PaperlayerView exact revision on arXiv

Source-linked AI summary

Error Bounds of Imitating Policies and Environments

Tian Xu, Ziniu Li, Yang Yu

arXiv:2010.11876v1cs.LG

TL;DR

The paper addresses the limited theoretical understanding of imitation-learning methods and their value gaps relative to expert policies. It derives error bounds for behavioral cloning and GAIL in policy and environment imitation, finding more favorable horizon and model-bias dependence for GAIL. These results support adversarial imitation as a promising direction for model-based reinforcement learning.

  • Problem

    The theoretical explanations for imitation-learning performance, especially GAIL’s policy-value behavior and environment-model imitation, remain incomplete.

  • Method

    The paper derives error bounds and sample-complexity results for behavioral cloning and GAIL while treating environment transition models as dual agents.

  • Results

    GAIL has linear effective-horizon dependence for policy and environment imitation, while behavioral cloning has quadratic dependence in the analyzed setting.

  • Takeaways & Limitations

    The analysis supports replacing BC-like transition learners with GAIL-style learners to improve generalization in model-based reinforcement learning.

  • Takeaways & Limitations

    The GAIL analysis focuses on discriminator generalization and leaves the policy’s computation and approximation abilities for future work.

Abstract

from arXiv · show

Imitation learning trains a policy by mimicking expert demonstrations. Various imitation methods were proposed and empirically evaluated, meanwhile, their theoretical understanding needs further studies. In this paper, we firstly analyze the value gap between the expert policy and imitated policies by two imitation methods, behavioral cloning and generative adversarial imitation. The results support that generative adversarial imitation can reduce the compounding errors compared to behavioral cloning, and thus has a better sample complexity. Noticed that by considering the environment transition model as a dual agent, imitation learning can also be used to learn the environment model. Therefore, based on the bounds of imitating policies, we further analyze the performance of imitating environments. The results show that environment models can be more effectively imitated by generative adversarial imitation than behavioral cloning, suggesting a novel application of adversarial imitation for model-based reinforcement learning. We hope these results could inspire future advances in imitation learning and model-based reinforcement learning.

1 Introduction

Imitation learning offers an efficient alternative to reinforcement learning by learning from expert demonstrations, but the theoretical reasons behind GAIL’s empirical advantage over behavioral cloning remain incompletely understood. This paper derives policy and environment imitation error bounds, showing more favorable horizon dependence for GAIL and implications for model-based reinforcement learning.

  • 1 Introduction: GAIL’s policy value gap grows linearly with the effective horizon, whereas behavioral cloning’s grows quadratically, formally supporting reduced compounding errors.The respective dependencies are 1/(1 −γ) and 1/(1 −γ)^2.
  • 1 Introduction: The paper analyzes both the policy value gap and sample complexity of behavioral cloning and GAIL.
  • 1 Introduction: GAIL can generalize well from incomplete trajectories, while discriminator complexity creates a trade-off between estimation error and policy value-gap control.
  • 1 Introduction: Treating the environment transition model as a dual agent enables imitation learning to analyze environment-model errors for model-based reinforcement learning.
  • 1 Introduction: Adversarial environment imitation has linear policy-evaluation error in model bias, improving previous quadratic results.

2 Background

The paper formulates imitation learning in discounted infinite-horizon Markov decision processes and evaluates imitation through policy-value gaps. It contrasts behavioral cloning’s action-level matching with GAIL’s state-action distribution matching.

  • 2 Background: An infinite-horizon MDP consists of states, actions, transitions, rewards, discounting, and an initial-state distribution, with policy quality measured by discounted cumulative reward.
  • 2 Background: The effective planning horizon is 1/(1 −γ), and it determines how imitation-learning error bounds depend on discounting.
  • 2 Background: Imitation learning trains a candidate policy from expert action labels and studies the value gap VπE − Vπ.
  • 2 Background: Behavioral cloning minimizes the KL divergence between expert and imitating-policy action distributions, equivalently as supervised maximum-likelihood estimation.
  • 2 Background: GAIL uses a discriminator to distinguish expert from generated state-action pairs while a policy maximizes discriminator-provided rewards.
  • 2 Background: At the discriminator optimum, GAIL minimizes the Jensen-Shannon divergence between expert and imitating-policy state-action distributions.

3 Related Work

Prior work established behavioral cloning’s compounding errors and developed alternative imitation methods, while GAIL’s policy-value theory remained incomplete. Related environment-learning work motivates analyzing adversarial transition-model imitation as a way to reduce model bias.

  • 3 Related Work: Finite-horizon analyses found that behavioral cloning produces an O(T^2) optimality gap, while DAgger reduces it to O(T) with additional expert queries.
  • 3 Related Work: Existing GAIL theory addressed generalization, discriminator-related properties, or convergence, but did not provide the policy-value-gap bound targeted here.
  • 3 Related Work: Apprenticeship learning infers a reward function from demonstrations and then learns a policy through reinforcement learning using that recovered reward.
  • 3 Related Work: Environment transition models can be treated as dual agents, but behavioral-cloning transition learning causes compounding model-bias errors in policies trained in the learned environment.
  • 3 Related Work: This paper applies GAIL’s distribution-matching property to environment learning because prior adversarial approaches lacked a clear explanation for their effectiveness.

4 Bounds on Imitating Policies

The paper bounds policy-value gaps for behavioral cloning and GAIL, showing quadratic horizon dependence for BC versus linear dependence for GAIL under distribution matching. It also analyzes GAIL generalization through discriminator complexity and identifies a limitation in approximation and optimization analysis.

  • 4.1 Imitating Policies with Behavioral Cloning: O(1/(1 −γ)2) is unavoidable for behavioral cloning in the worst case, because small training errors can compound as the policy visits states outside expert demonstrations.The bound is tight up to a constant using a hard deterministic MDP example.
  • 4.2 Imitating Policies with GAIL: GAIL minimizes state-action distribution discrepancy, whose temporal structure supports fewer compounding errors than directly minimizing action discrepancies.The analysis applies to f-divergences and connects distribution matching to policy-value bounds.
  • 4.2 Imitating Policies with GAIL: GAIL’s policy-value gap grows linearly with the effective horizon, 1/(1 −γ), multiplied by the square root of the f-divergence error.This improves the horizon dependence from BC’s quadratic form, although it does not establish that GAIL is always better overall.
  • 4.2 Imitating Policies with GAIL: The generalization analysis uses neural-network distance because restricted discriminator classes may not realize the original divergence, and JS divergence may fail to generalize even with sufficient samples.The neural-network distance is an integral probability metric and depends on the discriminator class.
  • 4.2 Imitating Policies with GAIL: Controlling discriminator complexity reduces estimation error, while simpler discriminator classes may enlarge the compatible coefficient, creating a complexity trade-off.Gradient penalties empirically control model complexity, whereas reducing network depth did not significantly affect GAIL performance in the reported experiments.
  • 4.2 Imitating Policies with GAIL: The results do not deeply analyze policy-class approximation ability or computation under stochastic policy-gradient descent.This is stated as a limitation of the theoretical result.

5 Bounds on Imitating Environments

The paper treats transition-model learning as imitation of a dual agent and compares behavioral cloning with a GAIL-style adversarial learner. It emphasizes that model-bias can compound during policy evaluation, while adversarial environment imitation yields a linear dependence on model-bias.

  • 5. Bounds on Imitating Environments: Environment learning recovers an MDP transition model from real-environment data and is central to model-based reinforcement learning.The transition model receives the current state and action and predicts the next state.
  • 5. Bounds on Imitating Environments: The analysis assumes the true reward function is available and notes that finite-sample bounds can be obtained by adding concentration measures.Learning an unknown reward is treated as a lower-order sample-complexity term relative to transition-model learning.
  • 5.1 Imitating Environments with Behavioral Cloning: Behavioral cloning minimizes one-step prediction errors when imitating environments, but the resulting model can still suffer compounding evaluation errors.The paper formulates environment BC as direct one-step prediction-error minimization.
  • 5.1 Imitating Environments with Behavioral Cloning: Model-based policy-evaluation error can blow up with model-bias when the learned environment fails to capture the transition model globally.The bound contains both model inaccuracy under the data-collecting distribution and divergence between the evaluated policy and that policy.
  • 5.2 Imitating Environments with GAIL: A GAIL-style environment learner trains a transition model and discriminator on state-action-next-state tuples to match distributions generated by the true and learned environments.The transition model is treated as a policy that outputs a next-state distribution.
  • 5.2 Imitating Environments with GAIL: Theorem 3 indicates that adversarial transition-model recovery yields linear policy-evaluation dependence on model-bias, improving previous quadratic results.The paper presents this as evidence that GAIL-style environment imitation can mitigate model-bias when evaluating policies.

6 Experiments

Experiments compare imitation methods across effective horizons, discriminator regularization, and learned environment models. GAIL performs better than BC as the planning horizon increases, moderate gradient penalties improve GAIL, and GAIL-trained environments yield lower policy evaluation errors.

  • 6 Experiments: The evaluation uses MuJoCo locomotion tasks and compares BC, DAgger, GAIL, WGAIL, AIRL, FEM, and GTAL.The considered methods are evaluated on three OpenAI Gym MuJoCo benchmark tasks.
  • 6.1 Imitating Policies: As the planning horizon increases, BC performs worse than GAIL and possibly AIRL, FEM, and GTAL.Figure 2 reports relative policy performance under different discount factors, with expert demonstrations limited to three trajectories except for DAgger.
  • 6.1 Imitating Policies: Moderate gradient penalty coefficients λ=0.1 or 1.0 outperform large λ=10 or small λ=0 in GAIL.The gradient penalty controls discriminator complexity by favoring 1-Lipschitz functions.
  • 6.2 Imitating Environments: GAIL-trained environment models have smaller policy evaluation errors than BC-trained models on all three environments.BC tends to overfit, and its policy evaluation errors do not decrease on HalfCheetah-v2.

7 Conclusion

The paper develops error bounds for BC and GAIL in policy and environment imitation. It finds linear effective-horizon dependence for GAIL versus quadratic dependence for BC, with implications for model-based reinforcement learning, while leaving policy computation and approximation analysis for future work.

  • 7 Conclusion: GAIL achieves linear effective-horizon dependence, whereas BC has quadratic dependence in the presented policy and environment imitation bounds.The conclusion summarizes the infinite-horizon analysis of both methods.
  • 7 Conclusion: Replacing BC-like transition learners with GAIL-style learners may improve environment-model generalization in model-based reinforcement learning.The authors connect this possibility to reducing model bias and learning useful environment models for sample-efficient reinforcement learning.
  • 7 Conclusion: The analysis focuses on discriminator generalization and leaves policy computation and approximation ability for future work.This bounds the scope of the paper’s theoretical treatment.

Broader Impact

This section states that the work has no direct societal consequence and focuses on theoretical analyses of imitation learning for policies and environments.

  • Broader Impact: The work presents no direct societal consequence because it focuses on theoretical understanding of imitation learning for policies and environments.
  • Broader Impact: The analysis includes error propagation, state-distribution discrepancies, policy value gaps, and behavioral-cloning sample complexity.

A.3 Tightness of Theorem 1

The constructed deterministic MDP demonstrates that behavioral cloning’s quadratic dependence on the effective planning horizon is tight in the worst case.

  • A.3 Tightness of Theorem 1: Figure 6 depicts the corresponding deterministic MDP with initial state s0 and absorbing states s1 and s2, labeling rewards on transitions.
  • A.3 Tightness of Theorem 1: In the hard example, the imitated policy selects a2 with probability 0.15 and has value gap γ/[10(1−γ)].
  • A.3 Tightness of Theorem 1: The worst-case policy value gap scales quadratically with the effective planning horizon, O(1/(1 −γ)^2), and this dependence is tight up to a constant.The example yields VπE −VπI = C · 1/(1−γ)^2 for a constant C.

B.2 Proof of Lemma 1

The proof establishes that GAIL using listed f-divergences yields a policy value-gap bound, while neural-network distance and discriminator complexity support its finite-sample analysis.

  • B.2 Proof of Lemma 1: GAIL with the listed f-divergences has a policy value gap bounded by the corresponding divergence, yielding linear horizon dependence.
  • B.2 Proof of Lemma 1: Because practical discriminators have limited representation, the analysis uses neural-network distance and controls generalization through boundedness and Rademacher-complexity arguments.
  • B.2 Proof of Lemma 1: The value-gap proof reformulates policy value through state-action occupancy distributions and assumes the reward lies in the discriminator class’s linear span.

C Analysis of Imitating-environments

This section analyzes environment imitation by treating transition models as policies, decomposing evaluation error, and applying adversarial imitation to learn transition models.

  • C Analysis of Imitating-environments: Environment-model learning through adversarial imitation achieves a policy-evaluation error linear in model bias, improving prior quadratic results.
  • C Analysis of Imitating-environments: The evaluation-error analysis decomposes error into model error and policy-distribution discrepancies, then bounds the resulting terms using transition operators and divergence inequalities.
  • C Analysis of Imitating-environments: Algorithm 1 alternates model rollouts, discriminator-based rewards, model updates, and discriminator updates before outputting the learned environment model.
  • C Analysis of Imitating-environments: Wasserstein GAIL restricts the discriminator to 1-Lipschitz functions and uses a gradient-penalty objective with mixing distribution ˜ρ and regularization coefficient λ.

E Experiment Details

The experiments compare imitation methods on OpenAI Gym tasks using standardized networks, demonstrations, evaluation procedures, and three random seeds, with separate protocols for imitating policies and environments.

  • Imitating Policies: All policy-imitation methods use 2-layer MLPs with 100 hidden units and tanh activation, while most receive three expert trajectories of length 1000.DAgger instead continually collects samples and queries expert action labels.
  • Imitating Policies: The study evaluates learning curves and final discounted returns on true environments, using 10 evaluation trajectories and three random seeds.Figure 7 reports means and standard deviations over the seeds; Table 6 lists final returns.
  • Imitating Environments: Environment imitation augments the state with information such as robot position, assumes the true reward is known, and evaluates data-collecting policies in learned models.Robot positions are normalized for model learning, while rewards are not normalized.
  • Imitating Environments: For environment imitation, BC and GAIL retain the policy-imitation configuration, but model outputs are normalized using statistics from the demonstrations because their output space is unbounded.Samples come from a sub-optimal policy executed in true environments.
  • Imitating Policies: The policy-imitation comparison includes DAgger, GAIL, AIRL, WGAIL, FEM, GTAL, and BC.These methods are shown together in the learning-curve comparison.
Loading 2010.11876v1…