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How Generalizable Is My Behavior Cloning Policy? A Statistical Approach to Trustworthy Performance Evaluation

Joseph A. Vincent, Haruki Nishimura, Masha Itkina, Paarth Shah, Mac Schwager, Thomas Kollar

arXiv:2405.05439v2cs.ROcs.AIcs.LGstat.AP

TL;DR

Small numbers of costly rollouts make behavior-cloning policy performance difficult to assess, especially under distribution shift. The paper develops confidence bounds on the full performance distribution and evaluates them in simulation and on hardware. Experiments validate the bounds and show, at 95% confidence, that RT-2 outperforms VC-1 in each tested setting.

  • Problem

    With few costly real-world rollouts, empirical success rates or average rewards can be difficult to interpret, and average performance may be insufficient for safety and reliability needs.

    Source: §I · p. 1

  • Method

    The framework computes worst-case bounds on behavior-cloning policy performance from few rollouts, using bounds for binary success and continuous-reward distributions.

    Source: §I · pp. 1–2

  • Results

    Experiments validate the bounds in simulation and assess OOD generalization; at 95% confidence, RT-2 outperforms VC-1 in each tested setting.

    Source: §I · p. 2, C. Comparing the Generalization of Two… · p. 6

  • Takeaways & Limitations

    The approach can assess OOD generalization and compare policies while also bounding each policy’s individual performance.

    Source: §I · p. 2, C. Comparing the Generalization of Two… · p. 6

Abstract

from arXiv · show

With the rise of stochastic generative models in robot policy learning, end-to-end visuomotor policies are increasingly successful at solving complex tasks by learning from human demonstrations. Nevertheless, since real-world evaluation costs afford users only a small number of policy rollouts, it remains a challenge to accurately gauge the performance of such policies. This is exacerbated by distribution shifts causing unpredictable changes in performance during deployment. To rigorously evaluate behavior cloning policies, we present a framework that provides a tight lower-bound on robot performance in an arbitrary environment, using a minimal number of experimental policy rollouts. Notably, by applying the standard stochastic ordering to robot performance distributions, we provide a worst-case bound on the entire distribution of performance (via bounds on the cumulative distribution function) for a given task. We build upon established statistical results to ensure that the bounds hold with a user-specified confidence level and tightness, and are constructed from as few policy rollouts as possible. In experiments we evaluate policies for visuomotor manipulation in both simulation and hardware. Specifically, we (i) empirically validate the guarantees of the bounds in simulated manipulation settings, (ii) find the degree to which a learned policy deployed on hardware generalizes to new real-world environments, and (iii) rigorously compare two policies tested in out-of-distribution settings. Our experimental data, code, and implementation of confidence bounds are open-source.

I. INTRODUCTION

The paper proposes statistical bounds to evaluate behavior-cloning policies from few rollouts, covering both binary success and continuous reward. The framework bounds the full performance distribution and is used to study policy generalization in out-of-distribution settings.

I. INTRODUCTION

  • Higher confidence and tighter bounds require more policy rollouts, so the method specifies the minimum sample count for user-selected criteria.Confidence is the probability that the bound holds; tightness measures how close it tends to be to the unknown quantity.Source: §I · p. 2
  • Binary evaluation bounds task-success probability, while continuous evaluation bounds the reward CDF, which characterizes the full reward distribution.For binary metrics, the paper uses a tight confidence bound; for continuous metrics, it uses a tighter DKW bound.Source: §I · p. 2
  • Experiments apply the framework to OOD generalization, compare bounds with common alternatives, and compare two policies.The authors also use hardware experiments to bound a visuomotor diffusion policy in new environments.Source: §I · p. 2
  • Worst-case bounds use a small number of rollouts to bound the entire performance distribution in an arbitrary environment.The framework targets user-specified confidence and tightness while minimizing the number of rollouts.Source: §I · pp. 1–2
  • The paper introduces a method for computing maximum expected shortage and releases an open-source implementation of its binomial bounds.Maximum expected shortage is the worst-case average degree of underestimation of the true success rate.Source: §I · p. 2

II. RELATED WORK

The paper contrasts formal evaluation with statistical black-box validation, then situates its approach among statistical methods for learned policies. It emphasizes bounding the full performance distribution, quantifying tightness, and limiting the required rollout count.

A. Formal Evaluation of Learned Policies

  • Formal methods can certify rich closed-loop behaviors but typically require particular policy and environment structures, assumptions this paper does not make.The paper therefore places its approach among black-box validation methods.Source: A. Formal Evaluation of Learned Policies · p. 2

B. Statistical Evaluation of Learned Policies

  • Prior statistical bounds often target scalar summaries and may overlook distribution tails, while this work bounds the CDF and quantifies tightness.The cited approaches may also require more rollouts than necessary for the specified confidence.Source: B. Statistical Evaluation of Learned… · p. 3
  • Conformal prediction bounds quantiles but can ignore distribution tails; common Clopper-Pearson and DKW bounds may be loose and use more samples than necessary.Unlike odds-ratio tests, the paper’s policy comparisons also provide confidence intervals for each policy’s success rate.Source: B. Statistical Evaluation of Learned… · p. 3

III. DISTRIBUTIONAL BOUNDS

The paper orders performance distributions by their CDFs and constructs confidence bounds that represent worst-case performance consistent with rollout data. Its statistical guarantees rely on i.i.d. rollout measurements.

A. Bounds

  • A CDF-based partial ordering treats a distribution with more probability mass on lower performance values as less preferable.When CDFs cross, the paper does not claim that one distribution is preferable to the other.Source: A. Bounds · p. 3
  • The framework bounds the entire performance distribution through its CDF; for binary metrics, an upper CDF bound is a lower bound on task-success probability.The bound is constructed from observed rollouts and represents the worst-case distribution at confidence 1−α.Source: A. Bounds · p. 3

B. Assumptions

  • The bounds require i.i.d. rollout measurements, so sampling rules and rollout counts should be set before data collection.Stopping after enough favorable outcomes or testing in a time-varying environment can violate the assumption.Source: B. Assumptions · p. 3

IV. CONFIDENCE BOUNDS - BINARY METRIC

The binary-metric section constructs lower confidence bounds on policy success from i.i.d. rollout outcomes, aiming to meet user-specified confidence and tightness with minimal rollouts. It uses maximum expected shortage to measure tightness without assuming the unknown success probability.

IV. CONFIDENCE BOUNDS - BINARY METRIC

  • Each rollout is modeled as an i.i.d. Bernoulli outcome, and the observed successes determine a lower confidence bound on the policy’s success rate.The section seeks the desired confidence and tightness with as few rollouts as possible.Source: §IV · pp. 3–4

A. Optimality Criteria

  • A uniformly most accurate (UMA) lower confidence bound minimizes underestimation for every possible value of the unknown parameter.Shortage quantifies the amount of underestimation.Source: A. Optimality Criteria · p. 4
  • Maximum expected shortage (MES) measures worst-case expected underestimation across parameter values, avoiding reliance on an unknown parameter value.Computing MES can be challenging when expected shortage is not concave, motivating the section’s tractable computation method.Source: A. Optimality Criteria · p. 4

B. Optimal Binomial Bounds

For the binary metric, the section constructs a randomized lower confidence bound from rollout success counts and computes its tightness using maximum expected shortage. A mixed-monotonic reformulation enables tractable global optimization, and the bound has lower MES than Clopper-Pearson.

B. Optimal Binomial Bounds

  • Adding a uniform random variable to the observed success count yields a randomized test statistic and a UMA lower confidence bound for the unknown success probability.The resulting statistic is almost surely distinct for a fixed observed success count; randomization is necessary for a UMA bound.Source: B. Optimal Binomial Bounds · p. 4
  • The method computes MES by globally optimizing a reformulation of expected shortage that is increasing in p1 and decreasing in p2.This mixed-monotonic formulation is described as the first tractable method that certifiably solves the required optimization.Source: B. Optimal Binomial Bounds · p. 4
  • Given any two of confidence level, MES tightness, and rollout count, the method can determine the third; when choosing n, it searches for the smallest count meeting the specified α and MES.MES decreases as n increases.Source: B. Optimal Binomial Bounds · p. 4
  • Our bound has lower MES than Clopper-Pearson, and is appreciably tighter at small sample sizes.The figure compares MES as n and α vary.Source: Figure 3 · p. 5, B. Optimal Binomial Bounds · p. 5

V. CONFIDENCE BOUNDS - CONTINUOUS METRIC

For continuous reward metrics, the method constructs an upper confidence bound on the unknown reward CDF using the minimum rollouts needed for specified confidence and tightness. Its offset is optimal, and the bound is tighter than DKW, especially with small samples.

V. CONFIDENCE BOUNDS - CONTINUOUS METRIC

  • The continuous-metric method constructs an upper confidence bound on the CDF of an unknown reward distribution using the minimum rollouts needed for specified confidence and tightness.The reward distribution may be continuous, discrete, or mixed.Source: §V · p. 5
  • ϵ∗ is the optimal offset from the empirical CDF; reducing it further would violate the bound, and bisection search computes it.To find the smallest sample size for a desired α and ϵ∗, the method searches over n.Source: §V · p. 5
  • The proposed bound is always tighter than DKW, with the advantage especially appreciable at small sample sizes.Fig. 4 depicts variation in ϵ∗, n, and α.Source: §V · p. 5, Figure 4 · p. 5

VI. EXPERIMENTS

The experiments evaluate behavior-cloning policies in out-of-distribution settings in simulation and hardware. Simulation tests check the bounds’ empirical guarantees, while hardware tests examine generalization; the approach requires i.i.d. performance scores.

VI. EXPERIMENTS

  • All experiments evaluate policies in out-of-distribution settings because performance changes under shifts are difficult to predict.Without assumptions about the shift, policies must be tested directly in the OOD settings.Source: §VI · p. 5
  • Simulation experiments compare empirical confidence and tightness with theory, while hardware experiments assess OOD generalization and compare two learned policies.The approach requires i.i.d. performance scores; the authors sought to avoid time-varying hardware conditions that could undermine this assumption.Source: §VI · p. 5
  • Across 100 trials, empirical confidence and expected-shortage estimates generally agree with theory, and more trials are expected to improve that agreement.Table I reports empirical estimates for binary bounds and continuous-bound confidence; continuous tightness is given by ϵ∗.Source: Table I · p. 6

A. Simulation Results

Simulation evaluates five diffusion policies under modified task environments and finds that the effects of distribution shifts vary by task. The bounds reflect these trends, while small-sample Monte Carlo estimates can be overly optimistic.

A. Simulation Results

  • Five diffusion policies trained for separate visuomotor manipulation tasks were tested after changes to environment colors.The modifications changed can and floor colors, cube colors in Lift and Transport, square color, wall color, and the Transport lid-handle color.Source: A. Simulation Results · p. 6
  • The evaluation used 1,000 rollouts per environment to estimate ID and OOD distributions, then 4,000 rollouts to construct 100 bounds from 40 samples each.The true distributions are typically unknown and are shown only to validate the bounds; in practice, one bound would be constructed.Source: A. Simulation Results · p. 6
  • Can and Square shifts were harmful to policy performance, whereas shifts in Lift, Tool Hang, and Transport were benign.The bounds tended to reflect these trends; 40-sample Monte Carlo distribution estimates were overly optimistic and unsuitable as bounds.Source: A. Simulation Results · p. 6

B. Generalization of a Single Policy

A diffusion policy trained to pour ice was tested in two OOD settings, one hypothesized harmful and one benign. The observed successes and confidence bounds supported those hypotheses and distinguished whether a 60% success-rate claim was justified.

B. Generalization of a Single Policy

  • The ice-pouring policy had 4/50 successes in the hypothesized harmful setting and 38/50 in the hypothesized benign setting.The harmful setting added another red object; the benign setting had no other red objects.Source: B. Generalization of a Single Policy · p. 6
  • At 95% confidence, the lower bounds were 0.035 for the harmful setting and 0.641 for the benign setting, with MES = 0.118.The low harmful-setting lower bound was attributed to frequent grasp failure despite movement toward the red mug.Source: B. Generalization of a Single Policy · p. 6
  • The bounds support claiming the benign setting meets a 60% success-rate requirement at 95% confidence, but not the harmful setting.The authors suggest retraining with data from the harmful setting and reevaluating.Source: B. Generalization of a Single Policy · p. 6

C. Comparing the Generalization of Two Policies

The authors compare RT-2 and VC-1 using confidence bounds on success rates. In all three evaluation settings, the bounds support an RT-2 advantage at 95% confidence.

C. Comparing the Generalization of Two Policies

VII. CONCLUSION

The paper presents statistical bounds for evaluating policy performance and OOD generalization from few samples, including bounds on the full performance distribution. Its experiments visualize bound behavior and compare two policies, while the stated guarantee applies to the environment from which samples were drawn.

VII. CONCLUSION

  • The primary contribution is using statistical bounds to assess OOD policy generalization from few samples, bounding the entire performance distribution for binary and continuous metrics.Source: §VII · p. 6
  • The bounds appropriately cover the OOD reward distribution about 95% of the time, while Monte Carlo estimates are overly optimistic about 50% of the time.The ID distributions indicate harmful shifts in Can and Square and benign shifts in the other environments.Source: Figure 6 · p. 7
  • RT-2 outperforms VC-1 in every evaluated setting, with disjoint success-rate confidence intervals supporting the comparison at 95% confidence.Source: Figure 8 · p. 7
  • The paper also contributes MES computations for UMA and Clopper-Pearson bounds, open-source UMA-bound software, and an approach applicable to any black-box policy.Source: §VII · p. 7
  • The bounds are valid for the environment from which the samples were drawn; distribution-shift sensitivity or robustness requires an extension.Source: §VII · p. 7

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