Smoothed SGD for quantiles: Bahadur representation and Gaussian approximation

TL;DR

This paper introduces a smoothed SGD algorithm for quantile estimation that ensures non-crossing quantile curves, providing theoretical guarantees and demonstrating strong finite sample performance.

Contribution

It proposes a novel smoothed SGD method for quantiles, establishing non-asymptotic bounds, Bahadur representation, and Gaussian approximation results.

Findings

Non-crossing quantile estimates achieved.

Strong finite sample performance demonstrated.

Theoretical bounds and Gaussian approximation established.

Abstract

This paper considers the estimation of quantiles via a smoothed version of the stochastic gradient descent (SGD) algorithm. By smoothing the score function in the conventional SGD quantile algorithm, we achieve monotonicity in the quantile level in that the estimated quantile curves do not cross. We derive non-asymptotic tail probability bounds for the smoothed SGD quantile estimate both for the case with and without Polyak-Ruppert averaging. For the latter, we also provide a uniform Bahadur representation and a resulting Gaussian approximation result. Numerical studies show good finite sample behavior for our theoretical results.