Provably Efficient Generative Adversarial Imitation Learning for Online and Offline Setting with Linear Function Approximation

TL;DR

This paper introduces provably efficient algorithms for generative adversarial imitation learning (GAIL) in both online and offline settings with linear function approximation, providing theoretical guarantees on regret and optimality gap.

Contribution

The paper develops the OGAP and PGAP algorithms for online and offline GAIL with linear functions, achieving near-optimal regret and gap bounds with rigorous proofs.

Findings

OGAP achieves $ ilde{O}(H^2 d^{3/2}K^{1/2}+KH^{3/2}dN_1^{-1/2})$ regret.

PGAP attains the minimax lower bound for offline GAIL.

Under sufficient coverage, PGAP achieves $ ilde{O}(H^{2}dK^{-1/2} +H^2d^{3/2}N_2^{-1/2}+H^{3/2}dN_1^{-1/2})$ optimality gap.

Abstract

In generative adversarial imitation learning (GAIL), the agent aims to learn a policy from an expert demonstration so that its performance cannot be discriminated from the expert policy on a certain predefined reward set. In this paper, we study GAIL in both online and offline settings with linear function approximation, where both the transition and reward function are linear in the feature maps. Besides the expert demonstration, in the online setting the agent can interact with the environment, while in the offline setting the agent only accesses an additional dataset collected by a prior. For online GAIL, we propose an optimistic generative adversarial policy optimization algorithm (OGAP) and prove that OGAP achieves $\widetilde{\mathcal{O}}(H^2 d^{3/2}K^{1/2}+KH^{3/2}dN_1^{-1/2})$ regret. Here $N_1$ represents the number of trajectories of the expert demonstration, $d$ is the feature dimension, and $K$ is the number of episodes. For offline GAIL, we propose a pessimistic generative adversarial policy optimization algorithm (PGAP). For an arbitrary additional dataset, we obtain the optimality gap of PGAP, achieving the minimax lower bound in the utilization of the additional dataset. Assuming sufficient coverage on the additional dataset, we show that PGAP achieves $\widetilde{\mathcal{O}}(H^{2}dK^{-1/2} +H^2d^{3/2}N_2^{-1/2}+H^{3/2}dN_1^{-1/2} \ )$ optimality gap. Here $N_2$ represents the number of trajectories of the additional dataset with sufficient coverage.