Stopping Rules for Monte Carlo Methods of Martingale Difference Type

We establish a practical and easy-to-implement sequential stopping rule for the martingale central limit theorem, focusing on Monte Carlo methods for estimating the mean of a non-iid sequence of martingale difference type. Starting with an impractical scheme based on the standard martingale central limit theorem, we progressively address its limitations from implementation perspectives in the non-asymptotic regime. Along the way, we compare the proposed schemes with their counterparts in the asymptotic regime. The developed framework has potential applications in various domains, including stochastic gradient descent methods. Numerical results are provided to demonstrate the effectiveness of the developed stopping rules in terms of reliability and complexity.