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# On Query-efficient Planning in MDPs under Linear Realizability of the Optimal State-value Function

*Proceedings of Thirty Fourth Conference on Learning Theory*, PMLR 134:4355-4385, 2021.

#### Abstract

We consider the problem of local planning in fixed-horizon Markov Decision Processes (MDPs) with a generative model under the assumption that the optimal value function lies close to the span of a feature map. The generative model provides a restricted, “local” access to the MDP: The planner can ask for random transitions from previously returned states and arbitrary actions, and the features are also only accessible for the states that are encountered in this process. As opposed to previous work (e.g. Lattimore et al. (2020)) where linear realizability of all policies was assumed, we consider the significantly relaxed assumption of a single linearly realizable (deterministic) policy. A recent lower bound by Weisz et al. (2020) established that the related problem when the action-value function of the optimal policy is linearly realizable requires an exponential number of queries, either in $H$ (the horizon of the MDP) or $d$ (the dimension of the feature mapping). Their construction crucially relies on having an exponentially large action set. In contrast, in this work, we establish that $\poly(H,d)$ planning is possible with state value function realizability whenever the action set has a constant size. In particular, we present the TensorPlan algorithm which uses $\poly((dH/\delta)^A)$ simulator queries to find a $\delta$-optimal policy relative to any deterministic policy for which the value function is linearly realizable with some bounded parameter (with a known bound). This is the first algorithm to give a polynomial query complexity guarantee using only linear-realizability of a single competing value function. Whether the computation cost is similarly bounded remains an interesting open question. We also extend the upper bound to the near-realizable case and to the infinite-horizon discounted MDP setup. The upper bounds are complemented by a lower bound which states that in the infinite-horizon episodic setting, planners that achieve constant suboptimality need exponentially many queries, either in the dimension or the number of actions.