分布式变分不等式的认证残差拟牛顿方法

中文摘要 AI 辅助

光滑单调变分不等式上的二阶方法达到最优速率 $O(T^{-3/2})$，但分布式精确雅可比矩阵的通信成本是算子值的 $d$ 倍。我们证明相似性可以免费完成部分工作：如果服务器的雅可比矩阵与全局雅可比矩阵的差异至多为 $\eta$，则使用它在一阶通信成本下给出 $O(L_1D^3T^{-3/2}+\eta D^2T^{-1})$ 的速率。残差雅可比矩阵 $\ abla F-\ abla F_1$ 的拟牛顿近似，由已通信的割线构建，改进了模型，但无法消除 $T^{-1}$ 项，因为雅可比误差的任何一致界都会使其保留在速率中。因此，我们仅沿候选步认证代理：一次雅可比-向量乘积测试它，失败的测试被重用为精确校正。这实现了精确速率 $O(L_1D^3T^{-3/2})$，同时仅传输向量。在 LIBSVM 和合成实例上的实验测量了精度与通信的权衡。

英文摘要

Second-order methods for smooth monotone variational inequalities reach the optimal rate $O(T^{-3/2})$, but a distributed exact Jacobian costs $d$ times more communication than an operator value. We show that similarity does part of the work for free: if the server's Jacobian differs from the global one by at most $β$, using it gives $O(L_1D^3T^{-3/2}+βD^2T^{-1})$ at first-order communication cost. A quasi-Newton approximation of the residual Jacobian $\nabla F-\nabla F_1$, built from secants already communicated, improves the model but cannot remove the $T^{-1}$ term, because any uniform bound on the Jacobian error leaves it in the rate. We therefore certify the surrogate only along the candidate step: one Jacobian-vector product tests it, and a failed test is reused as an exact correction. This attains the exact rate $O(L_1D^3T^{-3/2})$ while transmitting only vectors. Experiments on LIBSVM and synthetic instances measure accuracy against communication.

发表机构

• Moscow Institute of Physics and Technology（莫斯科物理技术学院）
• FusionBrain Lab（FusionBrain实验室）
• Innopolis University（伊诺波利斯大学）
• Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）

机构由 AI 辅助整理，请以论文原文为准。