  Despite the empirical success of the actor-critic algorithm, its theoretical
understanding lags behind. In a broader context, actor-critic can be viewed as
an online alternating update algorithm for bilevel optimization, whose
convergence is known to be fragile. To understand the instability of
actor-critic, we focus on its application to linear quadratic regulators, a
simple yet fundamental setting of reinforcement learning. We establish a
nonasymptotic convergence analysis of actor-critic in this setting. In
particular, we prove that actor-critic finds a globally optimal pair of actor
(policy) and critic (action-value function) at a linear rate of convergence.
Our analysis may serve as a preliminary step towards a complete theoretical
understanding of bilevel optimization with nonconvex subproblems, which is
NP-hard in the worst case and is often solved using heuristics.
