New Theoretical Framework Established for Stochastic Gradient Descent Quantile Estimation
A new paper establishes the first central limit theorem (CLT)-type theoretical guarantees for quantile estimation using stochastic gradient descent (SGD) with a constant learning rate. The work treats the SGD iteration as a Markov chain to derive its stationary distribution and prove Gaussian convergence as the learning rate approaches zero. The results provide a rigorous statistical foundation for online quantile inference, including a recursive algorithm for constructing confidence intervals.
Researchers have developed asymptotic theory for quantile SGD estimators, addressing a longstanding gap caused by the quantile loss function being neither smooth nor strongly convex — properties that standard SGD analyses typically require. By reframing the SGD iteration as an irreducible, aperiodic, and positive recurrent Markov chain, the authors derive the exact stationary distribution via its characteristic function and establish tight bounds on its moment generating function and tail probabilities. The central result is that the centered and standardized stationary distribution converges to a Gaussian as the learning rate tends to zero, yielding the first CLT-type guarantee for this class of estimators. Building on this theory, the paper proposes a recursive online algorithm for constructing statistically valid confidence intervals without storing the full data stream. Numerical experiments confirm strong finite-sample performance of both the estimator and the inference procedure. The authors note that the theoretical tools developed — particularly the Markov chain framework for non-smooth, non-strongly convex objectives — are broadly applicable to other SGD settings beyond quantile estimation.
What's missing
The paper does not discuss computational cost or scalability comparisons against batch quantile estimation methods, nor does it address how performance degrades under heavy-tailed or highly dependent data distributions. The optimal or adaptive choice of the constant learning rate in practice is not fully resolved.
What different sources said
- arXiv stat.MLCenter
Central Limit Theorems for Stochastic Gradient Descent Quantile Estimators
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