  Warm-Start reinforcement learning (RL), aided by a prior policy obtained from
offline training, is emerging as a promising RL approach for practical
applications. Recent empirical studies have demonstrated that the performance
of Warm-Start RL can be improved \textit{quickly} in some cases but become
\textit{stagnant} in other cases, especially when the function approximation is
used. To this end, the primary objective of this work is to build a fundamental
understanding on ``\textit{whether and when online learning can be
significantly accelerated by a warm-start policy from offline RL?}''.
Specifically, we consider the widely used Actor-Critic (A-C) method with a
prior policy. We first quantify the approximation errors in the Actor update
and the Critic update, respectively. Next, we cast the Warm-Start A-C algorithm
as Newton's method with perturbation, and study the impact of the approximation
errors on the finite-time learning performance with inaccurate Actor/Critic
updates. Under some general technical conditions, we derive the upper bounds,
which shed light on achieving the desired finite-learning performance in the
Warm-Start A-C algorithm. In particular, our findings reveal that it is
essential to reduce the algorithm bias in online learning.
  We also obtain lower bounds on the sub-optimality gap of the Warm-Start A-C
algorithm to quantify the impact of the bias and error propagation.
