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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 in the span of a feature map that is accessible through the generative model. As opposed to previous work 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 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$(H, d)$ learning is possible (with state value function realizability) whenever the action set is small (i.e. O(1)). In particular, we present the TensorPlan algorithm which uses poly$((dH/\delta)^A)$ queries to find a $\delta$-optimal policy relative to any deterministic policy for which the value function is linearly realizable with a parameter from a fixed radius ball around zero. 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. The upper bound is complemented by a lower bound which proves 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.

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