A Regression-Based Prediction-Correction Method for Stochastic Time-Varying Optimization Problems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915681442725888 |
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| author | Kamijima, Tomoya Marumo, Naoki Takeda, Akiko |
| author_facet | Kamijima, Tomoya Marumo, Naoki Takeda, Akiko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Regression-Based Prediction-Correction Method for Stochastic Time-Varying Optimization Problems Kamijima, Tomoya Marumo, Naoki Takeda, Akiko Optimization and Control 90C15, 90C25, 65K05 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. |
| title | A Regression-Based Prediction-Correction Method for Stochastic Time-Varying Optimization Problems |
| topic | Optimization and Control 90C15, 90C25, 65K05 |
| url | https://arxiv.org/abs/2512.15205 |