Smoothed SGD for quantiles: Bahadur representation and Gaussian approximation
This addresses quantile estimation for statistical modeling, providing incremental improvements in algorithm stability and theoretical guarantees.
The paper tackles quantile estimation by smoothing the stochastic gradient descent (SGD) algorithm to prevent quantile curve crossing, achieving non-asymptotic tail bounds and Gaussian approximation results with numerical validation.
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.