  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 $(n+d)^3/\varepsilon^2$
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 $T^{2/3}$ regret, improving upon a recent result of
Abbasi-Yadkori et al.
