Preconditioned Truncated Single-Sample Estimators for Scalable Stochastic Optimization

TL;DR

This paper introduces PTSS estimators, a novel family of stochastic Krylov methods with preconditioning and truncation, providing low-variance, stable solutions for large-scale stochastic optimization tasks involving linear systems and log-determinants.

Contribution

The paper develops PTSS estimators combining preconditioning with truncated Krylov methods, offering a new efficient approach with theoretical guarantees for variance reduction and stability in stochastic optimization.

Findings

PTSS achieves lower variance than existing estimators.

PTSS provides stable and accurate solutions in large-scale problems.

Numerical experiments demonstrate improved efficiency and stability.

Abstract

Many large-scale stochastic optimization algorithms involve repeated solutions of linear systems or evaluations of log-determinants. In these regimes, computing exact solutions is often unnecessary; it is more computationally efficient to construct unbiased stochastic estimators with controlled variance. However, classical iterative solvers incur truncation bias, whereas unbiased Krylov-based estimators typically exhibit high variance and numerical instability. To mitigate these issues, we introduce the Preconditioned Truncated Single-Sample (PTSS) estimators--a family of stochastic Krylov methods that integrate preconditioning with truncated Lanczos iterations. PTSS yields low-variance, stable estimators for linear system solutions, log-determinants, and their derivatives. We establish theoretical results on their mean, variance, and concentration properties, explicitly quantifying the variance reduction induced by preconditioning. Numerical experiments confirm that PTSS achieves superior stability and variance control compared with existing unbiased and biased alternatives, providing an efficient framework for stochastic optimization.