AI 中文总结
该研究针对时变网络下的去中心化随机梯度跟踪问题，通过构造时变二次范数将窗口压缩转化为一步李雅普诺夫恒等式，得到误差递推关系，在两类凸目标下均匹配集中式小批量性能且实现线性加速。
AI 中文摘要
我们在满足均匀窗口混合条件的N个智能体组成的时变网络上，研究去中心化随机梯度跟踪问题。τ个连续双随机混合矩阵的乘积能将分歧度压缩至λ<1，即使单个矩阵无需严格压缩分歧度，且单个通信图可能不连通。我们构造一个时变二次范数，将该窗口压缩转化为精确的一步李雅普诺夫恒等式，进而得到质心误差与分歧误差的耦合一步递推关系，无需展开通信窗口上的动力学。对于光滑强凸目标，主导随机项为$\tilde{\bigO}(1/(NK))$；对于光滑凸目标，主导随机项为$\bigO(1/\bigsqrt{NK})$。两者均匹配集中式小批量对应项，且在网络相关的暂态后实现线性加速。
英文摘要
We study decentralized stochastic gradient tracking over a time-varying network of $N$ agents under a uniform window-mixing condition. Products of $τ$ consecutive doubly stochastic mixing matrices contract disagreement by a factor $λ<1$, although individual matrices need not contract disagreement strictly and individual communication graphs may be disconnected. We construct a time-varying quadratic norm that turns this window contraction into an exact one-step Lyapunov identity. This leads to coupled one-step recursions for the centroid and disagreement errors, without unrolling the dynamics over communication windows. For smooth strongly convex objectives, the leading stochastic term is $\widetilde{\mathcal O}(1/(NK))$; for smooth convex objectives, it is $\mathcal O(1/\sqrt{NK})$. Both match their centralized mini-batch counterparts and yield linear speedup after a network-dependent transient.
