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Reinforcement and Imitation Learning via Interactive No-Regret Learning

Stephane Ross, J. Andrew Bagnell

arXiv:1406.5979v1cs.LGstat.ML

TL;DR

Imitation-learning methods often do not use action-cost information, despite the importance of distinguishing catastrophic from recoverable errors. The paper develops AGGREVATE and extends its cost-sensitive no-regret strategy to reinforcement learning, yielding NRPI and theoretical support for online approximate policy iteration. It also frames existing imitation and reinforcement-learning techniques within a unified algorithmic family.

  • Problem

    Existing interactive imitation-learning approaches address learner-induced input distributions but neither require nor benefit from information about action costs, limiting their treatment of errors with different long-term consequences.

  • Method

    The paper introduces AGGREVATE, which combines demonstrations with cost-to-go information, and extends cost-sensitive no-regret learning to approximate policy iteration through NRPI.

  • Results

    The approach provides statistical regret guarantees for imitation learning and reinforcement-learning approximate policy iteration, while unifying existing techniques across both settings.

  • Takeaways & Limitations

    The framework provides theoretical support for the commonly observed success and stability of online approximate policy iteration and explains related imitation-learning practices as special cases.

  • Takeaways & Limitations

    The practical trade-offs among the many available no-regret learners remain largely open, and identifying the most effective methods requires future work.

Abstract

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Recent work has demonstrated that problems-- particularly imitation learning and structured prediction-- where a learner's predictions influence the input-distribution it is tested on can be naturally addressed by an interactive approach and analyzed using no-regret online learning. These approaches to imitation learning, however, neither require nor benefit from information about the cost of actions. We extend existing results in two directions: first, we develop an interactive imitation learning approach that leverages cost information; second, we extend the technique to address reinforcement learning. The results provide theoretical support to the commonly observed successes of online approximate policy iteration. Our approach suggests a broad new family of algorithms and provides a unifying view of existing techniques for imitation and reinforcement learning.

1 Introduction

The paper argues that imitation learning should account for the long-term costs of errors, then introduces a cost-sensitive no-regret framework covering imitation and reinforcement learning.

  • Imitation learning methods that only match demonstrator actions can treat severe and minor errors equally, producing poor trade-offs when mistakes have different long-term costs.Even crude cost-to-go information can distinguish recoverable actions from catastrophic ones.
  • SEARN incorporates cost-to-go but may be impractical because it uses current-policy rollouts and requires stochastic policies.
  • AGGREVATE extends interactive imitation learning by combining correct demonstrations with cost-to-go information.
  • The paper interprets prior methods as special cases of a general no-regret strategy and provides statistical regret guarantees rather than statistical error-reduction guarantees.
  • The same cost-sensitive no-regret strategy extends to reinforcement-learning approximate policy iteration through NRPI, offering guarantees as strong as available results and a unified view of imitation and reinforcement learning.The framework also supplies theoretical support for the observed success and stability of online policy iteration.

2 Imitation Learning with Cost-To-Go

AGGREVATE extends interactive imitation learning by training policies against expert cost-to-go rather than immediate action agreement. Its no-regret analysis links learned-policy performance to cost-sensitive learning regret, while highlighting limits from partial information, policy-class mismatch, and regression reductions.

  • AGGREVATE procedure: AGGREVATE collects cost-to-go estimates after learner interaction, then trains policies as an online cost-sensitive classification problem.Each iteration executes the current policy up to a random time, explores an action, and lets the expert continue to estimate future cost.
  • AGGREVATE procedure: Expert cost-to-go distinguishes actions that preserve recoverability from actions that make high future cost unavoidable, such as falling off a cliff.This relies on the expert being good and the policy class containing similarly good policies.
  • Limitations: Partial-information exploration remains a limitation because uniform action selection is inefficient and more sample-efficient contextual-bandit strategies impose restrictive assumptions.Relevant assumptions include a finite policy class or full realizability for some contextual-bandit algorithms.
  • Analysis and guarantees: With sufficient iterations, AGGREVATE can approach demonstrator performance when the policy class contains policies with low aggregate cost-sensitive classification regret.The guarantees also connect task performance to regression regret when cost-to-go is learned through squared-loss regression.
  • Analysis and guarantees: Any no-regret online learner can update the policy sequence, yielding a statistical regret reduction rather than merely statistical error reduction.The reduction relates task performance directly to online cost-sensitive classification regret on the aggregate dataset.
  • Limitations: When the expert substantially outperforms every policy in the class, its cost-to-go can be overly optimistic and favor actions that weaker policies cannot safely execute.The paper gives a narrow-road example where this mismatch can lead to eventual failure despite safer policies being available.

3 Reinforcement Learning via No-Regret Policy Iteration

NRPI reduces sample-based approximate policy iteration to no-regret online cost-sensitive learning, using exploration distributions to collect current-policy cost-to-go estimates. Its guarantee depends on distribution alignment and supports a single stationary policy for execution.

  • Algorithm: NRPI learns a single policy rather than one policy per time step, potentially improving generalization and efficiency when the horizon is large or infinite.This is presented as an advantage over PSDP, while requiring no-regret online cost-sensitive classification.
  • Algorithm: NRPI collects cost-to-go examples by sampling a time and state from exploration distributions, taking an exploratory action, then executing the current policy.The resulting examples are aggregated for cost-sensitive policy learning.
  • Algorithm: The method can use either aggregated datasets with a cost-sensitive classifier or an online learner applied sequentially to iteration-specific datasets.The online-learning formulation defines losses as expected cost-to-go for acting once and then following the current policy.
  • Guarantee: NRPI performs as well as any policy whose state distribution stays close to the exploration distribution on average over time.If the exploration distributions match an optimal policy’s state distributions, the theorem guarantees convergence to an optimal policy in the class.
  • Guarantee: The reinforcement-learning result is a reduction of model-free reinforcement learning to no-regret online learning, but performance is strongly limited by exploration-distribution quality.The guarantee is directly related to online regret in the cost-sensitive classification problem.

4 Discussion and Future Work

The discussion connects no-regret learning to stability in approximate policy iteration and clarifies practical trade-offs of cost-to-go information. It also identifies computational limitations, unresolved bounds, and open choices among no-regret learners.

  • Discussion: Online approximate policy iteration is argued to be more effective and stable than convergence theory and counterexamples might suggest because of no-regret or similar stability properties.The claim is offered as theoretical support for an empirical observation.
  • Discussion: SEARN’s expert-cost-to-go heuristic is interpreted as a special case of AGGREVATE using online Frank-Wolfe, while deterministic variants may outperform stochastic mixing in practice.The discussion also characterizes stochastic mixing as only one route to good online performance.
  • Discussion: Training on cost-to-go information from previous policies, not only the latest policy, is suggested to reduce oscillations and divergences in batch procedures.The resulting learned policy is intended to be good across many states.
  • Limitations: Cost-to-go methods can be impractical because each state-action estimate may require executing an entire trajectory, whereas DAGGER can collect T action labels per trajectory.The cost is less serious in structured prediction when expert cost-to-go is quickly computed; combining imitation loss with later AGGREVATE refinement is proposed.
  • Limitations: The reinforcement-learning bound is meaningful only under a condition involving dt and T Qmax, which is generally O(T^2), and extending it to a stronger Bayes-optimal multiplicative guarantee is not obvious.The current guarantee is stated for an arbitrary policy class, while prior methods provide the stronger fully observed-MDP comparison.
  • Future Work: Future work must examine the practical trade-offs among the wide variety of available no-regret learners.The paper describes this empirical comparison as largely open.

Appendix: Proofs and Detailed Bounds

The appendix supplies proofs and detailed analyses for the imitation-learning and reinforcement-learning algorithms developed in the main text.

  • Appendix: The appendix contains proofs and detailed analysis of the paper’s imitation-learning and reinforcement-learning algorithms.

Lemmas

The lemmas bound distribution mismatch and policy-performance changes used in the analysis. They relate mixed expert execution to state-distribution distance and establish the relevant performance-difference framework.

  • Distribution bounds: A general distribution lemma bounds differences in expected bounded losses using the L1 distance between two state distributions.The lemma is used to transfer expected-loss bounds across distributions.
  • Distribution bounds: The state-distribution mismatch between the learner and an expert-mixing policy is bounded by 2 min(1, T βi).The bound follows from the probability of using the expert at least once over T steps.
  • Performance bounds: The performance difference lemma bounds the change in total cost-to-go between any two policies.Its proof uses a non-stationary policy that follows one policy for t steps and then switches to the other.
  • Performance bounds: The analysis uses time-averaged state distributions and a uniform distribution over time indices when comparing policy behavior across the finite horizon.These quantities support the paper’s distribution-based performance arguments.

AGGREVATE Reduction Analysis

AGGREVATE’s guarantee bounds the learned policy’s performance using classification regret, class error, and a term controlled by the exploration schedule. Under no-regret learning and suitable decay of β_i, the extra term vanishes asymptotically.

  • AGGREVATE Reduction Analysis: As N → ∞, no-regret learning drives ε_regret toward zero, while the additional exploration term is likewise controlled.The asymptotic conclusion follows directly from ε_regret → 0 and the decay condition on β_i.
  • AGGREVATE Reduction Analysis: The proof defines n_β as the largest iteration with β_n > 1/T because β_i is non-increasing.This separates early iterations with larger exploration probabilities from later iterations.
  • AGGREVATE Reduction Analysis: The minimum performance among learned policies is no worse than their average performance, enabling selection of the best iterate.The argument uses min_i J(π̂_i) ≤ (1/N)∑_i J(π̂_i).

Finite Sample AGGREVATE with Q-function approximation

The finite-sample analysis applies AGGREVATE with Q-function approximation, using regression on noisy expert cost-to-go samples and empirical no-regret updates. Its guarantee separates approximation, online-regret, and sampling effects, with an almost-sure asymptotic result.

  • Finite Sample AGGREVATE with Q-function approximation: AGGREVATE can estimate expert cost-to-go with features such as f(s, a, t), including a linear regressor Q̂^{T−t+1}(s, a) = w^⊤f(s, a, t).The learned policy selects the action minimizing the estimated cost-to-go.
  • Finite Sample AGGREVATE with Q-function approximation: The finite-sample theorem guarantees performance with probability at least 1 − δ using class regret, online regret, and concentration terms.The bound is stated after N iterations with m regression examples per iteration.
  • Finite Sample AGGREVATE with Q-function approximation: With a no-regret online algorithm selecting regressors, the finite-sample guarantee converges almost surely as N → ∞.The asymptotic statement is made for the sequence of learned regressors.
  • Finite Sample AGGREVATE with Q-function approximation: The regression analysis relates policy cost-sensitive regret to regression regret relative to the Bayes-optimal cost-to-go predictor.The Bayes-optimal regressor predicts the expected expert cost-to-go Q* from noisy trajectory observations.
  • Finite Sample AGGREVATE with Q-function approximation: At each iteration, the method collects m state-action-time-cost samples and runs the no-regret learner on empirical squared loss.The empirical loss compares predicted and observed expert cost-to-go values.

NRPI Reduction Analysis

NRPI extends the interactive no-regret reduction to reinforcement learning using state exploration distributions. Its performance guarantee compares the best learned policy with any reference policy, adding regret and distribution-mismatch terms.

  • NRPI Reduction Analysis: NRPI samples from state exploration distributions ν_1:T and returns the best policy found across iterations.The learned policies’ cost-to-go values are assumed nonnegative and bounded by Q_max.
  • NRPI Reduction Analysis: The proof uses an alternate performance-difference lemma to connect policy losses with the learner’s cost-sensitive online-learning losses.The lemma compares two policies through the value and Q-value functions of one policy.
  • NRPI Reduction Analysis: For any reference policy π′ ∈ Π, the guarantee bounds the selected policy by J(π′) + T ε_regret + T Q_max D(ν, π′).The distribution term measures the average L1 distance between exploration distributions and the reference policy’s state distributions.
  • NRPI Reduction Analysis: The best-iterate guarantee follows because the minimum policy cost is no greater than the average cost across learned policies.This converts the average-regret analysis into a guarantee for the best policy found.
  • NRPI Reduction Analysis: As N → ∞, the regret term vanishes when the online learner has ε_regret → 0.The asymptotic conclusion is stated directly for the NRPI iterations.
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