Recursive Quantile Estimation: Non-Asymptotic Confidence Bounds
Likai Chen, Georg Keilbar, Wei Biao Wu; 24(91):1−25, 2023.
This paper considers the recursive estimation of quantiles using the stochastic gradient descent (SGD) algorithm with Polyak-Ruppert averaging. The algorithm offers a computationally and memory efficient alternative to the usual empirical estimator. Our focus is on studying the non-asymptotic behavior by providing exponentially decreasing tail probability bounds under mild assumptions on the smoothness of the density functions. This novel non-asymptotic result is based on a bound of the moment generating function of the SGD estimate. We apply our result to the problem of best arm identification in a multi-armed stochastic bandit setting under quantile preferences.
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