Asymptotic Distribution of Stochastic Approximation Procedures

By J. Sacks. 1958

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SEQUENTIAL ANALYSIS: SOME CLASSICAL PROBLEMS AND NEW CHALLENGES (2001, background): “Asymptotic normality of the KW scheme (5.9) has also been established by Sacks (1958).”
Evaluation of Optimization Methods for Nonrigid Medical Image Registration Using Mutual Information and B-Splines (2007, background): “More details can be found in [11], [17], [35], and [36].”
Efficient Robbins-Monro procedure for binary data (2004, methodology): “They recommended the simple choice a n = c/n for some constant c. Based on the results of Chung (1954), Hodges and Lehmann (1956), and Sacks (1958), the procedure is fully asymptotically efficient with a n = {n ˙ M (θ)}”
On stochastic gradient and subgradient methods with adaptive steplength sequences (2011, unknown): “Sacks [49] proved that, under suitable conditions, a choice of the form a/k (where k is the iterate index) is optimal from the standpoint of minimizing the asymptotic variance.”
An Overview of Stochastic Approximation (2015, background): “Given the appropriate parameters, this algorithm converges asymptotically at a rate of O(n−1/2) [35].”