New Theoretical Framework Established for Stochastic Gradient Descent Quantile Estimation
Researchers have developed asymptotic theory proving that stochastic gradient descent (SGD) with constant learning rates converges to a Gaussian distribution for quantile estimation, providing the first central limit theorem guarantees for this setting. The work treats quantile SGD as a Markov chain and derives tight bounds on moment generating functions and tail probabilities, addressing challenges posed by the non-smooth and non-strongly convex nature of quantile loss functions. This theoretical advance enables construction of confidence intervals with statistical guarantees and provides tools applicable to broader classes of SGD algorithms.
A new paper on arXiv develops rigorous asymptotic theory for quantile estimation using stochastic gradient descent with constant learning rates. The key innovation is viewing quantile SGD iterations as an irreducible, periodic, and positive recurrent Markov chain that cyclically converges to a unique stationary distribution regardless of initialization. The authors analyze the characteristic function of this stationary distribution and derive tight bounds for its moment generating function and tail probabilities. They prove that as the learning rate approaches zero, the centered and standardized stationary distribution converges to a Gaussian distribution—the first central limit theorem-type guarantee for quantile SGD with constant learning rates. The paper also proposes a recursive algorithm for constructing confidence intervals with theoretical guarantees and validates the approach through numerical experiments. The theoretical techniques developed are positioned as having broader applicability to other SGD algorithms in non-smooth and non-strongly convex settings.
What's missing
The paper does not discuss computational complexity or scalability to high-dimensional problems, nor does it compare convergence rates with alternative quantile estimation methods or discuss practical guidance for learning rate selection in finite-sample settings.
What different sources said
- arXiv stat.MLCenter
Central Limit Theorems for Stochastic Gradient Descent Quantile Estimators
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