Online Inference for Quantiles by Constant Learning-Rate Stochastic Gradient Descent

TL;DR

This work addresses online quantile estimation under non-smooth, non-strongly convex objectives using a constant learning-rate SGD. By modeling the SGD sequence as an irreducible, positive recurrent Markov chain and applying Foster's theorem along with Tweedie-type MGF bounds, the authors prove the existence of a unique stationary distribution and establish a CLT-type limit for the last iterate as $\eta\to 0$, yielding a Gaussian limit with variance $\tau(1-\tau)/(2f_i(\theta_i(\tau)))$. They further develop online inference via recursive kernel density estimation to estimate the stationary density at the quantile and construct confidence intervals for the quantile estimator. Numerical simulations on asymmetric and heavy-tailed distributions demonstrate strong finite-sample performance and valid online coverage. This framework provides the first CLT-type guarantees for the last iterate of constant-learning-rate SGD in this non-smooth setting and offers broad tools for analyzing general Markovian kernels in online stochastic approximation.

Abstract

This paper proposes an online inference method of the stochastic gradient descent (SGD) with a constant learning rate for quantile loss functions with theoretical guarantees. Since the quantile loss function is neither smooth nor strongly convex, we view such SGD iterates as an irreducible and positive recurrent Markov chain. By leveraging this interpretation, we show the existence of a unique asymptotic stationary distribution, regardless of the arbitrarily fixed initialization. To characterize the exact form of this limiting distribution, we derive bounds for its moment generating function and tail probabilities, controlling over the first and second moments of SGD iterates. By these techniques, we prove that the stationary distribution converges to a Gaussian distribution as the constant learning rate $η\rightarrow0$. Our findings provide the first central limit theorem (CLT)-type theoretical guarantees for the last iterate of constant learning-rate SGD in non-smooth and non-strongly convex settings. We further propose a recursive algorithm to construct confidence intervals of SGD iterates in an online manner. Numerical studies demonstrate strong finite-sample performance of our proposed quantile estimator and inference method. The theoretical tools in this study are of independent interest to investigate general transition kernels in Markov chains.

Figures (3)

• Figure 1: Quantile loss function (left panel) and score function (right panel).
• Figure 2: Asymptotic normality, convergence of mean squared error, and empirical coverage for quantiles of Beta $(2,3)$ distribution.
• Figure 3: Asymptotic normality, convergence of mean squared error, and empirical coverage for quantiles of Cauchy $(0,2)$ distribution.

Theorems & Definitions (10)

• Theorem 1
• Proposition 2
• Remark 3
• Lemma 5
• Corollary 6
• Corollary 7
• Theorem 8
• Corollary 9
• Remark 10
• Corollary 12