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 $η\rightarrow0$. 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 guarantees. Numerical studies demonstrate the effective 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.

翻译：本文发展了带常数学习率的随机梯度下降（SGD）分位数估计的渐近理论。分位数损失函数既非光滑也非强凸。突破常规视角与技术，我们将分位数SGD迭代视为不可约、周期且正常返的马尔可夫链，该链无论初始值如何任意固定，都会循环收敛至其唯一平稳分布。为推导平稳分布的精确形式，我们通过利用平稳方程分析其特征函数的结构。同时导出了其矩母函数（MGF）和尾部概率的紧界。综合上述方法，我们证明当学习率$η\rightarrow0$时，中心化与标准化后的平稳分布收敛至高斯分布。这一发现首次为常数学习率下的分位数SGD估计量提供了中心极限定理（CLT）型的理论保证。我们进一步提出一种递归算法，用于构建具有统计保证的估计量置信区间。数值研究展示了在线估计量与推断程序在有限样本下的有效性能。本文发展的理论工具对于研究一般SGD算法（尤其是非强凸与非光滑情形）的马尔可夫链形式具有独立参考价值。