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

TL;DR

This paper introduces the TensorPlan algorithm, which achieves polynomial query complexity for local planning in fixed-horizon MDPs under linear realizability of the optimal state-value function, assuming a small action set.

Contribution

It relaxes previous assumptions by only requiring linear realizability of a single policy's value function and provides the first polynomial-query algorithm under these conditions.

Findings

TensorPlan uses polynomial queries in (d, H, 1/δ) for near-optimal policies.

Linear realizability of a single value function suffices for polynomial planning complexity.

Exponential query lower bounds are established for infinite-horizon settings with many actions.

Abstract

We consider local planning in fixed-horizon 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 local access to the MDP: The planner can ask for random transitions from previously returned states and arbitrary actions, and features are only accessible for 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. 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 open question. We extend the upper bound to the near-realizable case and to the infinite-horizon discounted setup. We also present a lower bound in the infinite-horizon episodic setting: Planners that achieve constant suboptimality need exponentially many queries, either in $d$ or the number of actions.