分布式学习中双向块分裂的预测-校正分析
A Prediction--Correction Analysis of Two-Way Block Splitting in Distributed Learning
- School of Mathematics and Statistics, Yunnan University（云南大学数学与统计学院）
- Yunnan Key Laboratory of Statistical Modeling and Data Analysis, Yunnan University（云南大学云南省统计建模与数据分析重点实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过预测-校正框架分析双向块分裂算法在分布式学习中的收敛性，建立降维实现与全空间更新的等价性，并给出全局收敛与复杂度保证。
AI中文摘要：
本文通过何-袁预测-校正框架重新审视了Parikh-Boyd双向块分裂算法在大规模分布式学习中的收敛性。同时的行-列划分也与混合联邦学习相关，其中数据可能在样本和特征上均存在异质性。我们将降维迭代提升到等维乘积空间，并重构了其实现中省略的原始和对偶坐标。所引入的正交补结构建立了与已发表更新的逐次迭代精确等价性。混合变分不等式表示随后导出了基本下降不等式、全局收敛性、遍历复杂度界以及当前迭代残差估计，这些均在标准凸性、可解性、精确子问题和不变初始化假设下成立。分析还表明，降维状态递归是一种度量近端点迭代。未施加强凸性、可微性或满秩条件。推导阐明了哪些代数初始化条件允许降维实现在不修改其局部更新的情况下继承全空间收敛性和复杂度保证。
英文摘要：
This note revisits the convergence of the Parikh--Boyd two-way block-splitting algorithm for large-scale distributed learning through the He--Yuan prediction--correction framework. Simultaneous row--column partitioning is also relevant to hybrid federated learning, where data may be heterogeneous in both samples and features. We lift the reduced iteration to an equal-dimensional product space and reconstruct the primal and dual coordinates omitted by its implementation. The induced orthogonal-complement structure establishes exact iteration-by-iteration equivalence with the published updates. A mixed variational-inequality representation then yields a fundamental descent inequality, global convergence, an ergodic complexity bound, and current-iterate residual estimates under standard convexity, solvability, exact-subproblem, and invariant-initialization assumptions. The analysis also shows that the reduced state recursion is a metric proximal point iteration. No strong convexity, differentiability, or full-rank condition is imposed. The derivation clarifies which algebraic initialization conditions allow the reduced implementation to inherit the full-space convergence and complexity guarantees without modifying its local updates.
