Proximal Point Imitation Learning

This work develops new algorithms with rigorous efficiency guarantees for infinite horizon imitation learning (IL) with linear function approximation without restrictive coherence assumptions. We begin with the minimax formulation of the problem and then outline how to leverage classical tools from optimization, in particular, the proximal-point method (PPM) and dual smoothing, for online and offline IL, respectively. Thanks to PPM, we avoid nested policy evaluation and cost updates for online IL appearing in the prior literature. In particular, we do away with the conventional alternating updates by the optimization of a single convex and smooth objective over both cost and Q-functions. When solved inexactly, we relate the optimization errors to the suboptimality of the recovered policy. As an added bonus, by re-interpreting PPM as dual smoothing with the expert policy as a center point, we also obtain an offline IL algorithm enjoying theoretical guarantees in terms of required expert trajectories. Finally, we achieve convincing empirical performance for both linear and neural network function approximation. From the technical point of view, the most important related works are the analysis of REPS/Q-REPS [90, 14, 89] and O-REPS [124] that first pointed out the connection between REPS and PPM. We build on their techniques with some important differences. In particular, while in the LP formulation of RL, PPM and mirror descent [15, 47] are equivalent, recognizing that they are not equivalent in IL is critical for stronger empirical performance. As an independent interest, our techniques can be used to improve upon the best rate for REPS in the tabular setting [89] and to , where E π s denotes the expectation with respect to the trajectories generated by π starting from s 0 = s. The optimal value function , is known to characterize optimal behaviors. Indeed V ⋆ c is the unique solution to the Bellman optimality equation V ⋆ c (s) = min a Q ⋆ c (s, a). In addition, any deterministic policy π ⋆ c (s) = arg min a Q ⋆ c (s, a) is known to be optimal. For every policy π, we define the normalized state-action occupancy measure µ , where P π ν0 [•] denotes the probability of an event when following π starting from s 0 ∼ ν 0 . The occupancy measure can be interpreted as the discounted visitation frequency of state-action pairs. This allows us to write ρ c (π) = ⟨µ π , c⟩.