Bilinear Exponential Family of MDPs: Frequentist Regret Bound with Tractable Exploration & Planning

We study the problem of episodic reinforcement learning in continuous stateaction spaces with unknown rewards and transitions. Specifically, we consider the setting where the rewards and transitions are modeled using parametric bilinear exponential families. We propose an algorithm, BEF-RLSVI, that a) uses penalized maximum likelihood estimators to learn the unknown parameters, b) injects a calibrated Gaussian noise in the parameter of rewards to ensure exploration, and c) leverages linearity of the exponential family with respect to an underlying RKHS to perform tractable planning. We further provide a frequentist regret analysis of BEF-RLSVI that yields an upper bound of Õ( √ d 3 H 3 K), where d is the dimension of the parameters, H is the episode length, and K is the number of episodes. Our analysis improves the existing bounds for the bilinear exponential family of MDPs by √ H and removes the handcrafted clipping deployed in existing RLSVI-type algorithms. Our regret bound is order-optimal with respect to H and K. * https://redaouhamma.github.io/ Preprint. Under review. Table 1: A comparison of RL Algorithms for MDPs with functional representations. Algorithm Regret Tractable Tractable Free of Model, assumptions exploration planning clipping Thompson sampling √ d 2 H 3 K ✗ ✓ N.A Gaussian P [RZSD21] (Bayesian) Known rewards EXP-UCRL √ d 2 H 4 K ✗ ✗ N.A Bilinear Exp Family (BEF) [CGM21] (Frequentist) known rewards SMRL [LLS + 21] √ d 2 H 4 K ✗ ✗ N.A BEF, known rewards UCRL-VTR [AJS + 20] √ d 2 H 4 K ✗ ✗ N.A Linear mixture model F -PHE-LSVI [ICN + 21] poly(dEH) √ KH ✓ ✗ ✗ Eluder dimension, Tabular PHE-LSVI (linear-RL) √ d 3 H 4 K Anti-concentration UC-MatrixRL [YW20] √ d 2 H 5 K ✗ ✗ N.A Linear factor MDP OPT-RLSVI [ZBB + 20] √ d 4 H 5 K ✓ ✓ ✗ Linear V BEF-RLSVI (this work) √ d 3 H 3 K ✓ ✓ ✓ Bilinear Exp Family