A Regression-Based Prediction-Correction Method for Stochastic Time-Varying Optimization Problems

TL;DR

This paper introduces a regression-based prediction-correction algorithm for stochastic time-varying optimization that improves tracking accuracy and reduces computational costs without requiring Hessian inversions or derivative estimates.

Contribution

It proposes a novel regression-based prediction step for stochastic optimization, avoiding complex derivative and Hessian computations, and provides theoretical error bounds and empirical validation.

Findings

Improved tracking accuracy over existing methods.

Reduced computational cost due to avoidance of Hessian inversions.

Theoretical error bounds established under standard assumptions.

Abstract

In many real-world applications, optimization problems evolve continuously over time and are often subject to stochastic noise. We consider a stochastic time-varying optimization problem in which the objective function $f(x;t)$ changes continuously and only noisy gradient observations are available. In deterministic settings, the prediction-correction method that exploits the time derivative of the solution is effective for accurately tracking the solution trajectory. However, a straightforward extension to stochastic problems requires an estimate of $\nabla_{xt} f(x;t)$ and the computation of a Hessian inverse at each step--requirements that are difficult or costly in practice. To address these issues, we propose a prediction-correction algorithm that uses a regression-based prediction step: the prediction is formed as a linear combination of recent iterates, which can be computed efficiently without estimating $\nabla_{xt}f(x;t)$ or computing Hessian inversions. We prove a tracking-error bound for the proposed method under standard smoothness and stochastic assumptions. Numerical experiments show that the regression-based prediction improves tracking accuracy while reducing computational cost compared with existing methods.