of candidate policy à over this sample defined as:
) from Pdemo- The empirical risk
JEMP (f; Sw)
| (x 1
u (0)
(B.1)
where the form of के (x (x) u -1:-1: t) depends on policy parameterization (e.g. Markovian or chunked (3.1)). Notably, Es,, [JEM (#; S.)] JDEMO, Pdemoo. Our work is independent of how A is derived, as we are concerned only with how the on-expert error (i.e. imitation generalization error) J T (t; )demo) translates to closed-lood trajectory error ( However, to conceptual- ize how J JDMMO,T( (f: scales in terms of n, we may consider the quintessential Empirical Risk Minimization (ERM) algorithm:
(6)
arg min, JEMP, T (#;; Sn); notably, this corresponds to the ob-
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jective of vanilla behavior cloning. Since J (it; Fdemo) is simply the expected error over the same distribution Pdemo that generated the data, one can apply standard supervised learning bounds for ERM, for example, JDEMO, T (ft; IP demo) S n= corresponding to parametric/ fast" rates and JDEMO, ( demoo) S n n=a a E (0, 1) corresponding to "non-parametrie" scaling, see e.g. Bartlett & Mendelson (2002); Shalev-Shwand ences, and Tu et al. (2022); Pfrommer et al. (2022); Simchowitz et al. (2025) for discussion specific imitation learning.
& Ben-Davvd (2014); Wainwright (2019) for standard refer-
We further note that the above proxies for the scaling of JDEMO, ( IP Pdemo) ignore the axis of tra- jectory horizon T; for long trajectories that experience varying degrees of stationarty/ergodicitty JDEMo.T (f;; ( ) plausibly also improves with ''' despite the temporal dependence within a trajec- tory. Various tearning-theoretic works linear and nonlinear control demonstrate this formally un- der various dependence structures, see e.g. (Simchowitz et al., 2018: Ziemann & Tu, 2022; Ziemann et al., 2024) for discussion surrounding the related problem of system identification, and (Zhang et al., 2023) for specification to linear imitation learning. As these works only affect the theoretical scaling of J (f; IP dedmo), our analysis is independent of these rearning-theorete discussions.
As final remark, we mention that action-chunking by definition induces different policy class Iehunnk than the class of "base" Markovian policies II: for example, given an input state, the for- mer outputs f Fcttonn rather than one. We surmise the additional statistical burden of predicting with a chunked policy class is negligible especially considering it mitigates exponential compounding tra- jectory error. Firstly, Theorem I implies the requisite chunk length is small (logarithmic in system parameters), so even if Lehunkk. is treated naively as an 4-step predictor class X' - u the differ- ence from inflating the output dimension between Lehunkk and II is similarly small. Furthermore, I is not a naive f-seep predictor (though it often suffices to implement action-chunking as such practice; see Appendix G), as constrains the output to be rolled out through the candidate dynamics. As such, this constraint intuitively should reduce the asymptotic variance of IIchmnk, compared to direct f-step prediction X' + 2 though establishing this concretely remains a rela- tively unexplored problem; see Somalwar et al. (2025) for initial studies on the related problem of system-identiiicaiion in linear systems.
C CONTROI--THOOK PRELIMINARIES
Before proceeding to the technical statements and proofs, we provide here a primer to some funda- mental intuitions, objects, and motivations in control theory.
We start with the most basic model and task: a linear time-invariant system regulating to the origin 0. A linear system obeys the transition dynamies:
X = Ax; + Bu.
Here, we can already introduce some of the common terms used in control. When B = O, the system is said to evolve autonomously: X Ax, fundamental fact about autonomous (discrete-time) linear systems is that they are stable if and only if p(A) < 1, where p(-) denotes the spectral radius. The cases p(A) > and p(A) = are referred (exponentially) unstable and marginally unstable, respectively.
In many cases, the autonomous evolution is unstable, and requires controller to, e.g., stabilize to the origin. Open-loop control generally refers to when the control input at time t does not depend on the state at that time, e.g.,
X++1 = Ax; + Bu, given U1,..., u, predetermined.
The general understanding in controls engineering is that prolonged open-loop control is undesir- able", as small model mismatches or unseen disturbances can shift the optimal control sequence away from the predetermined one, and may even render the system unstable. Therefore, stabiliza- tion is often performed via closed-loop control, which yields control inputs that condition on the observed state(s). The canonical example is (lincar) feedback controller:
X = Ax, + Bu, u, (x,)) 2 Kx
Which is why action-chunking, by intentionally executing chunks of inputs in open-loop. may he some- what surprising practice controls engineer.
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We note that the system evolution under feedback controller be equivalently written system autonomously evolving with dynamics X,+1 = (A + BK)x, Just as p(A) determines the exponential stability of the autonomous system O, say controller stabilizes the system, or alternatively, renders the system closed-loop stable if p(A + BK) < 1.
Remark C.1 (Open- vs. Closed-loop Stable). We remark that the above discussion leads to the gen- eral terming system being open-loop or closed-loop stable. In particular, open-loop stability generally refers to system satisfying given definition of stability without the need of feed- back controller (e.g.. p{A) < 1), where closed-loop stability refers to a system achieving a given definition of stability in closed-loop with feedback controller (e.g.. p(A + BK) < 1).
With the definition of feedback control and (linear) stability hand, we consider notions of the steerability of a system.
Definition C.1 (Controllability and Stabilizability (Kailath, 1980).. A linear dynamics Rd, Xd, Rd, is controllable if and only if the matrix C - A has rank dz- This is equivalent to saying, given any initial state and goal state Xg. there exists sequence of inputs {u..... ud_ J such that executing them steers the state to Xd, = Xg, i.e., every state is eventually reachable.
pair (A, B) E
B AB
A dynamicc pair B) is stabilizable if for any eigenvalue of A with |^] > 1, have that [A AI BJ is full row-rank. This is equivalent to: there exists K such that p + BK) < 1. In other words, system is stabilizable if all uncontrollable modes are autonomously stable.
Stabilizability is the minimal condition under which stable closed-loop control is possible, since any uncontrollable unstable mode is impossible to stabilize. Controllability is somewhat stronger, saying that (barring unmodeled disturbances) any state can be achieved under an appropriate con- trol sequence one of the key technical innovations in our ensuing analysis is bypassing relying (linearized) controllability, see Appendix E.4. Both controllability and stabilizability are binary con- ditions, and do not describe, e.g., which directions are "more" or "less" controllable. This motivates the controllability Gramian.
Definition C.2 ((Time-invariant) Controllability Gramian). Given dynamics pair (A, B), the time-t controllability Gramian given by:
W Σ A B (A*-1)T.
The controllability Gramian admits a few equivalent interpretations. In particular, the controllability Gramian the covariance matrix of the state x, under zero-mean, identity-covariance inputs, which exposes the state directions that are relatively casier harder to excite, corresponding larger or smaller eigendirections of WH., We also note that, particularly relevant to the setting, we may similarly define the controllability Gramian of closed-loop system in feedback with given controller K:
1-1
WH
A 1:t
Σ A 1=1
AB (A A
We note that despite the storied history of linear control, our setting contends with nonlinear dynam- ics and policies. In particular, the dynamics is now governed by possibly nonlinear transition
X = f(x, ur).
To build up to the incremental input-to-state stability we consider (see Definition 2.1), we start first with the same task as in the above linear case, where we want to stabilize to the origin. There, well-known notion of nonlinear stability is input-to-state stability.
Definition C.3 (Input-to-Si) State Stability (Khalil, 2002; Sontag, 2013)). A system X1+1 = fX;,, u,) ttettet stable if there exists class-KCC and class-K functions 3(2, t) and 7(2) such that for any initial state X, and control sequence {u,}, )=21, we have for all f. 2 1:
|Xx|| S ([|x t +
max kSt-1
|
'A function 7(2) is class A, if is continuous, increasing in and satisfies ((0) 0. and function 3(2, f) class K.C if it is continuous, (3(: t) is class K. for each fixed f, and ((2,-) is decreasing for each fixed
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Input-to-stan stability states that the dependence of future state's magnitude the initial state decays at some rate determined by 3. and bounded inputs have bounded effect on the state across time. This notion of stability is very general, such through choices of norm ||, and moduli 3, 7, and further admits many extensions, e.g. locality. To reduce the distracting technical baggage to a minimum, do not discuss the many extensions, and make the stability quantitative via exponential stabiliiy, where B(z,t) t) A P's 2, and y is exponential convolution of past [u across time, Definition 2.1.
However, input-to-state stability invariably considers tasks regulating the state fixed point. In general, this is not the case in imitation learning. Therefore, to capture that the stability we care about is not necessarily a prescribed equilibrium point, but to trajectory, we consider incre- mental stability (Angeli, 2002; Tran et al., 2016), where "incremental" refers to the fact that rather than contending with the state itself, we consider the state/trajectory difference. The definition of incremental input-to-state stability naturally follows.
The above concepts serve as the core background behind the control-theoretic terminology used in our presentation and analysis. However, beyond what is presented here, many technical complica- tions arise our nonlinear incremental setting. For example, the "incremental" part (i.e. trajectory- tracking) means many intuitions and specialized tools for the canonical task of stabilizing to pre- scribed fixed point no longer hold. Further, even for the control task of regulating time-invrrian nonlinear system f(x, u) to O, iteratively linearizing the trajectory yields time-varying linearized system. Furthermore, even the linearized controllability Gramian W is rank-deficient, this does not mean directions lying in its null-space nonlinear component of the dynamics. Contending with these complications arising from the gen- erality of our setting -wicch is core presenting sufficiently general descriptive framework-2 the subject of the ensuing theoretical analysis.
umexcitable/umreachhbb due to the contribution of the
D PROOFS AND ADDITIONAL DETAILS FOR SECTION 3
We first introduce some additional definitions: Definition D.1 (Additional Error Definitions). Given p 2 1, define the p-th power errors:
JTRAL & Ea
min {1, ||x - x; 11 + u|u u IP
J. (ft; Pdemo) A Egsema
| U1=-1, u. 11m
Note that JTNU2T JTRNTT JDEMO,P.T
JEMO, We further define the trajectory state error:
T
TR E min (1, -xx=xxx- xx'1 1r]
We now state some elementary results.
Lemma D.1. Assume t is a Markovian, L-pLisschitz policy. Then:
JIRAET (+t (1 (2L,)))) 2 ((t) + 2" J Pft:
Proof. Following the definition of TrAne,T we may add and subtract |uu! - #(x?") + #(x7 u I| and apply convexity of || || to yield:
T
JIRAST((1))) E .
Σ min { 1, 1|x
x7 IP (x # (x; IPP 2111 x: (xx (1PP)
Applying Lipschitzness to the second term, and observing the last term is precisely the sum- mand in J completes the proof.
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Lemma D.2 Cintttrovvic-Ruuiisttee Define the norm RR x B.. ([xx, u)llx,u & ||x Then, define the class of cost functions c(x, u) E Cláp(1) that is 1-Lipschitz in -lxx, Then, we have the following:
|||||
1 S
sup
The above is straightforward application of Kantorovich- 2009) by pulling out a conditional expectation over x x follows due the clipping at the definition of J 1,A',,
strong duality (Villani et al., on both sides. The inequality then
We will often use the following bound on triangular Toeplitz matrices.
Lemma D.3. Given E [0, 1), define the matrix:
0
0
P
1
0
(p)
p²
P
1
O
E Rdxd
pa-2
p
Given 1 Sp oc, the following bound holds the induced @2 -> E operator norm of A()):
d-1
(p)
P
8=0
Proof. We may prove this straightforwardly from application of the Riesz-Thorin interpolation theorem (Stein & Shakarchi, 2011), which states that fixing Ap),, the mapping 1/p [|A(p) |1 p-t
log-convex for p [1, oo). In particular, by taking the convex combination 1/p = 1 (1/p) + (1 1/p), we find:
log |IA (1/p) log (p) 1--1 + (1 1/p) log (p)]
[A()) $ [AA 1/p A
We then utilize the basic fact that || (p)]h and A [p) correspond to the maximum col- umn and row of A (p), respectively, which completes the result.
We may now state the detailed version of Proposition 3.1.
Proposition D.4 (Full ver. of Proposition 3.1). Let Assumption 3.1 hold. Let (f, f) be policy-dynamics pair that is PE-EIS,, and consider the corresponding chunked policy म = (fu f, e). Then the closed-loop system the chunked policy induces on the true dynamics (#, f) is (C,p'-a)-ER where E 1) and Č a0' log(1/p))1 poly(Im, Cıas), long the chunk length is sufficiently long: f 2 a log(1/p) log(poly Cisa: 1/a)).
Proof of Proposition 3.1. Let us define the chunk-indexing shorthand 4 - (k 1)6 + 1, such that 1. Toward establishing EISS of the closed-loop chunked system, want show for sequence of input perturbations {u, Ju21 and two trajectories {x] h221, {x h2s evolving as:
x
X1
x = f (x ue),
X = X
there exist some constants C2 1, p E (0. such that:
T-1 C T-1-
| - - Cp - | X,
To do so, we prove the following "contractivity" result going between chunks.
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Lemma D.5. Fix some & 2 1. Recall the true dynamics is (Ciss: -ESS.... Then, the following holds:
Mx x S p² pxI Xix
C Σ
where p 2 pt-0, as long as a pplylogo( (1+ + Lix Cass, log (lp))((1/p)))///)))))))) a a). As a corollary, setting C (1+ 1888 for 1 S h S we have:
3ap (og(l1(111)
h-1 1
x x S [x
x ||
Cass
uss+sll-
Proof of Lemma D.5. Applying (Ciss, P)-EISS of the true dynamics f. we have
||x x |l S Cissp'llx - x + Cass
6-1- |
Uuto
urutell
S Cisap l|x| - -x, + Cass
uuu Th + Crss
0 1
|uutoll
==0
==0
where ui and Uh ts the s-th actions outputted by the chunked policy conditioned X! and x respectively. We consider the "simulated" dynamical system that generates u², u²:
x,+1 f$t (x,; 0) f(x,, At (xx)); s 0, e..
u + - म (xx); Xy x
8,+1 fa (x,, 0) = f(x,, # (xx))) = 0, f-1 A # (Xx), Xn x
Crucially, we observe that (#t f) is (Ciss, p)-EISS, and thus:
|x, x S Cras ||xo xoll = Cissp" x
Therefore, by the T Lpschhichssss of ft, we have:
| uol+ S L= - 8oll S Lop Cissp" x
Plugging this back above, we get:
f-1
[ -Xien || S Cissp' x|x
xf,l1 + La
Cass
Σ
- xf,11
8-1
+ Cass
1 U
==0
Cissp'
1|2,,
x + Lop C2
Cras
8=0
8-1
4
x
+ Cass
Σ p f-1-9
ulstel.
We solve for the requisite chunk-length by solving: (1 + C2g S pt, where p = pl-a Rearranging, this amounts to & N satisfying:
log ((1 + Cxs
log(1/p)
To remove the dependence the right-hand side, the following elementary result.
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Lemma D.6 (Cf. Simchowitz al. (2018, Lemma A.4)), Given 2 1, for any l E N, f 2 a log(E) as soon as 2 2a log(
We observe the above result holds if we add any term that does not depend on l on the right- side of both inequalities. Applying to the above, since log(1/p) z log(e) = 1, setting a (a (ag((()))))- we have that
l 22a 12a1+ log ((1 + La)
4 (og((log(1/p))
(lo((1/p)
alog(1/p)
implies af 2 + log (oss)) which turn implies (1 + Lu Case S p as re- log(t/p)
quired. For the corollary, we observe that the maximum value attained by - max + L hhsh""ppp /p" (1+ issh hp is upper bounded by (1+L,) (1ss) completing the result. 3aplog(1/p)
Toward bounding |x xl we define the number of full chunks traversed K - 1 [(T T)/EJ, and the remaining timesteps h T (K 1)8 - Further define the shorthands U₁ Ciss 2 5= [ +all for l E [K], and UK+1 = Cass Σi-0
Then, for &
satisfying Lemma D.5, we use Lemma D.5 to iteratively peel:
| x S GEsp"1IIE xxxx + Cras
Σ
1i+s uixts Mextell
==0
xxx
-x + U-U++
x (1xx/|1x - xx + +Cp'UU UK+1
K+1
S TpT-111x X1-I| +
p [k 1)
T-1 1
I - x111 + CC
Σ
This establishes that (f, is (CCiss, p² 1'=)), given the chunk length satisfies > a -1 log(poly( + Lay Ciss, log(1/p) 1)), and leveraging p 1/e, we complete the result.
Having established that the chunked policy on the true dynamics (#, f) to show this controls compounding eтrоrs when ă achieves low on-expert error to an expert policy n² This is a straightforward application of EISS. In particular, by treating the expert inputs as perturbations to closed-loop system induced by (T, we may relate T to J
f) is (Č.p'-a) stable, we want
Proposition D.7 (Full ver. of Proposition 3.2). Let Assumption 3.1 hold. Let chunked (R, f.()
Iehunn, and assume (ft f), (H,f) f) are (C, (C,P)-EISS. Then, the following bound holds:
Ö
Jx T (
S
JDEMO,P,T Pm)).
1 1-
We have subsequently:
JTRALP.T(HT) poly" (I,,, T2x) T (x;:).)..
Proof. Given xi = x ~ D, we define x;" up² and x", u as the states and inputs given by the
expert policy n" and chunked policy in closed-loop. We may then view (x7", .,") ) as the result- ing trajectory generated by appropriately defined "input perturbations" the closed-loop chunked
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system f" x (x" A 1, where define
Du. u chunk, (xT,,
)
and 4 = - and 8 = --1 mod l. Therefore, applying the p)-ISS of (i, we have:
|| 1xf-x; x5 11 I
J 1
1T;
The second line follows straightforwardly by summing both sides from and applying an expectation. To extend this bound from 1RN,I, to general TRALPTT we leverage Lemma D.3. We define the vectors u, RT-1:
[|xx x5
|| || ||
ERT-1 ER' RR---1
x II]
[l|au ||
We observe that defining A (p) as in Lemma D.3, we have u = ČAv. Taking the p-norm on both sides and applying Lemma D.3 yields: ||u||p I - l|v|ll-- Taking the p-th power and applying expectation over X1 ~ on both sides yields the desired bound on PRA.. in terms of JDEMO
To extend this a bound on JRAI.,,T we apply a similar bound to Lemma D.1. However, we require some alterations since not a Markovian policy, We may add and subtract to yield:
| u S ||u. chunk, [ (x7, Il
|| - chunk, [+] (xT, Ill-
Summing up the second term over yields J To analyze the first term, we recall that u, and chunk, [+] (x.",) result from conditioning on the state every ty timesteps, then playing the next actions generated by the simulated closed-loop system f). Since by assumption ftt is p),p)-
this means that for each & and o,...,
||tu+t Kestoll || S
x
Furthermore, since u +xxe++0) and similarly chunk, [x _ (xt,+*), applying Ippshhissssssssss of a yields:
where Xac+s (f*)*(x())) ) Sects (x*)*(xIT)) ).
1/2 8-1
3/4 8-
Σ
u t+ - chunk, [x T +11
LPCPP) -x7
L
p T
Σ ||x -x7
1P.
1- p
1=1
Putting these pieces together, we get:
JIRNI.P, s J JRAAI (T) + Σ min ||f u, [13}
2T, T
< Jx JiRsspp (T) + 2 JDEMO.p.TTi;PP..
Σ
min |x x; (10)
1 1-p
2L Jx (.R.A.p.( + 2 (Tr; (Fr; Px.).
< 1+
Plugging in the upper bound on JRRupp (+) completes the result.
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In particular, specifying this result to Proposition 3.2 follows straightforwardly by setting p 2. Therefore, combining Proposition D.4, which says chunking policies induces EISS, with Proposi- tion D.7, which says EISS chunking policies induce low compounding error, yields the final guar- antee.
Theorem 3 (Full ver. of Theorem 1), Let Assumption 3.1 hold. Given a E (0. 1). for sufficiently long chunk-length: l > a- logl lo..-oll(ppol (oly(TTr, Cas,, 1/a)), let स chunked R2, g, E) cTuuunk, such that (#, f) is ")),, with Č = a- (pooppy Csss) The following bound holds on the trajectory error induced by ã:
C P
JIRALP, (+)
1 1 ---pt-
E PROOFS AND ADDITIONAL DETAILS FOR SECTION 4
E.1 RL- VERSUS CONTROL-THEORETTE THEORETIC PERSPECTIVES
The RL-theoretic perspective. RL-theoretic notions of exploration often take an information- theoretic flavor, where it is captured by notions of "coverage" (Jin et al., 2021; Zhan et al., 2022; Amortila et al., 2024; Jiang & Xie, 2024). Coverage analyses rely on density ratios and thus the existence of densities. In continuous state-action spaces, expert (deterministic) policies typically do not have densities, and thus they can be induced by incorporating (possibly shaped) noise to the actions (Haarnoja et al., 2018; Schulman al., 2017). Crucially, this makes the policy itself poss--ooppae this to Practice 2, where the expert's 's recorded action is uncorrupted. When the noise is Gaussian, this practice turns imization. Hence, existing analyses of behavior cloning (e.g.. Foster et al. (2024)) ensure consistent imitation.
estimation (MLE) into square-error min-
However, this comes at the price of corrupting the demonstrations provided to the learner, which in turn, we show in Appendix E.7, leads to suboptimal rates of estimation. In particular, by reducing imitation learning to MLE over noisy data, the performance of IL is dictated by the capacity of the stochastic policy class, as measured by a covering number Nior (II,)) under, e.g., the log-loss. For
-u-scaled Gaussian noise, this equates to covering under the Euclidean norm at resolution Rs Vols. For non-parametric classes-such as the lower bound constructions leading Theorem A, this can introduce additional polynomial factors of oū in the estimation error. These factors of o. must then be traded off with the error induced by imitating a noisy expert rather than the true expert labels.
perspective. In the control- -theoretic literature, persistency of excitation (PE) The well-established sufficient condition for ensuring parameter recovery in system- idettificantnnonnn and adaptive control, which in turn yields performant policy synthesis (Bai & Sastry, 1985; Narendra & Annaswamy, 1987; Willems al., 2005; Van Waarde et al., 2020). input-sequence "PE" if it yields a full-rank sequence of states, which guarantees parameter recovery across all modes the system may encounter. Therefore, when an expert policy may output degenerate trajectories in closed-loop." natural approach achieve PE is inject excitatory noise into the inputs or directly into the system state (Annaswamy, 2023). More modern analyses of both the online linear- quadratic regulator (LQR) problem (Dean et al., 2018; Mania et al., 2019; Simchowitz & Foster, 2020) and of imitation learning Pfrommer et al. (2022); Zhang et al. (2023) have similarly turned toward PE ensure desirable learning behavior; üttherellyigg on process noise (i.c., non-degenerate noise entering additively to the state) to excite state variables, or assuming the ability to directly perturb states during expert demonstration. By contrast, our setting assumes neither the presence of process noise, nor direct access to the system state. Lastly, we do not even assume the system is controllable," 10 i.e.. we also cannot rely on input perturbations inducing the PE condition.
Comparisons to the RL and control perspectives. By combining ideas from RL and control, we arrive at conclusions that may be surprising from either perspective. Compared the RL. perspec- tive, 1. we do not have coverage in the usual sense, 2. we avoid accumulating mean-estimation error
See e.g., cases for linear systems under an optimal IQQR controller (Polderman, 1986; Lee et al., 2023). IIformally the ability of system to he steered from any state to another by applying appropriate control inputs, ef. (Kailath, 1980).
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from imitating noisy action labels, 3. using the mixture distribution Put,ou,a subverts the additive o error in Proposition 4.1. On the control-theoretic side, 1. imitating over .a.. removes the additive o error in Suboptimal Proposition 4.2, 2. we avoid any assumption of controllability as well as any dependence on the small eigendirections of the controllable subspace. In fact, by moving any additive du factor, our bound suggests that we should set the noise-scale du as large as permissible!
E.2 PROOF PRELIMINARIES
We first recall the definition of the lincarizations around expert trajectories from Definition 4.3.
A = Vx u B, Vul(x) u K. 2 ... (A, B,
Vx (x7
(E.1)
A (A.-1 B,-K (A B,
We also
recall the definition of the controllability Gramian: Σ A s- B,B, A T. For a noising distribution that is zero-mean with covariance
(x.
k.d
Σz, Ze D(0, Σ,), we further define the noise controllability Gramian:
i-1
(x. &
Σ
Ad A
Note that for z, sampled from the Euclidean unit ball, we have Eg (d.+2) Ias 2 3d,
I and
thus:
W W (x.
W W.e (x7')
3d,
The ensuing results are written for any noising distribution D(0, Σg) that are 1-bounded, mean-zero, with covariance Mn 0. unless otherwise stated.
We now establish that the linear (time-vayyin)) system induced by linearizations along expert tra- jectories inherits (Ciss: p)-SSSS. We note that though the original dynamics and expert policy the linearized system is in general not.
Lemma E.1. Let Assumption 4.1 hold. Given a nominal trajectory generated as
x = [(x uf"). uf = =x 121, x ~ P
and recall the linearizations in Eq. (E.1). Then, the following bounds hold:
|Ast I Cissp'-1, ||A Cresp'= IA I Cisspt-l-s for all 1 t.
An equivalent way to view Lemma 1. is: for an input perturbation sequence (Au, J22 the incre- mental trajectory (Ax, Jes1. Ax, &, xx" induced by linearizations around an expert trajectory {xF hea is (Crss, EISSS::
A A (A+ BA, Ax, + B,A., Acl Ait+ Ax,
Σ
A el la4 B.A..
Proof of Lemma E.I. Given the nominal trajectory {x. hez1 generated by x = f(x,", uu and the corresponding linearizations A,, B,, K= evaluated along the trajectory, consider the trajectory
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{ h221 generated X++1 =f f(x, 17" (Xx, u,,t). Expanding the Jacobian linearizations, we have
X141 x7+1 s(Xe, क -xx) + Uz () - [xx] , )x)))
x" = A x (xx))) + (x7 }) +0 (11 (T<( - x (x7
X
W
Ar7
= (A, + B,KK (x+ xxx)) B.(u, + +(|x, x; 112)) + r
Erl"
A 1 x ) Σ A s41:441 (B,(u, + r") + t.r))
(E.2)
We perform a simple sensitivity analysis to isolate A Defining the displacements o7 X; x," and setting = 0, t 2 1, we see that á 88; oz Ail, since we observe 67 = A 07 +
E (B,r"Brrr" +r²) ) is linear in of and the residuals r. r are higher-order by definition. On the other hand. by the (Ciss. p)-EISS of (I*, we know that ||67|| S 1|8 By definition of the operator norm, we have A llop = supy || A vl/llv/ and thus by a limiting argument o7 -> 0, we
| LAtellop
I lim | /// |881 S Cissp' o7 -0-0
To establish a similar bound on Ac we observe that x f(x;" , (x," (x]") )) is by definition a time-invariant closed-loop system, may apply (Ciss, (Ciss,))-EIS) P)-EISS starting from 83 as the initial displacement such that || s C'ssp"==||8811 Applying the same argument yields:
|| 8:t olp < lim | / 11881 S Cissp"
Now, instead setting x - x = and an impulse input {0,.... O, ug O, ..... } for some k, we have 67 = el
1+1 B + E. Ag s+1a (B,r" + ). By the same appeal to EISS of (m*, and limiting argument + O, we have: ||Ag| k+1.1 Bxllop I Cispp-i-h, Notably, this holds for any k and 2 k. completing the proof.
Given expert-induced trajectory X = = (x,), / [7-1), consider noise-injected trajectories (x,, u ~ IP as in Definition 4.1. Our next result demonstrates that the noise-injected trajectories well-described by the expert linearizations, up to higher-order term quadratic in the noise-scale ou-
Proposition E.2. Let Assumption 4.1 hold. Consider noise-injected expert trajectories (&,, n (x,))(x)))))) ~ Pat.on for given initial condition &, D: &
f(x,, m" (X,) +
ouze), 2t 1.dd D(0, Ex). Consider the linearizations along expert trajectory given in (E.1), setting xi x1. Define the linear and residual components of the noised state Ne:
x
Ad A.
X
2 xx
xc
t
(E.3)
Then, as long as ou < 2 Catab
V-40k
and defining C. A Cx + 4Crez(1 + 4Ck). we have
T S C3 Cro t > almost surely over Xn D and {2,) lid D(O, Ex).
Proof of Proposition E.2. Given the nominal trajectory {x] hez generated by x f(x²", uu and the corresponding linearizations A, B,, K (E.1) evaluated along the trajectory, consider the
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trajectory {%t h21 generated Xx+1 f(x,, TT" (xx) + ue,t), with x1 =x=- Then, following (E.2), we may write:
X1+1
x7+1 f(Xe, x² (Xx) + ue t) - f(x]" I (x(x
A 1 (x1 - x + Σ A 8414141 (B,(u, + +3))
A cl
s+1:+11 (Ba rr) +r)
where we recall r and r3" are the second-order remainder terms of the dynamics and policy outputs, respectively. By Assumption 4.1, these are bounded by:
| & Cn ||x - x; 11
| S Creg
|| x 1| + T (x))
T" (x + uell uell²)
$ Creg
|| - x 2 + 2 (Xt. - x² (x] I| + 2/ 2/|ue|1
S Creg
(1+40k) |x - x; 1| + 4||x;||| + 21(
< Creg
(1+ ) \ - xx" 11 + 40 x 1| + 2/
Defining Ex. [ xf Il. and u,
1.1 D(O are iid zero-mean, Eu covariance, ou- bounded random vectors, we want to bound the mean and covariance of x,. We note the presence of the quartic term 4C2= in our remainder term; we first impose Ex 1+4Ck absorb into the 4C
quadratic term, then show this constraint is obviated for sufficiently small ||u,ll-
Since (TT,,) (J) is (Ciss, -EESSS, we have Ex. S CIsS Ex -----1111 [uull < Therefore, we have:
Ciss max ||u,||-
T-p
|| S C, I C C2 max
||u
sSt-1
|| < Crog
((1 + 4C+) + +. + +2((
< Creg
max 2(1+40 )
||ual1 + 2//lul
< 4Creg (1 + 4
These hold long as ou is small enough such that 22, < C² Caaao
1+1Ck
which holds for 40
VI+ICk With these perturbation bounds in hand, we now move onto bounding the Ou < Catab Cx linear and residual components of x, We have immediately:
A s+ia (B, + +r,) + +1,)
| = |x
x
||
el st la (B,r + r++)
A cl
B Bzllop | + ||A + 1 ||
8- 1:
1=1
< Ca
(C= + 4Creg( +4C%)) o
20,
This completes the proof.
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We now proceed with the one-step controllable setting, where Wi: > AwI for all 2 2, leading up to Suboptimal Proposition 4.2, where we also purely on noise-injected trajectories, order to grasp the core ideas and the remaining key deficiencies.
E.3 ONE-STEP CONTROLLABLE CASE: PERSISTENCY EXCITATION
We consider settings where the controllability Gramians induced by linearizations around an expert trajectory are always full-rank.
Assumption E.1 (Linearized one-step controllability). Let 1i (xf') = Aw 2 w.p. over x D for some Aw > Consider the noise-controllabiiity Gramians Wi: as defined in Definition 4.3. Accordingly, there exists > > such that w.p. over x D, W (x.x(x")) z I,, 2 2.
Proposition E.2 in conjunction with Assumption E. implies the noise-injected expert states &, form This corresponds with the well-
a full-rank covariance around x for each timestep 1 = ., T. ,,, known notion of persistency of excitation from the control literature (Annaswamy, 2023). As consequence of Proposition E.2, we have the following excitation bound.
Corollary E.1. Let Assumption 4.1 hold and Cp be as defined in Proposition E.2. Recall the noise- controllability Gramian W1:: 1:t as in Assumption E. As long as:
(Wit(x7") Caea CF Cstab
/I+4CR
Tu Z min
On
the following holds almost surely over 81 =x²" Pxtt and {2a} iid D(O, Ez):
Ex xxxx (xx=x²]) xxx (Xe -x² ]
on w (x
(E.4)
2
Proof of Corollary I. Denoting C= D A 1+ B.u, and E E Ad $41a (B,r"" + +r²). we bound the second moment of &c - x7
Xx - xx (xx-x,") xx" ]
Eu
= Eu |C + EC + T
Eu [CC
+ Eu
EE
+ Eu
EC + CE
E.
[CC
E,,
EC T CE
By Weyl's inequality (Horn & Johnson, 2012), we have for each k 1,...., Cank(E, [CC]]):
A& (BBlu((C+) I) A (E 2| llop
Ciss Tu (C (C= + 4Creg + 4C +
2
2C stab stab Cro..
Rearranging the above yields, for each h: 1, I,..., rank,., [CC
A (E. + E)(C + E) 2 X (EC,(.CCT)) - 20 (Cn + 4Crog + ++C oi
Therefore, for sufficiently small ou such that:
ou < +t+
(En, (E., (CC Stab C-1,
where )mi(( (.) denotes the smallest positive eigenvalue, we have (E + EC + E) 2 do (En CCC I), & I,..., raan(EE [CCT] such that
En (xx x (x, xx-x77 (xx=x;+)
> Fu [CC]]
2
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1-1
2 lat B,
T
Proposition E.2 demonstrates that noise injection yields full-rank exploration around the expert tra- jectory that is essentially described by the controllability Gramian induced by linearizations around the expert trajectory. In this case, we show that a policy म attaining low on-expert error does not suffer exponential compounding error. The first ingredient is an adapted result from Pfrommer et al. (2022) that certifies low trajectory error as long as policies are persistently close in a tube around the expert trajectory.
Proposition E.3 (TaSIL (Pfrommer et al., 2022)). Assume the closed-loop system induced by (f" is (Ciss P)-EISS. For any (deterministic) policy it and initial state X1, let x = x" X1, and consider the closed-loop trajectories generated by & and n²:
x++1 = =f(x,,((xx) ()x
x = [ i7 (x 1), t21.
(E.5)
Then for any given e T N, as long as:
max ISST
sup ||
ll nt' (xx]" + +wwill S CatabE,
we are guaranteed maxigisT ||x] x S.
An elementary proof to Proposition E.3 can be found in e.g., Simchowitz et al. (2025, Lemma I.4). Our next ingredient demonstrates that if noise injection induces full-rank state covariances, closeness in a tube with radius proportional to the noise variance is certified, up to higher-order perturbations from smoothness.
Lemma E.4. Let Assumption 4.1 hold, and let Assumption E.1 hold with Da 0. Let (xT IT
(8e)t=1 be expert and noise-injected states initialized from given xx Let be any C--smooth (deterministic) policy. For sufficiently small noise-scale ou
x1
S
min { Catan V Cstab /1+ 1++00 O=1)
the following holds for each = L....,
11( - ( (x²"" +=w)l < xx [1|*(x)) x (xx:)111 + 9C Ctaabo
sup [w Is1
for any E S ou du Amain W. (x7 1)/2.
Proof. Toward upper-bounding the left-hand side of the desired inequality, we have:
sup [1( - n (x?""" + +wwll2
||w wi
sup 211( nt" (xp + + - "" (xx] ww| 8C3:4
||
x 4f x²) <x) 1|2 sup V - x (x} )w I2e + 8C
||||||
< 41 -²) (x, 1|2
4 V ( n" (x7 >|2 8C
(E.6)
where use the fact that क - nt is at worst 20, smooth. and repeatedly apply (a + 6)2 < 2a - 26 We now lower bound Ex, x [1] (x,) n (x | Recall the linear and residual decomposition
of x, = xlin + stTes from Proposition E.2. Applying the C+ smosoothsssss of and n², we have:
*(x;) - x - (xx) (5t n x*)
Vx (x) J (x! x""
x r.
where ||<|||| < 20 |x x;" 12 < 2C, Caaa oi by applying C₂ -mooohhhsssss and (Cres, P)-EISS (Definition 2.1) under ou bounded input perturbations. Therefore, may lower bound:
1 [( - n x))(x²² + Vx(# f x')(x" ) T(c!in x; )²) V (t - T Txx, ) pres
||*t)) - > 2 2 2
+ +r,l1
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1 2 2 (+ x" (x] + Vx - -f- (x] (x!! -x,"))
Vx क (x} x
8C xt stab Jn
Taking the expectation on both sides, we have:
Exg1x"" (xx) - x (x+) 2 2 |x
1xp"
[[ - x² (x²] + Vx(( (xx [(x))) - x )
|1 - x (x+ ) x - 8C
2E |x1
Notably, Ex x x;
0, and thus the first term on the right-hand side can be expanded
yield:
xx
|[( x²) (xF
Vx At x (x]
)T (xetin x7 {12]
xx
[[(t - x² (x}" )|
+ tr
V x²) (x? )T Bx ixe" [(x] xx )x! - ) xx ((x(())
Ex
[[(( - ) |||]
+ tr
Vxx(ft x xx7 ) Wi: (xT) - क x))(x)))
Vx
On the other hand, expanding the second term yields:
|x7" [x T| Vx - 7"' (xx') Ex,Ixx" |x]" V - x- (x, [|] = tr V (tr - x* xx7 ) TBx |x7'
I tr Vx - x* (xF V - +xx )) Caaat Cho
where we applied Proposition E.2 for the second line. Therefore, for sufficiently small noise level:
a S stab C-² Amin( (Win (x."(x))) )),
we may combine the first and second terms to yield:
E xe xf (1), n" (x,))1
2 [x
1
[I[CE - (xx [1r]] + er (xx(t (f-==xx)) (x) (x.
V -0====="xx x²)(x())) (xx'') -80 otabb
1
1x [[(
x (x] [P]] Amin (W )) V (x7 )1le) &C
2
where we used the elementary inequalities tr(PQ) Amnin (P) tr (1) Amin (P) Amar (Q), for any
P > 0, Q = 0. Notably, the validity of this inequality rests P (xx] ) > 0 granted by Assumption E. Rearranging (F.6) yields:
- (x7 11 V T)) 11pp=²
sup |( - T" (x; +2 117--20 20 ||
For g² < 2 ( (xx')) ))
o Amin (W (xT')x )) plugging this into the above sequence of z
inequalities yields:
1
1
Ex,/sT" (1(()))='(x))))
sup
11( (x) + 2C30" -CC
2
4
[wiss
1 0 - 803C sup |1 TT (xq + +ww)
16
||||s
2
We have trivially that x2 < yields the desired inequality:
1/2 Amin WT. (xT )) S d
C2 Caab and thus rearranging the inequality
2
sup 11( m²) (xT" +ew)ll < 16R x,,,, 1xp" [II((x)) (xx))1] + 9C, C C.
[w 1s
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Therefore, using Lemma E.4 to certify the tube condition in Proposition E.3 yields the (suboptimal) imitation guarantee.
Suboptimal Proposition 4.2. Let Assumption 4.1 hold, and let Wi. (xf' > Aa. 2 w.p. over xi ~ D for some Aw > Let be a Cs-smooth candidate policy. For o that satisfies Aw. we have:
oi x O.(poly(1/Cx:
1 Jo (f; o.)) + C2
JIRAIT(f) (T) Aw
Proof of Suboptimal Proposition 4.2. Using the identity for non-negative random variable Z sup- ported on [0, 1], [ = So I > E| de, we have:
F Ex
max
xx
PP
max IStST
||x]
x 113 >
de
xp
1SIST
J
//' P
max
|
- 113
de +
IP
max 1S1ST
||x
-x,' 112 > de
1StST
F
max 1SETT
x 11
de
+P
max ISAST
|
x 11
where we choose a splitting point T E [0, 1] to be determined later. Now, applying Proposition E.3 yields:
P
max
|x
x7 1 > E de
I
max ISEST
x
x 112
SSST
P
max IStST
sup |
||( ." (x+
vew))l?
Geabe de
sup |[ x" (x+ VFw)ll3 > EEarT
P
max ISEST
|| 511
For the first term, we have:
P
max
sup || |(+==))(x² (x²" + /2w) catab de
ISET
|w SS
J
P
max
sup | Wi
||( T"" (x"" + // > 2 Llab E
ds
ISSST
min
T, Ctab
C2
Exi
max ISIST
sup
|| - n" (x} + rw)l1
wi
where the last line arises from combining the trivial bound Jn PIZ > E] ds S and by perform-
ing the variable substitution = =c2 Ctaabe then applying the identity E[Z] = fo' Plz > e') de' Therefore, setting + S 8%. we apply Lemma E.4 to get:
11( (x7 + 1- de
P
max ISSST
sup [[]]s1
Ciss
min
C2
Cata Ex,
max
16E
&
1xp" [1|*(x)) - > (xx))1 + JC..
ISEST
< min
T, CC.o
+ 16C2 Σ Ex, [| न (2:))12)
For the second term, we apply Markov's inequality and similarly bound:
P
max 1StST w|
sup
||( - n (x] + =Fw) > 2aub
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I C stab? Trabb-EEx,
max 1815T
16E
Ny 1x7"
[1x((x)) - x [xxx)]1 + 902 Ct
stab
T
I C T
999 Sd + 16 Σ Ex, [1 - x (xx))1
à yields bound on maxigier ||x x" A in terms of 2. o.)) and an additive drift term. By summing over each 1 S t < T, we get bound on 2 (f), accruing T factor. Now, by Lemma D.I, we have:
Combining the two bounds and setting ExI x1"
JTRII (#) S (1 + 41 S (f)) + 4JDEMO, (i; P...)
It remains to relate 2 Pa.) to 2 (n; Par.ou). Since the injected noise is by def- inition -o-boudded, applying (Ciss, p)-EISS of (x*, yields w.p. I over any xi and {z} iid Ex);
t-1
| x7 S Ciss
p111-3 [
I out
In other words, for a given ~ D(0, En) we always have:
[|( x"" (x, }| & II x (x, ouzi)l + Cstabou
Squaring both sides and taking an expectation yields the following bound on J (f; Pa.):
2 (+; Pn.) S 2EMO, (+; (rtjPr..oul 1-++L, TLT TTL2 Ci
Putting the pieces together, we have:
JTR) S (1 + AL?) 2. ( + 4
< (1 + 4L2)
'T
2,M ( + C&Cd ot
+ Jumo, 2.7 (#; Pxt..))) TL,
When D(0, Ex) is the uniform distribution over the ball, we have X, ~ Aw/da, Lumping terms together, this completes the proof of Suboptimal Proposition 4.2.
This result says that if noise injection fully excites the state space, then the trajectory error bounded by the on-expert error evaluated the noise-injected law Prt.o,, plus higher-order error term from smoothness. Note that simply regressing on the expert trajectories without noise injection, even the smooth one-step controllable case considered here, can suffer from exponential compounding error (see Simchowitz et al. (2025, Theorem 4)). Though this is marked improvement upon vanilla be- havior cloning, this set-up leaves open couple deficiencies. Firstly, performing behavior cloning on Px..o. yields a drift term ~ o that persists even when J (f; (...), is small; this in- troduces a trade-off on the noise-scale, where larger ou benefits the excitation, but exacerbates the drift. We demonstrate in Appendix E.7 that this additive factor is fundamental. Secondly, one-step controllability and in similar vein persistency of excitation-is strong condition (e.g. requires d, = dz)); typically we do not expect inputs to be able to excite every mode in a system, let alone instantaneously.
E.4 DEPARTING FROM CONTROLLABILITY AND PERSISTENCY OF EXCITATION
We now consider the case where lack controllability, one-step otherwise. In other words, the linear controllability Gramians need not be full-rank: rank W 1:r (xf promised the body, we hope to lift the inverse dependence on the smallest positive eigenvalue of controllability Gramian, including when it is rank-nefic On the technical front, few barriers
de. Furthermore, as
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present. Firstly, the state-covariane bound in Corollary E.1 imposes constraint du scaling with the smallest positive eigenvalue of - this can exponentially small in dg various cases. Secondly, Proposition E.3 requires certifying that á and x² match on a (full-dimensional) ball around the expert trajectory, and subsequently the "expectation-to-unforn""" bound in Lemma E,4 requires full-rank covariance.
Given these technical difficulties, we introduce the notion of the "reachable subspace" under the linearized system under the expert.
Definition E.1. Fix any x D. Recall the expert linearizations from Eq. (E.I), Define the reachable subspace of the expert closed-loop system at time t:
t-1
R²
A A B.
{u,}aul Rd
The following facts hold:
RT is linear subspace of IR
Given any positive- definit E > 0, the associated controllability Gramian satisfies
rank == A T AA T) dim(RT7) for each t 2 1.
Let {(Ai,) v=, be the eigenvalues and vectors of W I: t 2 2.
Let us further define the
reachable subspace truncated A:
R7 (A) A span(va. A,, 2 {}:
as well as the corresponding orthogonal projection matrix PR. '(A)- We also abuse notation and denote R, (A) as the subspace component of " orthogonal RT (A).
In line with the body, we will consider D(0, Σn2) = Unif(BB=(1)), such that W 2 3d W
l:t- As
previewed in the body, the main guiding intuition moving forward as follows: 1. by smoothness of the dynamics, most of the eтror should be contained in the (linearized) reachable subspace, 2. the small eigendirections of the controllability Gramian are precisely those that are hard-to-excite, and thus should accumulate compounding errors slowly enough "ignore" them. We start by proving a restricted "Jacobian sketching" result (cf. Proposition 4.4). We note that though we present Proposition 4.3 first in the body, we will in fact use an extended version of it that relies on the subsequent result.
Proposition E.5 (Full ver. of Proposition 4.4), Let Assumption 4.1 hold. For x ~ D, define RT (A) & span(vi. Ai, 2 }} and (RT as in Definition E. for some X 2 X HainWW (WH.(xT Then, for ou satisfying:
VI+
402
Gu S min Adz C Cstab
= O+(A).
Cx
we have the following bound for each 2:
du
(II(# - nt (x{ I1 + 1x 11( - 7' xx)(2)) + duo CfC
|| [PRI V n (x) 112p
We note that Proposition 4.4 is recovered by applying an expectation over x? on both sides of the inequality.
Proof of Proposition E.5. First, we consider the following adaptation of Corollary E. Corollary E.2. Let Assumption 4.1 hold and Cr as defined in Proposition E.2. Fix any 2. For X X+ (WL, )), set "PRq"(()) as in Definition E.I. As long as:
V1+CC
0u & min Ad i Caaah C-1, .stab
Cr
"Though we omit it for clarity, recall all these quantities implicitly condition x
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the following holds almost surely over X1 = xf"
D and {z+}
lid D(O, M2):
|xT"
[ 2 xxxxxx) x;
P
P
= ou 2
duww La (xT P.
(F.7)
The proof of Corollary E.2 follows from a one-line modification in the proof of Corollary E.1, where instead of requiring Weyl's inequality to hold over all positive eigenvalues k = 1,..., rank Ia). we need only to consider up to = l.....P. p dim (R, (A)), for which Xp(Wi. 2
d, (r''Ap(TT.. 2 d, X.
We proceed by applying the Cr -moothhessss of á and RT, we have:
A(x;) - π. (x:) (f x' T' Vx(# (xf (x] x + x + r..
where || S 2C |از x 1 < 2C, C2 stab a by applying C₂ smoothness and (Ciss p) -EISS (Definition 2.1) under ou-bounded input perturbations. Therefore, we may lower bound:
(xx)) #² (xx)) V (t x² (x}] ) (5: xf" I1| 11 (x] r,
2 V i - x (x,"" ) (R: x," IIl 11 -x) ||| 2C, Stab ou
Rearranging the above incquality, squaring both sides, and applying the inequality (a + b + cc² S 3(a² 6² + c²) we have:
| x"' (x7 (x, x; 11 (1(#- (xx -12 1| - x (xx,)1 Stabb
Taking an expectation over the noise injection on both sides, we may apply Corollary E.2 on the left-hand side: for ou satisfying the requirements therein, we have:
Ex xx
|
x (x7)) ) (xx - -x,² )
tr Vx T²) (x] Bxx xx- [18e- xx [ - x5 )T (x - (x((==+'(x()))
tr (vx(t - (x] )7 {T
PRI (A) Bxi1xT (8x - xx ( xx )7] PR Vx(# - >x)
0
tr (Vx(# (xx) T (A) (x ) (P Vx(# (x7()
2
2 od PRE((A) (2) Vx(+ - x²" (x. d
where applied Corollary E.2 the second-to-last line, and for the last line we used by defi- nition - (PRF"()) W1.x(x] (()) 2 d= At (PR Wh (x. (PR((()) Z d= X. Thus, re- arranging the inequalities, we have
PR (2) (A) Vx(+ - π (x] 11sp
のgo (I( [1(#==='( xxx I11 Ex [|=(()) T" (xx)11) + d.o
du
x
c# stabs
X
which completes the result.
In light of Proposition E.5, we have demonstrated that small estimation error along both un-noised and noise-injected states implies a first-order closeness of से and π along a subspace of our choos- ing. However, by choosing the excitation threshold À that guarantee closeness above, do not track: 1. error in the reachable subspace below the À threshold, 2. error for non-linearity. As stated, Proposition E.3 requires uniform closeness on a e-scaled unit ball, which Proposition E.5 does not grant. Our next step is to prove the full version of Proposition 4.3.
Proposition E.6. Let Assumption 4.1 hold. For any initial state X2, let x xx" X1. and consider the closed-loop trajectories generated by and n². Define the constant Crem A 2C (3
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2C + )- Fix any sequence {Ne Y where each de min (xT )), Amax (WT: (xF )
Then for any given E E 10, T E N. as long as:
I1( - -x] + +ew + log(1/p) Vaeer + Crame frome?v]l S Cstabe,
Estab
max 151ST
sup [el]sL (At) 1HIS (A+)* liv
we are guaranteed maxisisT |x]
x7 S.
Proof of Proposition E.6. We prove this result by induction. Fix any [0, 1]. Define the quantity
1-p C1 CLas log(1/n)" Further define the shorthands P. (RR,-(A)) (Xi), and the relative orthogonal
component P. -A)) Let us for each timestep define the set
V. 2 EW + VACCLET + C [w]],
|v/l 1 1, w E RT (1)), r RT (A0)+)
In addition to the statement of Proposition E.6, we claim that x x²¹ E V/ for each I,..., T. Considering the base-case T 2: since x x by construction, and thus ft +(xf) (x] ) by
assumption this satisfies A [x]] n (xT )| S Lee.. By applying (Ciss (Ciss:))EEIS) p)-EISS, we have
Cass
1-P (x ||| S CLss
S Cass | (xi
CiSs
Furthermore, recalling the definitions in Lemma E. we apply the -mooshhhss the dynamics f and take second-order Taylor expansion around (x, u) (x] x (x] )) to yield:
x Bi x" (x 1) + r
We observe this implies B1 ( - x (xf) E RT", and Lemma E. implies |B1l l| I Cass. On the other hand, since Wi.
1:2 = B Bi T, we know ||P|B S Vd. Since xf = x" we have:
PP - (xf) - - (xf)) Ciss 1-P $ Se
Ciss
||P B, (+ +x-=+)(x+)) VAil( "" >xf(x)l S VLL VTiL-Bes VAICIE
Ciss
|| < Greg (1xx x 11+111((]] - (x ))p < Creg
(1-P-)
< Creme?
Crss
which implies x²- X5² E V/2 This completes the base-case.
Now for T > 2, we assume the statement holds for T 7-1; 1; in particular, we have max 1IxTIisT--
x"" S e and x xi E V1 fort t E [ - 1]. Then, by (Cies. -EISSS we have:
|| x || S Cass
T- [[(xx) - x" (x+xil
1=1
< Cass
1r-1-t1(A ग (x7 + Ax.)ll
(1-8 Pg Cass
S Ciss
(Inductive hypothesis)
where Ax, x x7" and the last line uses the induction hypothesis that each Ax, E V/, t e T 1]. This completes the first part of the induction step. It remains to show x Xi E Vr. From the definition the lincarizations Eq. (E.2), we may write:
x
x
c B. +*) + A
(E.8)
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T-1 l
T-1 B. (x5 - PT ++
PT Σ
cl
Σ
cl St.rB, xxx(x)) + A s
1:T
8=1
where r are the second-order remainder terms from linearizing the dynamics around (x""""")))) for E [T 1]. We first observe by definition Σ.= A B - (x{{)
Ryy i.e. the first term on the first line lies the reachable subspace. Focusing on the first term of the second line, we may trivially bound:
T-1
Pr
Σ A B, +x5
S
| at 1:7 B,llll - x (x3 + Ax,))l
11 Se
Cissp-11 1-s
Ciss
where we used Lemma E.1 and the induction hypothesis for the last line. For the second term, we el
first observe that since WH: lir (xT") _ E A B,B, T A 1: T, we have // Ag s 1.T B,
S
IIP WH (xf yP/ | < VA- Alternatively, we always have by Lemma E.1 |P A cl
B,
S
Cisap" T- Therefore, picking any It E [T 1], we have:
P7
A 1+11 1:7 - (xx))
π <
PT A
LTT Blll
Ax.)ll I|
1
k AT Σ
Cissp-
Cles
CLas
(1-B8)
1
kv AT + p 1-
Ciss
Now, by solving for the optimal truncation point: ming21 liv AT p the resulting value by:
Crza,
we may upper bound I-p²
-k Cass
VxT
VAr(1 p) Crss log(1/1 1/p) ))
min h AT A21
1 (1+log log
1
log(1 1/p)
Vxr
log(1/1)
where for the last line we observe that vAT S
v Amaxx (WLT
S
Cras
by Lemma E.1, and thus T-p
log
(1r(1-p) s 0. Therefore, may plug this back in to yield:
CCiss log(1/p)
T-1
< VAT
1 (-Pe) VATCLe.
P+
A cl
B. (f (=x)==((((x))
log(1/p)
Ciss
As for the last remainder term, we have:
| 1
1-8 Ti--
(Lemma E.1)
|
I Creg
(|x;" -x [|x(xx) T'' (x. 012))
(Assumption 4.1)
622 + 2 n²' (x5 Ax.))1 + 21l + + - x" -x,5 I1²
Creg
< Creg
(1 1:22ca V (x ) ]Ax, + +1² e²
S Creg
(1 + 20 + 2Ck + 2C?c?) E²
< 20rog
(3 + 2C + 2C
(E.9)
T
Cass 2 2 Srams
Σ
el
1
< Crssp T T-1
< 2Creg (3 + 20
+ 20
1
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Therefore, putting all the pieces back into Eq. (E.8), we have:
T-1
PT
A cl
I.T
Ba
x" (x
+ PT
Ag At - - xx)(x,) + A
8=1
S sw
ATT Creme
|| |||| Iv]l 1 w VERi re R7 (AT)'
E Vr.
we have demonstrated x
xi- E
Vr, completing the induction step and thus the proof.
To review, we have established two key tools in Proposition E.5 and Proposition E.6, correspond- ing to Proposition 4.4 and Proposition 4.3 in the body, respectively. The first states that, fixing our attention to the component of the reachable subspace that is excitable above threshold À (to be de- termined in hindsight), we may bound the first-order, i.e. Jacobian error between and x in terms of their error the mixture distribution PT au.a. The second states that, fixing excitation level, as long as we ensure matches n sufficiently well on the set V. for each t, which decomposes into the "excitable", (linearly) reachable component in R= (>)),, the low-excitation (lincarly) reachable component in RD (A,)+, and a generic second-order term, the resulting closed-loop trajectories will remain close.
We are now ready to prove our main guarantee for noise injection.
E.5 GUARANTEES WITHOUT CONTROLLABILITY: PROOF OF THEOREM 2
We dedicate most of the effort into establishing the following result.
Proposition E.7. Let Assumption 4.1 hold. Let Cr be defined as in Corollary E.1 and Crem as in Proposition E. Let the noise-scale Tu 0 satisfy
{
log(1/p) Saab
log(1/p)² c
v1 + 4C% V1+40R/
Ou S min
(V
Cstab
(E.10)
C
L2
Cr
Cx
Consider a candidate policy ā. Defining Caraj & be + 60 (1 (1+ 1 Epaa + -rem) we have the following bound on the expected (clipped) trajectory error:
E
ExI~P
max ISST
|x
x7
A1
x5
dn E
S 2
Praj C2 Caab
[x "x(""" )11 Fx 11x - x -xx))1
log(1/p)o3)
max 1ST-1
T.
< Cgiab
++
1 +
Praj C2
JDEMO (ir .a.a).
log(1/p)o0
Proof of Proposition E.7. Let us define the shorthands for the per-timestep trajectory and estimation errors:
trai (#,
| x" 11
est (ft mT A |x
x (x7 1|1
r #*) A E xx,
|| (x.) - म (1)
|xp
As in Proposition E.6, let define sequence (D. where each X (Atii (x.,(xT]) }), Amaa (WP, (xT }),, as well as the truncated subspaces and projection ma- trices: RJ (dte, P. = (Ac). By Proposition E.5, noise injection certifies norm bound on Vx(* n" (xT")) restricted to Rt (Ac), for each > Accordingly, we define the event:
(c) A
max 6
|P, V ft - x *x Ilop S {c)
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We may decompose the desired quantity into:
x max 1S7-1
Tt
trai (1, x*)
Exx"
T (f, (Cstab)) 1ST-1
max
+ xx
max
rt !rai (T, x
(TT,"))1(88 1 (Cstab)
1ST
In addition to the requirements on ou in Proposition E.5 for X = X assume that du satisfies across CRC
Sinue such that d.o. t 2 2: ou Z V du C,
stab S
c² stab
Since ri !rai ( x*) S 1, we may then
x
bound 22 by:
T3 = Ex
x7'
max
Tt
traj (T, x (Gta)
SP IP
max (ST
|PVV - -T) Mlop 2 Cstab
du
S
max (ST
(T: ; + ri ( (+); Z Saaa
A
du maxt X
2 stab
B
est 1ST-1
max
It (it; It est (#; #*)
o
where the second line arises from applying Proposition E.5 and the noise-scale condition du x
de Sana and the last line comes from Markov's inequality. As for Tí, we set the decomposition
V tu Cr
for given (0, to be determined later:
T <
max
tra) (f, nt' ) (Cambt)) > de +P
max T1 STT-1
traj (f, xt" x1)
( >
Tf
Tf
First, writing out the requirement of Proposition E.6, casting E ++ VE, we have:
sup [w SL.ERR (Ax) LIET (Aa)+ |
[1( nt" (x] Vew
log(1/p)
Catab
Viver Cremev)
< [l( - x"' (xF
V - ( (VEw log(1/p)
Cstab VAvVer + Cremev)]
sup
[w Is,,eR (Ae))
IIIEEE (Ae)*
1vls
+ 2C 2=2xxl VVV +
(1p) V Av Jever + Clemev|2
Catab
< [I( x" xx)(x²") I| +
Gstab v VAVE + 20 L,FFE V - - (xx T w VE 2L
sup
log (1/p)))
s (Ar)
|w/l
6r 1+
Gstab
Crem
log(1/p)
< |1 - "" (x, Il x - N'" xx)(xx)) (x) Ve
de + Cpem ))
Cstab + 1og(1/) ToVe+ 22 Tor 6C,, ( log(pp)
Cstab
Let us interpret what this yields. On the last line, the first term is the on-expert error term yest (it; ' ) the second term is controlled by Proposition E.5, and the rest the terms are the errors for which do not guarantee control. To leverage Proposition E.6, it suffices to have the last line bounded by Catab Intuitively, the higher order error term scaling as è automatically satisfies this for sufficiently small e, which leaves the error term scaling as Vdye. This is where we set the excitation levels {X₂ } in hindsight. Observing the above, it suffices to set:
1 レ log(1/))) 129 2.
X
16
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In other words, for components of the controllability Gramian below this X,, the excitability is low enough such that we do not need to guarantee कt, x² match on them. For convenience, us now define the quantity:
Ctraj _ re + 60 1
1 Cstab
+ C
4 Lus
Therefore, setting T x Craaj stab stab we may bound 7ī by applying Proposition E.6:
T
P
max S
(i, n 1 (cstab)) de Pt
traj
F
max 1ST
sup || | (As) 1,,E (A.)* 1|v
(ff=x')(x) + VEW + Cstah
VEr + CremEv)]
0
Laa
(G
Seaab de
P
max IST
cast (ft;
||PeVx (xx Hep = + CZoje" (( 2 Stabe de
P
max It (#; (a; Z LiabE de SST-
(Def. of & (Cttab): Orajs S caab for ST
Stah
E
max rt (+:
IST
The bound on T follows similarly:
T max rf (#t; 2 oa stab?
S Caa Blubbey
max 1o0 (ft; (it:
(Markov's) 's)
SST
Putting everything together, we get the final bound:
X7'
max 15T-1
Pe
traj (ft 73))
< TT + + T2
d, max; Ar
< C2 stab
(1 Ciay 22 (tab)
max est (+::T*)
E Ex
(fr; क
KTT-1
o d, 12
1ST-1
C2
(1 Cray
C2 )
EST 1i (+)=
F xxp
max 1ST-1
est (#:
(#:: 5
log 1/p)
d. log(1/))20)
C
1 Crg Ciraj Oatab
Exy
max BST-1
0st (+t ((: π + (#::
which gives the desired result.
Therefore, by using the trivial bound J 2 < TExx> mmaxisesT ||x xx 1|2 ^ 1] and
applying Lemma D.I to translate to JIRMI,,T()) we get the final result.
Theorem (Trajectory error bound; full ver. Theorem 2). Let Assumption 4.1 hold. Let C, be defined as in Corollary E.I and Crem as in Proposition E.6. Let the noise-scale ou 0 satisfy
0u min {
log(1/p) caaab
log(1/p)² Aaab C,
+I+
Lydd Cz
D2d
Cstab
C+
(E.11)
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Consider a candidate policy it Defining Craj _ 20remLa + 6Cn (1+ 1 Guale + CPem) we may bound the trajectory error by the on-expert error the mixture distribution P as::
- I Le
dgL2 ( (1 Grn + log( log(1/p)2o3
J (tr
= (T) o, JDemo (f;;; Pxt,PP a...s..
We conclude this section with a few technical remarks. Remark E.1 (Horizon 7 dependence). We note the linear-in-horizon T dependence arises from naive conversion between max and E We note that Proposition E.7 can actually be interpreted as bounding JR, ( "o""max"-omma"" which does not exhibit any horizon dependence. We expect more fine-grained analysis, e.g. leveraging Lemma D.3, to similarly remove the T dependence from JTRALP,T and p. with the main technical barrier in extending Proposition 4.3 (Proposition E.6).
DEMo..... (ft::="",,,...55 for appropriately defined
Remark E.2 (Noise-scale du dependence). We note that the final bound in Theorem 4 has ou dependence. Firstly, we note that, by removing additive factors of ou (as in Sub- optimal Proposition 4.2 or Proposition 4.1), we do not need to trade-off du with the on- expert error JDEMO 2.1 (f;; 1"; .o..s... and can fact set o, as large as permissible up to the smoothness constraints, turning the dependence (1).). However, observing where du arises in the proof of Proposition E.7, it comes solely from applying Markov's inequality on the 2 stab We can envision instead applying
event P maxxgr-1 X.ok
du (r22 (#;=**) m²* r (+;; 2
Chebyshev inequality. For example, if we square both sides, we raise the estimation error to quartic in [|( x" (x)]l If the estimation error satisfies semeent-quivvlence conditions, such as (4-2) hy- percontractivity conditions that have appeared in prior lcarning-for-conte literature (Kakade et al., 2020; Ziemann & Tu, 2022), this pushes the au dependence an additive higher-order term. This crystallizes the intuition that the noise-level du actually enters the trajectory errоr as higher-order term (or equivalently, the burn-in), explaining why huge differences in du scale have similar ef- fects on the final performance (see Figure 1). We avoid introducing these technical conditions in the body for clarity. Similarly, we note the proof of Proposition E.7 also reveals that the on on-expert error on the mixture distribution Prr..... enters only via the term depending the number of noised trajectories need not scale proportionally to n. This explains why the final performance of imitator policy is often not sensitive to the exact proportion of noised trajectories a in the training data, as long as some trajectories are noised and some are clean; see Figure 3.
du. and thus similarly
E.6 GUARANTEES FOR ANY JIRNIP.TT p E (o)
As stated above, by nature of Proposition E.7, setting p * oo our trajectory error guarantee JTRAL in Theorem 4 naively accumulates linear-in-horizon T dependence. However, this horizon- dependence may seem qualitatively conservative; since the expert-induced system EISS, one might hope that past "mistakes" are forgotten exponentially. Determining this rigorously requires some additional effort, as we cannot rely on our linchpin result in Proposition E.6, which translates to per-rimestep control of on-expert error max,21 key recursion.
We first establish the following
Lemma E,8 (Key Recursion). Consider non-negative sequences (Arhe (A he2 that satisfy A = A = 0 and for all 2:
A
Cppt -1- C2pt-1-07 Cs p + Capt-1- + -1p--1--1
t-1
4
Cip 111-- Ca C2 p- + Cap--1-0A2
for constants Ci 2 1, Cz, Cs, C., C1 > 0, p E 10, 7 E [0, 1), and non-negative sequences
(Ta (Ea)021. Then, long as the following conditions hold:
(1 (1 C1)
C1
Eg S Emax I
To S Tmax S
Vs Ys21
C1 1
SC 1-p
Cz
(2(1
5C, I-p
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(1 (1- (-
7
C3 1
5CL 1-p
we have that A satisfies A S E 1 Es. t2 1, where T Ci
p = Ltp 2
Proof of Lemma E.8. Toward establishing the result, we posit the existence of a sequence (Ar) that admits form H, a C C.E --1- 2 2, where A _ A. 2 A, for all 2 1. We also Ess
posit corresponding sequence (4, h221, where AJ - 0, A C1 Σ pt Ee,, satisfying A 2 AF for all 2 1. We will determine PE E (p. 1), 0,01 2 C1 in hindsight. As in the statement, we further impose the constraints E, S maa Ts S Tmax: s 2 1, where Emax: Tmax: Y will be set in hindsight. It remains to determine that A, 2 4 for all t 2. For the base-case
t = 2, since A = A = O, we have trivially A₂ I Ce S Ce = Σ2. and 4 S Cie1 = A Now, given A. S A, for all s = I,...,t t 1, we seek to establish the induction steps A, S A., A A. Starting with A., we may plug in A, S I., s S t into the bound on D yield:
Cipt-1~gg, + T. C3 min p - =A. + Cap A-- -1p-11-A3
(-1
Cipt-1-88. + C2 T, + C3 m2 1 + Cap Capt-1-As, Cipt-1-923 Σ
We now treat each summand corresponding to C₁ C2 C3 C4, C1 separately. The first term in Ci straightforwardly satisfies X A, since p S p and C₁ S C. Toward bounding the second term, we expand:
c2p T. = Capt-1
k=1
C2OT maxx
p
Ek
1-2
C2Ormman max
Ek
1-1-p----
k=
8=k+1
1-k-
CTUTTaR max
EKP
(p/p)
j=0
Co. Tmax
(p' k-1
p² - -- )
P
C2 Tmax
E..
p p
C2 ensures the second summand satisfies X T TΣ p- = Te. We may treat the second-order term corresponding to C4 similarly: since by assumption 6x S Emax, 8 2 1, we have A, S I- Emax for all s 2 1. Thus, we follow similar steps to bound:
Therefore, setting Tmax sufficiently small Tmax I
(-1
Ca Emax
1-1
Σ Cip 1- A v 1 p
Σ
- A
[-1 C,Cxmax
Σ p
<
P) p) a=1
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Therefore, setting Emax sufficient small Émax X (1-P)(p-p)) ensures the last summand satisfies X C -= p p A. It remains to bound the third term. We first observe the following elementary inequality: given a, b E [0, 1], minfa, a, b} I a²b(I=s) for any E [0. 1]. Applying this to min(7 p² p-=1=*,, setting c 1 log (t2)) log(p) E (0, 1). we have:
Σ C3 1 Ozs
Cf
Es
p ptp/2
of 1-1
p pp-a-1es
Es.
p p-p
In particular, this suggests that as long as 7 & ((p - p) p)/(C() 1/ the third term satisfies X CE Ir. Lastly, given the inductive hypothesis on (A, for 1, I.... t 1, we may bound the C1 term:
Σ Cip D SCIC1
C10 P
Now, to complete the induction step on A, and At we determine values of C and C in hindsight. We first bound A h. Leveraging the bounds on the C1, Cz. and C4 terms above, we have:
Oip E 20307f p 00-1-EEa + (1 (p
C5Dxmax p p-1--
A
p-p
(C 2C2Cge
C4Csmax p
p-p (1-p)(p-1))
We now set C1 2C1 and
l+c. Recalling that = 1 log (2+e) / log(p) = 1
log( 1+30) / log(p), we may verify by calculus or software that c is a monotonically decreasing
function of p. attaining a limit from above of limp-t1_ = 1/4, such that yp I 7/4 for all p E (0, (), > 1. Therefore, setting:
6 (0- 1D-PIC )=)
(0-DC))) <1
2030
(1 P)²
Emax
8C,0
we have A I Σt=1 20 s- Es = C1 C1
& A
completing the induction step
A < A Given C. 20, and p = _ we return to the bound on A., where we may collect ...
all the bounds on the ..,... C1 terms to get:
C2C
2030na +
C.
+ C1C1
A S Ci
Σ
1-
p-p
p-p (1 (1- ()-p((--)) p-p1
(E.12)
2_ -- (CrCTmex + 20300214 1/4 2C,T5mmax + 22,01 =(c + 1-p
1-1
E..
1-p
remains to set bounds on Emax, Tmax, Y and set C such that the RHS satisfies S C C p--1-a Intuitively, we may tune Emax, Tmax, Y such that the C2, Ca, Ca terms are as small as needed; how-
ever, the C1 term cannot be further shrunk. Thus, setting C = Ci (1+ ) we may set the
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constraints in hindsight:
(1-P)CiC,
(1 p)1
max
CC4
C. (1+
C₁C1
C1
maax
C2C
C2
(1+
I-p
-(
CiC1 )²
C1
$
C3C
C3
1 +
T-p
Collating these constraints with (E.12), we have that under the constraints:
(1-p) Tmax
C1
(1 - (1-
Emax
C4
5C1 I-p)
C2 1
5C1
C3 +
5C 1-p
1-p
we have the desired bound:
C1
1+ t-1-8
A S
Ci
1
1
2
a=1
completing the induction step A, S A, and the full proof.
To instantiate Lemma E.8, we recall the decomposition of x - x," into the linear reachable and non-linear components (E.8), and the first-order Taylor expansion of 5 =xf?) xt" (x.)) around x
A la Bf - =x))(x)))
el
(x ( [x Vx -
(xf
r.
where r r the higher-order remainder terms. Further recalling the projection matrices P, & PRE (A) onto the top A; 2 eigenspaces of W and the orthogonal complement P.. (relative to the reachable subspace R. we may write:
el 1411
( B -f (x - xx") + (ASt.. Brr"" +A
[-]
s+1d B, ( (+ P. Add 2t1. B. x*)(x)) (x -x5')
+ P cl
s+1l Bg - TT, - xx,) PPAd B.Vx - x²) Tpp p- x5
+ (A #+1:1
Brr + A A
(E.13)
P. (x x
Σ P. - BB xx] + P.- Ac
B& Vx(a T') [x xx ) + P.t (A Bar Ag
(E.14)
We parse the expressions in (E.8) term by term.
1. First term: A A.t.B.B Ba x²) xx)((x)) corresponds to the contribution of the on-expert regression error.
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2. Second term: B, +xx(xx) (xf xx²" corresponds to the first-order policy error in the low-excitation subspace (i.e. orthogonal complement of R.T (A) for some À deter- mined later).
3. Third and fourth terms: P.A
a+ V xxx (x.')) xx + i: II V
xT (x"") TP x5" correspond to the time-l reachable component, decomposed further into the time-s reachable and low-excitation components. Intuitively, Proposition E.5 ensures P. x - (x, ) is small, while the P.' component is automatically small by virtue of lying in the low-excitation subspace, whose evolution is tracked in (E.14).
4. Fifth term: (A 1 + A corresponds to the second-order residual error controlled by smoothness (Assumption 4.1).
We now work to match (E.13) to the terms in Lemma E.8 Firstly, we recall by definition of P above that | 1P,A++:0 ++1. B Blop ||Act B.ll S Cissp-i-=e, s+1 loop I Csapp-1-s and
lIP, el Bollop I min VV, Cissp' 1-1 (ef. Lemma E. We then denote A - |x - xx I|.
++1.
A I (xf -x E1 _ |[( (xF IlI. T. A I|P x (x7 Ilop. and y vA/Ciss By Lipschitzness and smoothness (Assumption 4.1), we have IP, x (xF 'llop 2L
|r < reg (3 (3+2Ck- + 220 A3 (E.9), || s 2C, A3 Plugging these definitions and bounds into (E.13) and (E.14), we have:
A
Cis 1 =, + 2C min Vvainn(///C/sss 1-1---------- + 20 Lan -
+ 20 rss Lan 1 + (2Crog (3 + 20 + 2C? + +2C,) A?
4
Σ Cissp' Cissp'-1~8 -- 8, 20 min VX/Ciss p' ----- + (2Grog (3 20 + 2C + +C2,) A
Under the conditions of Lemma E.8, we have A < for t 2 1. Instantiating the constants in
Lemma E.8, we set C₁ = Cress C2 = 2C Lim, C3 = 20 L,y C4 = 2Crg (3+2C% 3 (
+
C1 = 2Ciss Tix, which gives the following bound.
Lemma E.9. Let Assumption 4.1 hold. For any initial state X1, let x xi = X1, and consider the closed-loop trajectories generated by â and n*. Defining the projections onto the reachable subspace P.= (X) and the corresponding orthogonal complement Pt relative to RT (Defi- nition E. As long as the on-expert quantities and excitation-lewe satisfy:
p13
1,
1
(1 p)
II( (x] II|
|P.Vx(* - xx=mm==))xx (xx Mlop S CissLe
Yt2 A
Creg(1 + L2 +2)
C L7
18s
then we have the following bound on the trajectory error:
Il, VE 2 1, C Cise + CrssLa)
1 1+p
| - x7 1 C Σ (+
1 p
2
Notably, by applying Lemma D.I and Lemma D.3, we get for any p 2 1:
1/v
1/8
x Cise (1 + Ciss
Σ x |P)
Σ
I( 1 x (x²7 1P))
(1 p)2
We note that Lemma E.9 bounds the trajectory error in terms of the on-expert regression error over the un-noised expert distribution. In particular, the only reliance on the noise-injecte expert distri- bution enters through ensuring x* - (x?" 'llop is sufficiently small via Proposition E.5. Intuitively, convert Lemma E.9 to a bound in terms of JTRNIT and JDCMO,T, we convert the re- quirements on [1 xxx(x" Il and V ग (xF Mlop into additive error bounds.
Proposition E. Let Assumption 4.1 hold. Let C₂ be defined as in Corollary E.1 and Crem as in Proposition E.6. Let R. (A), > be the truncated reachable subspaces (Definition E.I), setting
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(1-p)² Ciss L
Recalling Cr _ CT + 4Crog (1 + 4CK)) n let the noise-scale du > 0 satisfy
Aduu
VI+
++4C
ou S min
Stab Stab CT Catab
K
OA)
C CisLss
Consider candidate policy it. Define the probabilities:
(1 (1-p)s
p()) 2 P
r1 "t; ) (reg(1 + 122 + C + C+
du
p( P
(f; at") + rest (+; (+; 2 CissLex
[V
oo
Then, for any p 2 1, the order-p trajectory error may be bounded as:
1/p
C
-TT- + -P()))
T-1
( 1/8
(f;
Proof of Proposition E.10. Define the shorthands for the per-timestep trajectory and estimation rors:
S (A, m²)
|x x A1
: (#::T)) m- [x;," - x' (x7 )
(f; #
Bxx x [1[((X)) x (X,)
For given timestep t 2 2, define the event:
&, - -
|[ II+ -x)
(1 - p)
1
sE it et-
I Crog(1 + L2 + C2) + C
IP..V xx)(x" Mlop 2 C
in other words the burn-in in conditions described in Lemma E.9, up to time t 1. Then, we may write:
Ep
traj rt
(t, xt)
ED
uraj rt
(ft
+ Ep rt (ft x )
Ep
Pr
traj
(ft
1
+ Ep [1g.]
(-1
C
p
[1 (f: + +[E)),
where we applied Lemma E.9 to yield the last line, recalling C bound [P] we have via the union bound:
Crss(I+Crssbe) 1-p
A 1te To
2
(1-p) 3)
P[E]] <
[ - n (x I1
+P - nr }llop 2 Ciss
Crzr(1 + 12 + C2) C
(1 - p)3
du do
PP rost ( n 2 Cror(1 (1 + L2 C2) CT
+
[
(1est (1;3*();;)))
(+:
Ciss Lar
where we applied Proposition E.5 and the condition on au to yield the last line. Therefore, defining:
(1
dn oo
p( A
(
p!
A
(1 (#; (f;TT") + (#r; (+; 2 Ciss
Crog( (1 + 122 + C2) + C.
summing up the bound on Ep ( trai (+t) over / E [T] and applying Lemma D.3, we get:
1/P
(f)) S
C
Jt: (T;; +
T-1
(T - + +p!07)
1-
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1/0
Σ p(u + p(2)
1 DEMO (f; T1/p
We make few remarks. First off, setting p 2 and trivially upper bounding the triangular factor - S and applying Markov's inequality on p()) p(()) (squaring the arguments therein), we may recover the same scaling in Theorem 4:
JTRALp, (#) o "i (r: (.....0.s).
Notably, by the statement of Proposition E.10, we now clearly see that the dependence on and Pxt,au. solely comes from p( which from Lemma E.9 solely arises from the first-order on-expert policy estimation Vx(# - (xx(x"") Importantly, we observe that the horizon-factor T1/P only enters via the conditioning on the localization events, and in fact shrinks as p + co - this precisely lines up with the horizon-free scaling of the "max-norm to max-norm" bound J S (1) oe. Pm..o...))) were we to directly work with the "max-to-max" statements from TaSTI.-based guarantees such as Proposition E.3 and Proposition E.6, and the T1/2 scaling by square-rooting the bound in Theorem 4.
Shifting horizon-scaling to higher-order. By virtue of going through the effort of refining a TaSIL-based "max-to-max" argument to the direct sum-to-sum bound of Proposition E. 10, we have now isolated the error decomposition of J URA..P (i) into the regression error term JDEMo, (f; Px))) the local-
that is horizon-free, and the horizon- deeeedene probabilistic enror from conditioning ization conditions (viewed alternatively, the burn-in) of Proposition E. We see that we may apply any Markov-type inequality on p! and p\(); tion h:
for example, given a positive monotone scalar func-
P(U) S M(O((1)) Ep(M(rT* (+; (f;=*)]]
This necessitates controlling En (Mr **]]; without further assumption, the ability to do so typically a property of the learning algorithm (and loss function), c.g. square-loss regression EDll((** (x, 2[. However, certain statistical properties precisely convert between different loss functions. A prototypical example is hypercontractivity, such as the classic 4 - 2 hypercontrac- tivity (Wainwright, 2019), satisfied by various aub-Gaussian random variables.
Definition E.2. A scalar random variable X is 4 2 hypercontractivve there exists C4_22 > such that [| Cx
Under such an assumption, we may relegate the horizon-scaling localization terms to higher-order. Corollary E.3. Consider the assumptions and definitions in Proposition E.10. Assume rest (ft; x²) and est (#;#*) #t* satisfy 4 - 2 hypercontracttvity with constant C4-12 each E [T 1] over TP, and IP Po respectively. Then, we have:
(1- C
J
(f: (x4--27 ( (f; P.. 0..)
1-p)
We note that may optimize over moment-equivvlencce conditions; we refer to Ziemann Tu (2022) for various examples.
How fundamental is horizon-dependennee? A natural consideration is whether horizon- dependence should be present at all. In our analysis of Proposition E. the horizon-dependence arises from conditioning on the on-expert errors being sufficiently small for each time-step. We sketch an intuitive argument why horizon-dependencce may not be avoidable in general: on-expert regression necessarily only certifies that matches म around expert-trajectories. Since the nomi- nal dynamics need not be open-loop EISS, sufficiently large regression enrors (1) time-steps can induce closed-loop unstable dynamics, regardless ensuing on on-expert regression errors. Given regression oracle that only controls J (I=;:), and non-stationary expert trajectories, cannot without further assumption (c.g. algorithmic stability) guarantee error is delocalized across timesteps.
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E.7 LIMITATIONS OF PRIOR APPROACHES
One may wonder what a control-oriented analysis as above buys compared to instantiating prior guarantees in the imitation learning literature. In particular, recent work in LogLossBC (Foster et al., 2024) reduces imitation learning to estimation in the Hellinger distance, which is achieved by regressing in the log-loss. However, as observed in Simchowitz al. (2025), LogLos (and in the same vein earlier analyses (Ross & Bagnell, 2010; Ross et al., 2011) that rely on the {0, 1 loss) yields vacuous guarantees even for deterministic experts in continuous action spaces. Therefore, we consider fitting a noised expert and yield guarantees on the trajectory error of the resulting noisy rollouts. Contrast this with Theorem 4, where the trajectories used training may be executed noisily, but the trajectory error bound is measured rolling out the noiseless expert and candidate policies. As a last caveat, we note these works typically bound a cost suboptimality J(a) -J(x)) this is generally weaker notion than the trajectory error consider, which via the formalism of integral probability metrics (IPMs) upper bounds the cost gap (see e.g. Sec 2.3 of Simchowitz et al. (2018)). We introduce (stochastic) policies X (l1), where:
π (x) = oid, I..), II.
(E.15)
In other words, π encodes the deterministic policy и and ou-bounded voise-injection process Definition 4.1, where we specify to scaled isotropic Gaussian noise for convenient evaluation of dis- tributional distances. 12 In particular, π denotes the noisy expert policy. key step of LogLossBBC bounds the Hellinger error of maximum likelihood estimator via a log-loss covering. Define an e-log-loss-cover 11' of II: for all π E II, there exists ाí E Il such that for all x E X. u E U, log (I' ju x\/ IP ju x]) S Denote Niog (II, E) the smallest such cover. Then, the following guarantee on an MLE policy holds Foster et al. (2024, Prop. B.1).
Proposition E.11. Given trajectories of length T generated by the noised expert πT* define the maximum likelihood policy:
T
E arg maX
P
X 1
Then, with probability at least 1 1 8, the resulting generalization error of ft is bounded by
6 (6lgg(2Nng (II, 8)/8) + 4E
DH T²) S inf =>0
n
Now, we observe for Conditionall policies (E. ग. n², the log-likelihood ratio is given by:
dn log [1= x|/ Pr x]) = 2 (1]='()) - ull - [| - ull?).
Though the log-likelihood ratio unbounded over the support
Rdd,, we may truncate the
domain, wherein the scaling is similar to KL(T((x) T (x)), from which we have:
du
(L(T(xx) || (x))
[(x) (x)11
20
Notably, this implies an e-cover in maxxex (L(T(x) || ('))) is equivalent to a v 20 s- cover of II in d(T, T') & maxx [xx T' (x)))2 For parametric classes with parameters in Rdo, log (II, do log and thus converting between an E² and KL cover only introduces ad- ditional logarithmic factors of ou. However, for non parameetiiic classes such as those in the lower- bound constructions in Theorem (Simchowitz et al., 2025), log Na(II, E) z poly(1/s). and thus and introduces additional polynomial factors
converting to KL cover worsens the dependence of on and d,.. Contrast this with Suboptimal Proposition 4.2 or Theorem 4, where the dependence is always o,,² regardless of the statistical capacity of क,, x* E II, since we covering in l over the deterministic class, rather than in KL over the conditional-Gautt Gaaussinn class. In either case, we recall that this route of analysis ultimately only controls the rollout cost of noised policies. We now establish in the sequel, as insinuated by the upper bound in Suboptimal Proposition 4.2, imitating purely on noised expert demonstrations yields an unavoidable bias scaling with du-
This technically violates boundedness, but this is of minor concern by concentration of measure.
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Suboptimality of only regressing noise-injected trajectories To underscore the importance of imitating on both noise-injected and noiseless expert trajectories, we show via a simple example with maximally benign expert closed-loop dynamics that even perfect imitation trajectories necessarily incurs an additive factor in the trajectory error scaling with the smoothness of æt and the noise-level o². Consider the system X++1 X, + up, expert policy "t (xx) = -X,-
noise-injected
Proposition E.12 (Full ver. of Proposition 4.1). ) Let the horizon T and xf" be fixed with |xx || 1. Fixing any du (0.1) and Cx > O, let z Dz be any log-coneay distribu- tion with mean-zero and covariance sutisfying En 2 2dn Id.. and recall the corresponding noised expert states Definition 4.1. Then, there is class of policies P where any it E P satisfies: 1.
1 10=l((m** - (o = with probability Z 1 - n dx)) where z(() 1i Dz,, 2. (x) n² (x) for all ||x]| Ou 3. it is Cz-smooth. However, the trajectory error induced by rolling out & is lower-bounded by:
J 2 O
In other words, even when the candidate policy fits the expert perfectly on moise-injected expert tra- jectories, the trajectory error of the policies necessarily suffers drift proportional to the smoothness budget C, and noise-scale a i.e. policies E d and n² are indistinguishable under purely noise- injected trajectories. On the other hand, a single un-noised trajectory from and x can distinguish between the two policies perfectly.
Noting the expert closed-loop system here satisfies Creg O. CK 1, Cstab to the key coppectationtto-uifooo step Lemma E.4 in establishing Suboptimal Proposition 4.2, where this lower bound matches the drift in the upper bound of Lemma E.4.
may compare
Proof of Proposition E. We first write out the noiseless expert's trajectory:
X1,
x 0, x 0.
In other words, the expert reaches 0 in one timestep and stays there. Now consider the expert under the noising process n (x) au - un Dz: letting 21, Z2 i.dd Dz be two i.i.d. draws of noise, we have
x = 82 = (-X1 onZii o x o. ou
In other words, after timestep 1, since the expert policy always perfectly cancels out the previous state, the distribution of noised expert states identical the noise distribution o.z. Therefore, the intuition for the lower bound is as follows: by concentration of measure, any "usual" distribution (e.g. log-concave, subgaussian) that has con-anishiin excitation, as captured by the second moment Ez - Cd Id., necessarily concentrates on the ((1)a scaled unit sphere $d.. 13 Therefore, given independent trajectories, i.e. independent draws {z,] Z-2
(4) (i) }}, with high probability we do not see
any states x(", x(6) within an R ou radius of the origin. This is formalized in the following lemma (Paouris, 2006; Adamezak al., 2014).
Lemma E.13 (Paouris' Inequality (Paouris, 2006)). Let z be a log- con-conavve random vector that with zero-mean and identity covariance supported Rd. Then, there exists universal constant such that for any 7 2 1: IP |z|| 2 orvd < exp(->vdd)
Therefore, re-scaling z such that Σn
2d. La and setting Y
/2 this implies: 2c
[||||| 2 1/2] < exp(- v day) ss exp( - dd.)) Union bounding over I,..., n. we have 2e
P |
11 2 1/2, vi E [n]] ス n exp( d.). (d.))
Given that the noised expert states concentrate Ou away from the origin with overwhelming probability, we now task ourselves to constructing a family of candidate policies î that maximally deviate from the expert policy at the origin, given its smoothness budget C,. This can be achieved, for example, by straightforward bump function construction.
We note that when D2 is the uniform distribution on the unit sphere gdy, then we may interchange the high-probability guarantce with expectation E[][( x T')(ou)) = 0.
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Lemma E.14 (Bump function existence, c.f. Simchowitz et al. (2025, Lemma A.15)). For any E N, we may construct a function bump,(-) R + R. bump, C succ that the following hold:
1. (umpa(z) = for all
2 S
| 122 2. 3. For each p 2 1 and z E ER", Vybumpa(z)loo I Cp. where cp > is a constant depending on p 0 but independent of dimension d.
2. bumpa (z) = 0 for all
4. VPbumpy(z) = 0 for all [|]] 2.
In other words, we may construct function that always outputs 1 in the unit sphere, and outside of the radius sphere, and has bounded-norn derivatives between. Before proceeding with the construction, we observe that (x) -x is linear function, and thus satisfies V² (x) everywhere. For given > 0 and smoothness budget Cx >> 0, it therefore suffices to determine that satisfies the properties:
ÀT
1. A = O for all ||x|| 2 du/2.
2. V (x)) Cz.
We construct AT follows. Fix any E gd.,, and let bumpa, (-) be a function that satisfies the properties in Lemma E.14. We propose:
AT(x) - Lbumpa, 0u/4
x
(E.16)
where > is a constant to be determined later. We observe that by construction: AT for all |||| 2 / A L, and [x)A+(x)llpp L L (etpbump_(Ax//na)))
Lcp du
Therefore, to ensure A C,-smooth, this informs choosing a CT. Therefore, the resulting policy मे ग* + AT satisfies the following properties:
16cz
1. A is C -moooth.
2. [0)] Cx I0cs
3. f(x) = -² (x) for all ||x|| 2 ou/2. In particular, by Lemma E.13 that 1 E (#
n
x* (') = 0 with probability 2 1 - exp((-Vil dw)
Now, we roll out स and n² without noise injection. We have as aforementioned x xg = 0. On the other hand, since our construction of म, x3 x + (xx)
1, we have #(x π (x7") and thus x x 0. However, by 00) C.a (1602) -1, and thus:
|| - x
x3 O(1)C,o3
max 1.2.33
After squaring both sides, we see the left-hand side is precisely 2 which is trivially upper bounded by JTRAL2,T
We note extending the construction above general, possibly improper learners, follows by noting that á and n² are constrained to generate near- near-indisting bear-indistinguishabbbe trajectories on P..,,. we refer to Simchowitz al. (2025) detailed minimax formulations. This lower bound establishes the unavoid- able drift from noise-injection due to nonlinearity of the expert policy, thus highlighting the necessity of imitating on dataset consisting both noise-injected and clean expert trajectories; though, as discussed in the previous section, the exact proportion of each is not necessarily important.
F EXPERIMENTAL VALIDATION
Action Chunking. To validate our predictions about the stability benefits of action- chunking, we propose experiments on robotic imitation tasks in the robomimic framework (Man- dlekar al., 2022). In particular, we pre-train performant state-based deterministie expert policy on r robomime mi e data, which we then roll out to generate training data. We fit models of the same ar- chitecture except the final output dimension of varying prediction horizons. We then execute varying
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numbers of the predicted actions in open-loop and evaluate the resulting success rate. We observe the findings in Figure 2; all experiment details can be found in Appendix G. In short, we find that:
Executing action chunks matters more than simply predicting longer sequences of actions. This demonstrates the action- cihionnchunning is more than a simple consequence of representation learning, simulation of receding- horrizon control.
The merits of action-chunking remain showcased in deterministic, state-based control. This reveals that action-chunking still improves performance independently of partial observability or compatibility with generative control policies.
End-effector control enables the benefits of action-chunking. This because end-effector control renders the closed-loop between system state and end-effector prediction incrementally stable (Block et al., 2024). Hence, the low-level end-effector controller transforms imitating the position policy to taking place in open-loop stable dynamical system, precisely the regime where prescribe our AC guarantees. Accordingly, in MuJoCo tasks that lack this property, we find that naively action-chunking hurts, not helps, performance- see Figure 3.
We emphasize the above remarks are not to rule out the role of non-Markovianity and representa- tion learning; it is likely that these contribute further, e.g. AC can demonstrably prevent "stalling" from demonstrations with pauses. Rather, our results should be understood as stating that, instead of the benefits of action-chunking existing in tension with control-as controls folk-knowledge typ- ically cautions against open-loop execution- it can be naturally explained by a control-theoretic perspective.
Noise Injection. We seek to validate our hypotheses about the exploratory benefits of noise- injection, making partieular note the algorithmic suggestions that our theoretical analysis reveal. We propose experiments on MuJoCo continuous control environments, where we seek to imitate pre-trained expert policies. We observe the findings across Figure and 3. To summarize:
Noise injection as in Practice 2 provides the exploration necessary to mitigate compounding errors, increasing performance on par with iteratively interactive methods such as DAgger (Ross et al., 2011) 1) and DART (Laskey et al., 2017). We note Practice 2 collects data in one shot, without ever observing learned policy rollouts.
Larger noise scales ou (within tolerance) improve performance, in contrast to prior under- standing (cf. Proposition 4.1, Suboptimal Proposition 4.2) which necessitates on set proportional to JDEMo, (f;; P demo), i.e. very small for policies with low on-expert error.
A mixture of noise-injected and clean expert trajectories is beneficial, and the difference is small when provided more data, as suggested by Eq. (4.1). This matches the theoretical intuition that noise-injection is necessary up until Å is "locally stabilized" sufficiently well around x"" (Proposition 4.3 and 4.4), and thus only enters the trajectory error as a higher-order term, i.c., we only need "a sufficient amount" of noise-injection.
G EXPERIMENT DETAILS
G.1 SYNTHETIC "CHALLENGING EXAMPLE" (SIMCHOWITZ ET AL., 2025).
Model:
two-hidden layers of dimension 256 and GELU activations (Hendrycks & Gimpel, 2016). We remove layer biases to introduce mild inductive bias of the model outputting at the origin.
Optimizer and training:
the AdamW optimizer (Loshchilov, 2017) with cosine decay learning rate schedule (Loshchilov & Hutter, 2016), with initial learning rate of 0.001 and other hyperparameters set default. The models are trained for 4000 epochs with batch size of 64. Evaluation statistics of each model are computed on an independent sample of 100 trajectories.
For the action-chunkng experiment in Figure 1, we consider synthetic nonlinear system that is open-loop EISS, and closed-loop EISS under a deterministic expert, constructed in Appendix E. and of Simchowitz et al. (2025). In particular, we first consider matrices:
A
1 + f
2 1- 2/4
K
[
3
11)
0
11
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where we set μ 1/8. We then embed these matrices into 6-dimensional state and input space:
K
0 0
A
K
R6x6
These matrices are respectively embedded into smooth nonlinear dynamics f and expert policy x* as described in Construction E.1 (Simchowitz al., 2025). For the requisite smooth function g the embedding, we generate a randomly initialized neural network with 1-hidden layer of dimension 16, with weights following the Xavier normal initialization (Glorot & Bengio, 2010) and biases sampled entrywise from [nif([-1, 1J); note that we only generate this network once to complete the problem instance. Having generated "hard instance" indexed by (A, (A, 9) the training data is comprised of 100 independent trajectories of length 64 rolled out under the expert policy π². For Figure 1, we train behavior cloning agent; for each chunk length, the BC policy takes the form described above, with the sole difference being the output dimension, which equates to 6 x chunk_les Given the training recipe above, all BC policies across chunk lengths E {1, 4, 8, 163 achieve training error of at most 10-6, attaining near perfect imitation on the expert data.
Crucially, we note that, in contrast our formal prescription in Practice 1, we are not enforcing that the chunked BC policies accompany a simulated dynamics that it stabilizes, and purely treat the policy as an E-step action predictor. Beyond the soft inductive biases in ensuring the policies output 0 at the origin, we make no effort to enforce "simulated" stability, yet we still see the clear stabilization benefits of action-chunking in Figure 1.
G.2 ACTION-CHUNERE EXPERIMENTS ON ROBOMIMIC
Model: for the expert and learner policies, we use a proprietary flow-matching policy parame- terization being developed in concurrent work. In short, the policy backbone architecture adopted from the seminal Diffusion Policy (Chi et al., 2023), which is built on top of a 1-D U-Net (Janner et al., 2022) with FiLM conditioning (Perez et al., 2018) on the observation and flow-timestep t E [0. 1]. All feed-forward hidden-layer components have width 512 across the expert and learned policies. For expert trajectory collection and learned policy evaluation, we set the flow-policy to deterministic mode, i.e. setting the prior distribution to all-zeros.
"Chi-UNet"
Optimizer and training: we use the A optimizer (Loshchilov, 2017) with a cosine decay learning rate schedule (Loshchilov & Hutter, 2016) decaying across the training horizon, with initial learning rate of 0.001 and other hyperparameters set as default. The horizon E {4, 8, 16) models are trained for 1000 epochs with a batch size of 1024, and the horizon E [32, 64) models trained for 2500 epochs to account for the larger output dimension and thus more difficult prediction problem. For a given training trajectory budget (e.g. 50 or 100), we collect trajectories where the task is successfully executed.
Evaluation: For the plots in Figure 2, we train three independent models per configuration and record success over 50 evaluation trajectories each. We then plot the percentile boostrap estimators (median, shaded 10-90 percentile) across the 3 x 50 evaluations as informed by Agarwal et al. (2021).
We recall prior hypotheses about action-chunking's effectiveness include: 1. robustness to non- barkovianityjartill-bseewabb
2. amenability multi-modal prediction. 3. improved represen- tation learning, and 4. simulation of receding-horizon control. We consider the following set-up: first train a flow-matching (i.e., generative) position control policy on full-state robomin mi i data to yield performant expert policy, after which we generate imitation data by rolling out the expert policy deterministic mode. By construction, this ensures the imitation data comes from jully-observable, deterministic, Markovian expert policy, ablating away contributions from the first two points. This leaves improved representation learning and simulating receding-hone horizon control as the remaining alternate hypotheses.
On the other hand, our analysis in Section 3 suggests that: 1. executing open-loop chunks of actions is key, 2. performant chunk-lengths can be relatively short: our theory predicts logarithmic in system parameters is sufficient (Theorem 1). The second point is important, as slow-gro now-growing benefits of action-chunking would come into conflict with the perils of open-loop control. We remark that imitating position control (i.e., end-effector control) as opposed to joint/torque control crucially aligns with our key condition for prescribing action-chunking: stability of the open-loop dynamics
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(Assumption 3.1). Position control is implemented by providing mid-level position commands that are tracked by figh-frequency joint/torque controllers- this low-level tracking ensures that given a plan of desired positions, reasonable differences in commanded versus realized positions do not lead to diverging trajectories (Definition 2.1). This hierarchical set-up is the backbone of modern robot learning, and failure modes of direct imitation sans tracking/stabiizzttonn well-documented; e.g., (Block et al., 2024; Mehta et al., 2025).
To test our hypotheses, given the collected expert data, we consider training multi-step imitators with the same architecture as the expert except the output dimension, that predict varying horizons of actions horizon {4, 16, 32) conditioned on the current state, and evaluate them at varying chunk lengths {1, 2, hori 2on] measuring task success. We display the results in Figure 2. Though there some gain from longer prediction horizons, (Venkatraman et al., 2015; Somalwar et al., 2025), it is less evident in lower-data regimes (see Figure 2, right) and is counteracted longer horizons due to greater strain on a given architectural capacity. On the other hand, the greater gain regardless of hori zon come from longer evaluation chunk lengths up to a point before decaying predictably due to open-loop control- 32 actions 20Hz control frequency corresponds ~ 1.5 seconds in open-loop.
line with bearning-theorett work
We also note that noise-injection, while preseribed for the open-loop unstable setting, is also ap- plicable in this setting: Figure 2, right, where add = 0.05-scaled Unif(S noise when executing the expert's actions for half the training trajectories. However, we remark that for long chunks, action-chunking removes supervision from intervening states, thus training long-chunk pre- dictors on noise-injected trajectories turns beneficial local exploration into uncertainty, since the predictor must fit chunks of actions to seemingly noisy targets.
We note that though Theorem pairs a candidate policy with a simulated dynamics to perform chunking. here we are simply fitting a multi-step action predictor. Despite this, we still observe the stark benefits of chunking. This hints at the role of architectural inductive bias.
G.3 NOISE INJECTION EXPERIMENTS ON MuJoCo
Model: two-hidden layers of dimension 256 and GELU activations (Hendrycks & Gimpel, 2016). We also additionally place batch-norm layers after the input and first hidden layers.
Optimizer and training: we use the A optimizer (Loshchilov, 2017) with a cosine decay learning rate schedule (Loshchilov & Hutter, 2016), with initial learning rate of 0.001 and other hyperparameters set as default. The models are trained for 4000 epochs with a batch size of 100.
Evaluation: For the reward training trajectories plots across Figure and Figure 3, we train 5 independent models for each configuration, and compute evaluation statistics independent sample of 100 trajectories. We then compute the percentile boostrap estimators for the median and shaded 10-90 percentile.
For the noise injection experiments depicted in Figure (left) and Figure 3, we used the HalfCheetah-v5 and Humaniod-v5 environments through the Gymnasium library (Towers et al., 2024). The expert policy for HalfCheetah is a Soft Actor-Critic (Haarnoja et al., 2018) RI. policy pre-trained using the StableBasclines3 library (Raffin et al., 2021), downloaded from Huggingface [url], and the Humanoid expert is Truncated Quantile Critic (Kuznetsov et al., 2020) RL policy ob- tained similarly. When collecting expert demonstrations, we set determin mimiisiii isst st Trree Given noise-scale ou.
scaled spherical noise du Unif(844 ) as the noise-injesd injection distribution. We set the trajectory horizons at T = 300 timesteps. For each figure specifically:
Figure 1 (center + right): We sweep over noise-levels on E {0.0, 0.5, 1.0} for HalfCheetah and oa E {0.05, 0.1, 0.25) for Humanoid, fixing the proportion of clean trajecto- ries at 50%, equivalent to imitating over P.. .... from Practice 2. We note that noise-level 0.0 corresponds to vanilla behavior cloning. Since the Humanoid environment terminates early when the agent falls over, we crudely pick the upper noise-limit for Humanoid by setting it such that the total collected timesteps is 80% of the maximum possible #traj x 300. We similarly run DAGGER (Ross & Bagnell, 2010) and DART (Laskey al., 2017), where split given training trajectory budget into 5 equal rounds of expert trajectory collection model updates. We found a performant mixing parameter for DAGGER to be f 0.5.
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Published as a conference paper at ICLR 2026
Figure 3 (left): We consider for du {0.5, 1.0) the effect of recording clean versus noisy action labels. Recall that Practice prescribes executing expert actions noisily X,-1 = f(x, π (x,) + o, but records the clean action label u, T (X,). On the other hand, the "RL-theoretic" approach (Appendix E.7), in order to achieve density, also requires recording the noisy label u, n²" (Xt) JuZe. We fix the proportion of clean trajectories 0.0 for both set-ups for fair comparison.
Figure (center): We sweep over proportion clean trajectories a E {0.0, 0.5, 0.8, 1.0), holding noise-level du 0.5 fixed. We note that 1.0 (no noise-injection) corresponds to vanilla behavior cloning, and a = 0.0 corresponds to pure noise-injection Prr (see Proposi- tion 4.1).
Figure 3 (right): We consider fitting multi-step chunking policies. We naively extend the output dimension to chunk chunk_length
du and play the full chunk open-loop. We note this does not necessarily rule out some form of advanced action-chunking recipe from enabling performance; however, where naive multi-step predictors benefit in the robomimic set-up, they do not appear do here, likely due the open-loop instability of the environments (e.g. lacking low-level stabilizing controllers).
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Reviews:
[{"review_id": "CxJWBfrlyh", "reviewer": "Reviewer_Xggu", "summary": "This paper gives a novel and rigorous analysis connecting two widely-used heuristics in continuous imitation learning, (1) action-chunking and (2) expert action noise-injection, to control-theoretic stability and compounding-error guarantees. \nUnder the notion of an open-loop exponential incremental input-to-state stability (EISS) assumption on the true dynamics, \nthe authors prove that sufficiently long action chunks yield horizon-free trajectory error bounds (Theorem 1). \nWhen the true dynamics are not open-loop EISS, they advocate Practice 2: collect a mixture $P_{\\pi^\\star,\\sigma_u,\\alpha}=\\alpha P_{\\pi^\\star}+(1-\\alpha)P_{\\pi^\\star,\\sigma_u}$ (noise injected at execution but clean action labels).\nThis paper also show that this mixture certifies first-order matching on the controllable subspace, producing a trajectory bound that removes the worst-case exponential blowup (Theorem 2). \nEmpirical validation on robomimic and MuJoCo tasks supports the theory and compares the offline noise-injection approach to baselines.", "questions": "Q1. Could the authors recommend a practical criterion (estimable stability indicator, threshold) to decide when to apply action-chunking?\n\nQ2. Could you justify the specific choice $\\alpha=0.5$ used in Proposition 4.4 and Theorem 2?\n\nQ3. Line 463 states “Practice 2 collects data in one shot, without ever observing learned policy rollouts.” By “one shot” do you mean a single interaction of many trajectories (not literally one trajectory)?\n\nQ4. Beyond the offline protocol, Practice 2 could plausibly be applied interactively (i.e., inject noise into expert actions when the expert is queried during training, similar to DAgger-style interaction). Do the authors see this interactive variant as theoretically or practically different from the offline mixture?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "**1. A novel control-theoretic perspective on imitation learning.**\n\nThe paper reframes two widely used practical heuristics (1) **action-chunking** and (2) **expert action noise-injection,** in control-theoretic perspectives, centered on exponential incremental input-to-state stability (EISS). \nBy reducing compounding error to a stability property of the true dynamics $f$ and the policy–environment closed loop, the work gives precise conditions under which each heuristic is principled. \nThis perspective clarifies when a practitioner should prefer chunking (longer chunks $\\ell$ when $f$ is open-loop EISS) versus noisy expert rollouts, and it interestingly connects IL practice to established stability concepts rather than treating the heuristics as ad-hoc.\n\n**2. Mutually aligned theoretical and experimental results**\n\nTheoretical predictions, (i) action-chunking yields horizon-free trajectory error under open-loop EISS, and (ii) noise-injection facilitates first-order matching in controllable subspaces when open-loop stability fails—are supported by empirical results on Robomimic and MuJoCo tasks.", "weaknesses": "**1. Narrow applicability of action-chunking under EISS assumption**\n\nThe primary theoretical guarantee for action-chunking (Theorem 1 / Propositions 3.1, 3.2) is stated under the strong assumption that the **true dynamics $f$ are open-loop EISS**. \nThis crucial scope condition is only briefly noted in the text, but the paper lacks a clear characterization of the method’s behavior when that assumption is violated. \nTo be more useful to practitioners and convincing to readers, the manuscript should clarify the conditions where chunking is recommended (e.g., global vs. local EISS), show how the bound degrades when stability weakens, and provide either a weakened theoretical statement for local/probabilistic EISS or empirical sweeps that identify where chunking helps versus hurts. \n\n**2. Limited empirical scope and reproducibility concerns**\n\nThe experimental validation is limited to a small set of tasks, which makes it difficult to assess the generality of the claims. In addition, the authors have not released the code or full data-collection scripts, which prevents independent reproduction and detailed ablation.\nThe paper also omits results and practical guidelines for applying the two practices jointly; reporting combined-case experiments and decision rules would be valuable.\n\n**3. Overbroad use of the term “data augmentation”**\n\nThe use of the generic term **data augmentation** in the title and abstract overstates the scope of the empirical/theoretical contribution: the manuscript studies a specific augmentation (additive action noise executed during expert rollouts with *clean* action labels) rather than arbitrary augmentation families (state perturbations, visual augmentations, domain randomization, etc.). This wording risks misleading readers and reviewers. To avoid overclaiming, replace or qualify “data augmentation” in the title/abstract/introduction with a precise phrase (e.g., “expert action noise-injection”, etc.), and if broader augmentation types are intended, either provide evidence for them or explicitly mark such extensions as future work.\n\n**Minor comments** \n\n- The paper’s presentation is a quite dense and often assumes control-theory familiarity that may not be shared by the broader RL/IL audience (who is more common ICLR reader); this reduces accessibility and reproducibility. \nAdditionally, Figures 1,2,3, whose experiments are discussed on page 9, are currently shown much earlier (pages 5 and 7) and therefore disrupt the narrative; these figures should be relocated.\n\n- In Theorem A, $\\mathcal{P}^\\star_{\\mathrm{stab}}$ \nand $\\mathcal{P}^\\star_{\\mathrm{unst}}$ are defined in a circular way, which does not seem a proper definition.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ni7DiMGD2R", "reviewer": "Reviewer_zW76", "summary": "This paper provides theoretical justification for two major ways that people have done imitation learning in robotics: action chunking (where an agent chooses K > 1 actions in parallel to execute) and data augmentation (where one can learn a generative model to generate more useful synthetic expert data to imitate), particularly in a state-based continuous control setting with deterministic dynamics.\n\nTheir theory relies on the notion of EISS, in which perturbations to actions and next states have a bounded effect over future state distributions under the dynamics across time. In particular, there is a clear upper bound on the deviation between states far away in time, depending on the initial perturbation amount. This, combined with Lipschitzness of the policy class, yield upper bounds for compunding error that scale polynomially with respect to the Lipschitz coefficient of the policy and the EISS coefficient of the dynamics. Furthermore, their theory shows that one can achieve a strong compounding error bound with a chunked policy, where the compounding error coefficient is independent of the horizon of the problem, which is a strong result.\n\nIn the case where the dynamics are not open-loop EISS, then the authors propose noise injection, where one injects uniform noise to the actions selected by the learner at each step. In this particular case, the imitation learning procedure is to then imitate both over clean expert trajectories and \"noised\" expert trajectories.", "questions": "See my thoughts in the Weaknesses section -- I feel like this is more of a misunderstanding than anything.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "I am not a theoretical expert nor a control theory expert, but it seems that the theory is decently sound, at least from the control theory perspective. This was corroborated by looking into Simchowitz et. al. (2025), the closest work to this.\n\nThe content is also quite novel, as no one (at least in robotics) has looking into an analysis of data augmentation and action chunking (aside from Simchowitz et al. (2025) partly), as it seemed to be more of a practical trick than a deeply analyzed method. Moreover, the authors analyze different components of learning that of Simchowitz et. al. (2025), notably the policy parameterization (chunking) and data collection (e.g. noisy expert), different from the learning algorithm perspective that the prior paper takes.\n\nThe empirical results were also quite compelling, and showed that the combination of both chunking horizon (e.g. Figure 2, middle & right) and noising of both actions (e.g. Figure 2, right) and trajectory data (e.g. Figure 3) yield improvements in imitation performance beyond that of standard BC.", "weaknesses": "While I am not a control theory person, it was quite difficult for me to follow along with the theory in this work. For instance, I didn't know what the difference between \"closed-loop\" and \"open-loop\" was, particularly when it came to defining the EISS notion (e.g. Theorem A). As such, while I was able to semi-follow the theory on action chunking reasonably well, I still had some questions regarding definitions. The same went for the noising analysis, which I found difficult to read through. I think that presentation could be fixed there, and that would clear things up for anyone who reads the paper, theoretician or not.\n\nThere is a typo in Eqn. (2.2) -- need a T in the subscript on the LHS. Small nit.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qDr66x2h9U", "reviewer": "Reviewer_Y1gT", "summary": "The main point of the work seems to be that predicting a suitable sequence - chunk -  of actions ( 2, 4, 8, ...) instead of a single action can be beneficial for imitation learning of certain tasks. The same finding holds for adding noise to training data. This are related to the compounding error problem of offline learning (small deviations in the real world accumulate). In online learning these problems are solved by using the current policy in sampling new data (DaGGER). \n\nI believe these findings are somewhat known in the scientific community. The authors claim to derive \"theoretical guarantees\" (lines 46-47), but their writing is so difficult to follow that I cannot understand what the authors mean by those. Moreover, Sections 5 and 6 do not anymore discuss about the theoretical guarantees and their consequences which makes it confusing what really are the main contributions of this work.", "questions": "If the authors can restate their research question(s), main contributions, and findings in more concise manner", "limitations": "", "rating": null, "confidence": 3, "soundness": 1, "presentation": 1, "contribution": 1, "strengths": "Action chunking is still a rather new methodology in imitation learning and therefore makes it an interesting topic to study and investigate its properties and reasons why it is sometimes beneficial.", "weaknesses": "**Major weaknesses:**\n\n - This paper writing style is very complicated and therefore I cannot really follow what are the main research questions, contributions, and the main findings. The authors should simplify the text and clarify the contributions. In this form I cannot accept the work.\n\n**Moderate weaknesses:**\n\n - The main contribution is rather complicated to be easily understandable: \"We provide the first theoretical guarantees in continuous state-action IL for interventions that provably prevent compounding error without iterative expert feedback.\" This contribution leads to *two practices* that result two \"surprising\" takeaways. So contribution is limited to practices that result to surprising takeaways. Could this line of thinking be simplified to something easier to understand and follow?\n\n - Text is partially difficult to follow (perhaps too extensive use of AI language correction). For example, the second paragraph of Introduction: first sentence is 5 plus lines and the last sentence is more than 4 lines. Such monster sentences appear all over the manuscript.\n\n - Usage of figures is not good. For example, Figure 1 on the page 3 is not referred until page 9. Figures should flow with the text and not be randomly around the manuscript.\n\n - Main findings from the experimental part (page 9) are difficult to comprehend. For example, in Figure 2 it seems that prediction horizon and action chunk 16 result to the best result (should not come as surprise that the both are them same). By adding noise, some of the prediction horizons improve (e.g. 8), but on the other hand the best performing 16 performance degrades. So the effects of noise and chunking are not complementary always. [Btw. \"left\" and \"right\" in a figure with three images is confusing, why not (a), (b), and (c)? Moreover, marking 2³ instead of 8 is odd until the purpose really is to confuse the reader from the obvious]\n\n**Minor weaknesses:**\n\n - Citations should be updated. Some of them are listed as ArXiv papers despite that the paper have already been published in a conference or journal (e.g. Chi et al. 2023)\n\n - Use of colors to emphasize parts of the text is already a bit exhaustive. On page two there are normal text, bold text, blue text, and red text.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)