Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation

Traditional statistical inference for parameters is based on asymptotic/non-asymptotic coverage guarantees at fixed sample sizes. Such guarantees tend to become problematic in sequential experimental design due to the issue of "peeking", i.e., experimenters deciding whether to collect more data for further experiments after looking at current results in order to make the outcome more significant (Feller, 1940; Anscombe, 1954; Robbins, 1952). Recently, there has been an emerging literature on safe anytime-valid inference (Johari et al., 2022; Howard et al., 2021; Grünwald et al., 2020; Shafer, 2021; Pace and Salvan, 2020; Ramdas et al., 2023) that partially solve such problems. At a high level, they utilize a su-permartingale method coupled with Ville's inequality (Ville, 1939) to obtain time-uniform coverage guarantees for the quantities of interest, which is also equivalent to coverage guarantees for arbitrary stopping times (Ramdas et al., 2020): P ( t 1: θ