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# Computational-Statistical Gap in Reinforcement Learning

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:1282-1302, 2022.

#### Abstract

Reinforcement learning with function approximation has recently achieved tremendous results in applications with large state spaces. This empirical success has motivated a growing body of theoretical work proposing necessary and sufficient conditions under which efficient reinforcement learning is possible. From this line of work, a remarkably simple minimal sufficient condition has emerged for sample efficient reinforcement learning: MDPs with optimal value function V* and Q* linear in some known low-dimensional features. In this setting, recent works have designed sample efficient algorithms which require a number of samples polynomial in the feature dimension and independent of the size of state space. They however leave finding computationally efficient algorithms as future work and this is considered a major open problem in the community. In this work, we make progress on this open problem by presenting the first computational lower bound for RL with linear function approximation: unless NP=RP, no randomized polynomial time algorithm exists for deterministic transition MDPs with a constant number of actions and linear optimal value functions. To prove this, we show a reduction from Unique-Sat, where we convert a CNF formula into an MDP with deterministic transitions, constant number of actions and low dimensional linear optimal value functions. This result also exhibits the first computational-statistical gap in reinforcement learning with linear function approximation, as the underlying statistical problem is information-theoretically solvable with a polynomial number of queries, but no computationally efficient algorithm exists unless NP=RP. Finally, we also prove a quasi-polynomial time lower bound under the Randomized Exponential Time Hypothesis.