Model-free Low-rank Reinforcement Learning via Leveraged Entry-wise Matrix Estimation

Stefan Stojanovic, Yassir Jedra, Alexandre Proutiere

Keywords: Low-rank RL; Entry-wise matrix estimation; matrix completion; sample complexity;

Abstract: We consider the problem of learning an $\varepsilon$-optimal policy in controlled dynamical systems with low-rank latent structure. For this problem, we present LoRa-Pi (Low-Rank Policy Iteration), a model-free learning algorithm alternating between policy improvement and policy evaluation steps. In the latter, the algorithm estimates the low-rank matrix corresponding to the (state, action) value function of the current policy using the following two-phase procedure. The entries of the matrix are first sampled uniformly at random to estimate, via a spectral method, the *leverage scores* of its rows and columns. These scores are then used to extract a few important rows and columns whose entries are further sampled. The algorithm exploits these new samples to complete the matrix estimation using a CUR-like method. For this leveraged matrix estimation procedure, we establish entry-wise guarantees that remarkably, do not depend on the coherence of the matrix but only on its spikiness. These guarantees imply that LoRa-PI learns an $\varepsilon$-optimal policy using $\widetilde{O}({S+A\over \mathrm{poly}(1-\gamma)\varepsilon^2})$ samples where $S$ (resp. $A$) denotes the number of states (resp. actions) and $\gamma$ the discount factor. Our algorithm achieves this order-optimal (in $S$, $A$ and $\varepsilon$) sample complexity under milder conditions than those assumed in previously proposed approaches.

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Serve As Reviewer: ~Yassir_Jedra1

Track: Fast Track: published work

Publication Link: https://openreview.net/forum?id=HavKlV22xJ

Submission Number: 132