In developing efficient optimization algorithms, it is crucial to account for communication constraints -- a significant challenge in modern federated learning settings. The best-known communication complexity among non-accelerated algorithms is achieved by DANE, a distributed proximal-point algorithm that solves local subproblems in each iteration and that can exploit second-order similarity among individual functions. However, to achieve such communication efficiency, the accuracy requirement for solving the local subproblems is slightly sub-optimal. Inspired by the hybrid projection-proximal point method, in this work, we i) propose a novel distributed algorithm S-DANE. This method adopts a more stabilized prox-center in the proximal step compared with DANE, and matches its deterministic communication complexity. Moreover, the accuracy condition of the subproblem is milder, leading to enhanced local computation efficiency. Furthermore, it supports partial client participation and arbitrary stochastic local solvers, making it more attractive in practice. We further ii) accelerate S-DANE, and show that the resulting algorithm achieves the best-known communication complexity among all existing methods for distributed convex optimization, with the same improved local computation efficiency as S-DANE.

翻译：在开发高效优化算法时，充分考虑通信约束至关重要——这是现代联邦学习场景中的一个重大挑战。在非加速算法中，最著名的通信复杂度由DANE实现，这是一种分布式近端点算法，它在每次迭代中求解局部子问题，并能利用个体函数间的二阶相似性。然而，为了达到这样的通信效率，求解局部子问题的精度要求略低于最优。受混合投影-近端点方法的启发，本研究首先提出了一种新颖的分布式算法S-DANE。与DANE相比，该方法在近端步骤中采用了更稳定的近端中心，同时匹配了其确定性通信复杂度。此外，该算法的子问题精度条件更为宽松，从而提高了局部计算效率。更重要的是，它支持部分客户端参与和任意的随机局部求解器，使其在实际应用中更具吸引力。我们进一步加速了S-DANE，并证明所得算法在所有现有分布式凸优化方法中实现了最佳的通信复杂度，同时保持了与S-DANE相同的改进局部计算效率。