Of Moments and Matching: A Game-Theoretic Framework for Closing the Imitation Gap (2103.03236v2)

Abstract: We provide a unifying view of a large family of previous imitation learning algorithms through the lens of moment matching. At its core, our classification scheme is based on whether the learner attempts to match (1) reward or (2) action-value moments of the expert's behavior, with each option leading to differing algorithmic approaches. By considering adversarially chosen divergences between learner and expert behavior, we are able to derive bounds on policy performance that apply for all algorithms in each of these classes, the first to our knowledge. We also introduce the notion of moment recoverability, implicit in many previous analyses of imitation learning, which allows us to cleanly delineate how well each algorithmic family is able to mitigate compounding errors. We derive three novel algorithm templates (AdVIL, AdRIL, and DAeQuIL) with strong guarantees, simple implementation, and competitive empirical performance.

A Game-Theoretic Approach to Imitation Learning

The paper "Of Moments and Matching: A Game-Theoretic Framework for Closing the Imitation Gap" presents a comprehensive examination of imitation learning (IL) through a game-theory lens. The authors propose a novel classification of IL algorithms based on "moment matching," focusing on whether these algorithms attempt to align reward moments or action-value moments between a learner and an expert. This classification facilitates a clearer understanding of the performance limitations and benefits of various IL methods.

Main Contributions and Insights

The central contribution of the paper is the unifying framework for IL through the concept of moment matching. This framework categorizes algorithms into those matching reward moments and those matching action-value (or Q-value) moments. The paper argues that mismatches between the moments of a learner and those of an expert encapsulate divergence between the two, leading to sub-optimal learner performance.

Performance Guarantees: The authors derive strong theoretical results concerning upper and lower bounds on policy performance for these algorithm classes. They show that:

• Algorithms matching reward moments have an upper bound on the performance gap proportional to the horizon $T$.
• Off-policy Q-value moment-matching algorithms, which operate entirely offline, exhibit a quadratic compounding of errors in the horizon, imposing a potentially large performance penalty.
• On-policy Q-value moment-matching requires interaction with the environment and a queryable expert, but can achieve strong performance in Markov Decision Processes (MDPs) with recoverable moments.

Novel Algorithms: The paper introduces three new algorithmic templates: AdVIL (Adversarial Value-moment Imitation Learning), AdRIL (Adversarial Reward-moment Imitation Learning), and DAeQuIL (DAgger-esque Qu-moment Imitation Learning). These algorithms are derived to provide robust performance guarantees by effectively leveraging the moment matching game-theoretic framework.

Theoretical and Practical Implications

Recoverability: A pivotal concept introduced is "moment recoverability," the ability of an expert's policy to naturally compensate for deviations by the learner. This property is crucial in analyzing the vulnerability of algorithms to compounding errors, providing a lens through which various imitation tasks can be assessed.

Game-Theoretic Foundation: By modeling IL as a two-player minimax game, the authors achieve a form of strong duality, permitting both primal and dual algorithmic constructions. The approach draws on functional gradient descent and IPM (Integral Probability Metric) optimization to derive practical methods demonstrable in empirical settings.

Broad Applicability: Evaluating classic IL paradigms (like DAgger and GAIL) within this new framework highlights intrinsic strengths and weaknesses matched against varying moment classes. The application of IPMs enriches the framework's applicability, extending beyond standard distance metrics.

Future Directions

The framework proposed by Swamy et al. opens several avenues for advancing IL research:

• Further exploration of non-recoverable MDPs, where current guarantees may not yield performance improvements, could provide insights into additional algorithmic adaptations.
• Investigating alternative moment spaces that could better encapsulate expert policies, particularly in dynamic and non-zero-sum environments.
• Extending the game-theoretic framework to address multi-agent imitation settings, thereby broadening the applicability of these insights.

Overall, this paper offers a substantive theoretical contribution to IL research, equipped with a robust mathematical foundation and demonstrable algorithmic advancements. By employing advanced mathematical constructs such as IPMs, moment recoverability, and game-theoretic strategies, it provides both practitioners and theorists with a valuable framework for developing and evaluating future IL algorithms.