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Fast Compute via MC Boosting

Sarah Polson, Vadim Sokolov

TL;DR

The paper addresses the bottleneck of repeated linear solves in modern statistical learning by proposing Monte Carlo boosting, a framework that leverages Neumann-series representations and random-walk estimators to efficiently estimate selected solution components or linear functionals. It unifies forward and adjoint Monte Carlo estimators, and connects Halton-style sequential residual correction with IRLS-like updates used in data augmentation and EM/ECM algorithms. Through theoretical formulations and empirical scaling studies, the work delineates regimes where plain MC, exact sequential MC, and subsampled sequential MC offer compute advantages over traditional direct or stationary methods. The results demonstrate that sequential correction can dramatically reduce compute for large-scale problems or when only a subset of outputs is required, with variance reduction and trajectory reuse playing key roles in practical performance. This framework enables practitioners to trade accuracy for compute in a principled way and suggests concrete extensions to structured matrices and nonlinear/incremental inference workflows.

Abstract

Modern training and inference pipelines in statistical learning and deep learning repeatedly invoke linear-system solves as inner loops, yet high-accuracy deterministic solvers can be prohibitively expensive when solves must be repeated many times or when only partial information (selected components or linear functionals) is required. We position \emph{Monte Carlo boosting} as a practical alternative in this regime, surveying random-walk estimators and sequential residual correction in a unified notation (Neumann-series representation, forward/adjoint estimators, and Halton-style sequential correction), with extensions to overdetermined/least-squares problems and connections to IRLS-style updates in data augmentation and EM/ECM algorithms. Empirically, we compare Jacobi and Gauss--Seidel iterations with plain Monte Carlo, exact sequential Monte Carlo, and a subsampled sequential variant, illustrating scaling regimes that motivate when Monte Carlo boosting can be an enabling compute primitive for modern statistical learning workflows.

Fast Compute via MC Boosting

TL;DR

The paper addresses the bottleneck of repeated linear solves in modern statistical learning by proposing Monte Carlo boosting, a framework that leverages Neumann-series representations and random-walk estimators to efficiently estimate selected solution components or linear functionals. It unifies forward and adjoint Monte Carlo estimators, and connects Halton-style sequential residual correction with IRLS-like updates used in data augmentation and EM/ECM algorithms. Through theoretical formulations and empirical scaling studies, the work delineates regimes where plain MC, exact sequential MC, and subsampled sequential MC offer compute advantages over traditional direct or stationary methods. The results demonstrate that sequential correction can dramatically reduce compute for large-scale problems or when only a subset of outputs is required, with variance reduction and trajectory reuse playing key roles in practical performance. This framework enables practitioners to trade accuracy for compute in a principled way and suggests concrete extensions to structured matrices and nonlinear/incremental inference workflows.

Abstract

Modern training and inference pipelines in statistical learning and deep learning repeatedly invoke linear-system solves as inner loops, yet high-accuracy deterministic solvers can be prohibitively expensive when solves must be repeated many times or when only partial information (selected components or linear functionals) is required. We position \emph{Monte Carlo boosting} as a practical alternative in this regime, surveying random-walk estimators and sequential residual correction in a unified notation (Neumann-series representation, forward/adjoint estimators, and Halton-style sequential correction), with extensions to overdetermined/least-squares problems and connections to IRLS-style updates in data augmentation and EM/ECM algorithms. Empirically, we compare Jacobi and Gauss--Seidel iterations with plain Monte Carlo, exact sequential Monte Carlo, and a subsampled sequential variant, illustrating scaling regimes that motivate when Monte Carlo boosting can be an enabling compute primitive for modern statistical learning workflows.
Paper Structure (22 sections, 34 equations, 2 figures, 3 tables)

This paper contains 22 sections, 34 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Connection to Existing Literature
  3. Problem Setup and Neumann Series
  4. Solving Linear Systems
  5. Detailed Formulation
  6. Overdetermined Systems and Least Squares.
  7. Random-Walk Monte Carlo Estimators
  8. Forward estimator for linear functionals
  9. Adjoint method (many components from shared walks)
  10. Adjoint Method.
  11. Monte Carlo Boosting and IRLS Connections
  12. Sequential residual correction ("boosting")
  13. Halton method with iteratively reweighted least squares
  14. Stable Version of the M-Step.
  15. Numerical Experiments
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Runtime comparison of five methods for solving linear systems as matrix size $m$ increases from 1000 to 5000. Traditional iterative methods (Jacobi, Gauss-Seidel) exhibit $O(m^2)$ scaling. Sequential Monte Carlo methods achieve improved scaling through geometric convergence and subsampling. Plain Monte Carlo achieves $O(1)$ scaling in $m$ when estimating a fixed number of solution components.
  • Figure 2: Monte Carlo convergence for Halton's direct estimator on a fixed $m=1000$ problem. The dashed line indicates a reference $M^{-1/2}$ rate.