提出双循环算法实现分布式优化加速，突破单循环方法理论极限。
Distributed Optimization via Energy Conservation Laws in Dilated Coordinates
- 用双循环结构结合多项式一致性与外层加速更新
- 达到每轮迭代 $\mathcal O(k^{-2})$ 的目标差距收敛率
- 证明单循环方法无法实现 $\mathcal O(k^{-2})$ 收敛，适合通信受限场景
连续时间模型可揭示分布式优化中的加速结构，但其速率未必能通过直接离散化保留。本文提出一种二阶原始-对偶流用于光滑凸分布式优化，并构造一个精确守恒的能量函数，使得聚合目标差距和平方一致性误差均达到 $\mathcal O(t^{-2})$ 的收敛速率。随后证明，对于一大类单循环有限记忆原始-对偶离散化方法，存在 $\Omega(k^{-1})$ 的下界，排除了在该类别中实现 $\mathcal O(k^{-2})$ 聚合目标保证的可能性。受此限制启发，我们设计了一种双循环方法，将有限步多项式一致性与加速外层更新相结合。该方法每外层迭代仅需一次梯度计算和最多 $m-1$ 次通信轮次（$m$ 为智能体数），保持精确一致性，并实现 $\mathcal O(k^{-2})$ 的聚合目标收敛率。数值实验支持理论结果，并量化了加速带来的通信开销。
原文摘要 · Abstract (English)
Continuous-time models can reveal accelerated structures in distributed optimization, but their rates need not survive direct discretization. We introduce a second-order primal--dual flow for smooth convex distributed optimization and construct an exactly conserved energy that yields an $\mathcal O(t^{-2})$ rate for both the aggregate objective gap and the squared consensus error. We then prove a horizon-wise $Ω(k^{-1})$ lower bound for a broad class of single-loop finite-memory primal--dual discretizations, ruling out a $\mathcal O(k^{-2})$ aggregate-objective guarantee within this class. Motivated by this barrier, we develop a double-loop method that combines finite-step polynomial consensus with an accelerated outer update. It uses one gradient evaluation and at most $m-1$ communication rounds per outer iteration, $m$ being the number of agents, maintains exact consensus and achieves an $\mathcal O(k^{-2})$ aggregate-objective rate. Numerical comparisons with representative distributed methods support the theory and quantify the communication cost of acceleration.
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