On Query-efficient Planning in MDPs under Linear Realizability of the Optimal State-value Function
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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/δ)^A) queries to find a δ-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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