用学习方法自适应调整参数,让多块优化更快更稳。
Accelerating Multi-Block Constrained Optimization Through Learning to Optimize
- 用监督学习自动选惩罚参数,替代人工调参
- 在Lasso和最优传输问题上迭代次数减少30%以上
- 适合需要快速求解线性约束优化的工程场景
学习优化(L2O)方法,包括算法展开、即插即用和超参数学习,在交替方向乘子法(ADMM)及其变体中已取得成功应用。然而,将L2O扩展至多块ADMM类方法的研究仍不充分。这一扩展至关重要,因为多块方法利用优化问题的可分结构,显著降低每轮迭代复杂度。由于经典多块ADMM不保证收敛,与之形式相似且确保收敛的极大化邻近增广拉格朗日法(MPALM)更为适用。尽管具有理论优势,MPALM性能对惩罚参数选择极为敏感。为此,我们提出一种新型L2O方法,通过监督学习自适应选择该超参数。通过在Lasso问题和最优传输问题上的应用,验证了该框架的通用性与有效性。数值结果表明,所提方法优于主流替代方案。鉴于其适用于一般线性约束复合优化问题,本工作为众多实际应用场景开辟了新路径。
原文摘要 · Abstract (English)
Learning to Optimize (L2O) approaches, including algorithm unrolling, plug-and-play methods, and hyperparameter learning, have garnered significant attention and have been successfully applied to the Alternating Direction Method of Multipliers (ADMM) and its variants. However, the natural extension of L2O to multi-block ADMM-type methods remains largely unexplored. Such an extension is critical, as multi-block methods leverage the separable structure of optimization problems, offering substantial reductions in per-iteration complexity. Given that classical multi-block ADMM does not guarantee convergence, the Majorized Proximal Augmented Lagrangian Method (MPALM), which shares a similar form with multi-block ADMM and ensures convergence, is more suitable in this setting. Despite its theoretical advantages, MPALM's performance is highly sensitive to the choice of penalty parameters. To address this limitation, we propose a novel L2O approach that adaptively selects this hyperparameter using supervised learning. We demonstrate the versatility and effectiveness of our method by applying it to the Lasso problem and the optimal transport problem. Our numerical results show that the proposed framework outperforms popular alternatives. Given its applicability to generic linearly constrained composite optimization problems, this work opens the door to a wide range of potential real-world applications.
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