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Quantitative Error Bounds for Scaling Limits of Stochastic Iterative Algorithms

Xiaoyu Wang, Mikolaj J. Kasprzak, Jeffrey Negrea, Solesne Bourguin, Jonathan H. Huggins

TL;DR

The paper delivers non-asymptotic, quantitative error bounds for the scaling limit of stochastic iterative algorithms (SGD/SGLD) by aligning discretized dynamics with a continuous-time Ornstein–Uhlenbeck process via a functional Stein's method with exchangeable pairs. It introduces two intermediate processes and constructs a carefully designed exchangeable pair to bound pathwise discrepancies, yielding explicit bounds that depend on step size, batch size, and other algorithmic parameters. The results establish weak convergence under modest conditions and provide finite-sample bounds for the variance of iterate averages, as well as explicit LP and bounded-Wasserstein distance bounds to the OU limit. These contributions offer a modular, tractable framework for assessing scaling-limit accuracy and pave the way for multivariate extensions and more advanced stochastic approximation schemes.

Abstract

Stochastic iterative algorithms, including stochastic gradient descent (SGD) and stochastic gradient Langevin dynamics (SGLD), are widely utilized for optimization and sampling in large-scale and high-dimensional problems in machine learning, statistics, and engineering. Numerous works have bounded the parameter error in, and characterized the uncertainty of, these approximations. One common approach has been to use scaling limit analyses to relate the distribution of algorithm sample paths to a continuous-time stochastic process approximation, particularly in asymptotic setups. Focusing on the univariate setting, in this paper, we build on previous work to derive non-asymptotic functional approximation error bounds between the algorithm sample paths and the Ornstein-Uhlenbeck approximation using an infinite-dimensional version of Stein's method of exchangeable pairs. We show that this bound implies weak convergence under modest additional assumptions and leads to a bound on the error of the variance of the iterate averages of the algorithm. Furthermore, we use our main result to construct error bounds in terms of two common metrics: the Lévy-Prokhorov and bounded Wasserstein distances. Our results provide a foundation for developing similar error bounds for the multivariate setting and for more sophisticated stochastic approximation algorithms.

Quantitative Error Bounds for Scaling Limits of Stochastic Iterative Algorithms

TL;DR

The paper delivers non-asymptotic, quantitative error bounds for the scaling limit of stochastic iterative algorithms (SGD/SGLD) by aligning discretized dynamics with a continuous-time Ornstein–Uhlenbeck process via a functional Stein's method with exchangeable pairs. It introduces two intermediate processes and constructs a carefully designed exchangeable pair to bound pathwise discrepancies, yielding explicit bounds that depend on step size, batch size, and other algorithmic parameters. The results establish weak convergence under modest conditions and provide finite-sample bounds for the variance of iterate averages, as well as explicit LP and bounded-Wasserstein distance bounds to the OU limit. These contributions offer a modular, tractable framework for assessing scaling-limit accuracy and pave the way for multivariate extensions and more advanced stochastic approximation schemes.

Abstract

Stochastic iterative algorithms, including stochastic gradient descent (SGD) and stochastic gradient Langevin dynamics (SGLD), are widely utilized for optimization and sampling in large-scale and high-dimensional problems in machine learning, statistics, and engineering. Numerous works have bounded the parameter error in, and characterized the uncertainty of, these approximations. One common approach has been to use scaling limit analyses to relate the distribution of algorithm sample paths to a continuous-time stochastic process approximation, particularly in asymptotic setups. Focusing on the univariate setting, in this paper, we build on previous work to derive non-asymptotic functional approximation error bounds between the algorithm sample paths and the Ornstein-Uhlenbeck approximation using an infinite-dimensional version of Stein's method of exchangeable pairs. We show that this bound implies weak convergence under modest additional assumptions and leads to a bound on the error of the variance of the iterate averages of the algorithm. Furthermore, we use our main result to construct error bounds in terms of two common metrics: the Lévy-Prokhorov and bounded Wasserstein distances. Our results provide a foundation for developing similar error bounds for the multivariate setting and for more sophisticated stochastic approximation algorithms.
Paper Structure (57 sections, 32 theorems, 300 equations)

This paper contains 57 sections, 32 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution
  4. Paper Structure
  5. Approach
  6. Functional Method of Exchangeable Pairs
  7. Comparison Processes and the Exchangeable Pair
  8. Functional Error Bounds between Stochastic Iterative Algorithms and Ornstein--Uhlenbeck Processes
  9. Assumptions
  10. Error Bounds
  11. Functional Error Bound for Ornstein--Uhlenbeck processes
  12. Motivation and Setup
  13. Error Bound for OU processes
  14. Proof of the Error Bound for OU Processes
  15. Implications of the Error Bounds
  16. ...and 42 more sections

Key Result

proposition 1

Assume that $(\mathcal{Y}, \mathcal{Y}')$ is an exchangeable pair of $D([0,1])$-valued random variables, fix some $\Lambda_{} > 0$, and, for all $f \in M$, let the random variable $R_f = R_f(\mathcal{Y})$ be defined to satisfy where $\cEE{\mathcal{Y}}\defas \EE\cbra*{\cdot \mid \mathcal{Y}}$. Let $(z_1,\dots,z_{n})$ be a Gaussian vector with a positive definite covariance and, for $t \in [0,1]$,

Theorems & Definitions (59)

  • proposition 1: dobler2021stein
  • remark 1: Interpretation of the error bound
  • remark 2: Assumptions
  • theorem 1
  • remark 3: Interpretation of \ref{['thm:main-theorem-simple']}
  • remark 4: Extension to multivariate setting
  • remark 5: Comparison to other applications of functional Stein's method
  • theorem 2
  • remark 6
  • corollary 1
  • ...and 49 more