Finite-time Analysis of Approximate Policy Iteration for the Linear Quadratic Regulator

We study the sample complexity of approximate policy iteration (PI) for the Linear Quadratic Regulator (LQR), building on a recent line of work using LQR as a testbed to understand the limits of reinforcement learning (RL) algorithms on continuous control tasks. Our analysis quantifies the tension between policy improvement and policy evaluation, and suggests that policy evaluation is the dominant factor in terms of sample complexity. Specifically, we show that to obtain a controller that is within varepsilon of the optimal LQR controller, each step of policy evaluation requires at most (n+d)3/varepsilon2 samples, where n is the dimension of the state vector and d is the dimension of the input vector. On the other hand, only log(1/varepsilon) policy improvement steps suffice, resulting in an overall sample complexity of (n+d)3 varepsilon-2 log(1/varepsilon). We furthermore build on our analysis and construct a simple adaptive procedure based on varepsilon-greedy exploration which relies on approximate PI as a sub-routine and obtains T2/3 regret, improving upon a recent result of Abbasi-Yadkori et al.