Title: The Impact of Approximation Errors on Warm-Start Reinforcement Learning: A Finite-time Analysis. Abstract: 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, calling for  a fundamental understanding, especially when the function approximation is used. To fill this void, we take a finite time analysis approach to quantify the impact of approximation errors on the learning performance of Warm-Start 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 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. We also derive the upper bounds, which provide insights on achieving the desired finite-learning performance in the Warm-Start A-C algorithm.