提出自适应步长的去中心化优化方法，提升收敛速度与稳定性。
Adaptive Decentralized Composite Optimization via Three-Operator Splitting
- 基于三算子分裂与新型预条件度量设计，实现局部自适应步长调整。
- 凸性下收敛速率达次线性，强凸时可达线性收敛。
- 适合分布式机器学习、智能电网等需本地自适应的场景。
本文研究网络中去中心化优化问题，其中各代理最小化局部平滑（强）凸损失之和，并附加一个非光滑凸扩展值项。我们提出一种去中心化算法，通过结合轻量级极小共识协议与局部回溯过程，使代理自适应调整步长。该设计源于对问题等价重构的三算子分裂分解，重构引入新的BCV预条件度量（Bertsekas-O'Connor-Vandenberghe），支持高效去中心化实现与局部步长调节。理论分析表明，在仅凸条件下，方法具有次线性收敛率；在总目标函数强凸且非光滑分量部分光滑的假设下，可证明线性收敛。数值实验验证了理论结果，凸显所提自适应步长策略的有效性。
原文摘要 · Abstract (English)
The paper studies decentralized optimization over networks, where agents minimize a sum of {\it locally} smooth (strongly) convex losses and plus a nonsmooth convex extended value term. We propose decentralized methods wherein agents {\it adaptively} adjust their stepsize via local backtracking procedures coupled with lightweight min-consensus protocols. Our design stems from a three-operator splitting factorization applied to an equivalent reformulation of the problem. The reformulation is endowed with a new BCV preconditioning metric (Bertsekas-O'Connor-Vandenberghe), which enables efficient decentralized implementation and local stepsize adjustments. We establish robust convergence guarantees. Under mere convexity, the proposed methods converge with a sublinear rate. Under strong convexity of the sum-function, and assuming the nonsmooth component is partly smooth, we further prove linear convergence. Numerical experiments corroborate the theory and highlight the effectiveness of the proposed adaptive stepsize strategy.
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