Central Limit Theorems for Stochastic Gradient Descent Quantile Estimators
arXiv:2503.02178v2 Announce Type: replace Abstract: This paper develops asymptotic theory for quantile estimation via stochastic gradient descent (SGD) with a constant learning rate. The quantile loss function is neither smooth nor strongly convex. Beyond conventional perspectives and techniques, we view quantile SGD iteration as an irreducible, periodic, and positive recurrent Markov chain, which cyclically converges to its unique stationary distribution regardless of the arbitrarily fixed initialization. To derive the exact form of the stationary distribution, we analyze the structure of its characteristic function by exploiting the stationary equation. We also derive tight bounds for its moment generating function (MGF) and tail probabilities. Synthesizing the aforementioned approaches,
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Abstract:This paper develops asymptotic theory for quantile estimation via stochastic gradient descent (SGD) with a constant learning rate. The quantile loss function is neither smooth nor strongly convex. Beyond conventional perspectives and techniques, we view quantile SGD iteration as an irreducible, periodic, and positive recurrent Markov chain, which cyclically converges to its unique stationary distribution regardless of the arbitrarily fixed initialization. To derive the exact form of the stationary distribution, we analyze the structure of its characteristic function by exploiting the stationary equation. We also derive tight bounds for its moment generating function (MGF) and tail probabilities. Synthesizing the aforementioned approaches, we prove that the centered and standardized stationary distribution converges to a Gaussian distribution as the learning rate $\eta\rightarrow0$. This finding provides the first central limit theorem (CLT)-type theoretical guarantees for the quantile SGD estimator with constant learning rates. We further propose a recursive algorithm to construct confidence intervals of the estimators with statistical guarantee. Numerical studies demonstrate the satisfactory finite-sample performance of the online estimator and inference procedure. The theoretical tools developed in this study are of independent interest for investigating general SGD algorithms formulated as Markov chains, particularly in non-strongly convex and non-smooth settings.
Subjects:
Machine Learning (stat.ML); Machine Learning (cs.LG)
Cite as: arXiv:2503.02178 [stat.ML]
(or arXiv:2503.02178v2 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2503.02178
arXiv-issued DOI via DataCite
Submission history
From: Ziyang Wei [view email] [v1] Tue, 4 Mar 2025 01:37:42 UTC (2,414 KB) [v2] Thu, 2 Apr 2026 18:09:14 UTC (684 KB)
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