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# Provably Efficient Reinforcement Learning via Surprise Bound

*Proceedings of The 26th International Conference on Artificial Intelligence and Statistics*, PMLR 206:4006-4032, 2023.

#### Abstract

Value function approximation is important in modern reinforcement learning (RL) problems especially when the state space is (infinitely) large. Despite the importance and wide applicability of value function approximation, its theoretical understanding is still not as sophisticated as its empirical success, especially in the context of general function approximation. In this paper, we propose a provably efficient RL algorithm (both computationally and statistically) with general value function approximations. We show that if the value functions can be approximated by a function class $\mathcal{F}$ which satisfies the bellman-completeness assumption, our algorithm achieves an $\widetilde{O}(\mathrm{poly}(\iota H)\sqrt{T})$ regret bound where $\iota$ is the product of the surprise bound and log-covering numbers, $H$ is the planning horizon, $K$ is the number of episodes and $T = HK$ is the total number of steps the agent interacts with the environment. Our algorithm achieves reasonable regret bounds when applied to both the linear setting and the sparse high-dimensional linear setting. Moreover, our algorithm only needs to solve $O(H\log K)$ empirical risk minimization (ERM) problems, which is far more efficient than previous algorithms that need to solve ERM problems for $\Omega(HK)$ times.