This paper studies the communication complexity of convex risk-averse optimization over a network. The problem generalizes the well-studied risk-neutral finite-sum distributed optimization problem and its importance stems from the need to handle risk in an uncertain environment. For algorithms in the literature, there exists a gap in communication complexities for solving risk-averse and risk-neutral problems. We propose two distributed algorithms, namely the distributed risk averse optimization (DRAO) method and the distributed risk averse optimization with sliding (DRAO-S) method, to close the gap. Specifically, the DRAO method achieves the optimal communication complexity by assuming a certain saddle point subproblem can be easily solved in the server node. The DRAO-S method removes the strong assumption by introducing a novel saddle point sliding subroutine which only requires the projection over the ambiguity set $P$. We observe that the number of $P$-projections performed by DRAO-S is optimal. Moreover, we develop matching lower complexity bounds to show the communication complexities of both DRAO and DRAO-S to be improvable. Numerical experiments are conducted to demonstrate the encouraging empirical performance of the DRAO-S method.

翻译：本文研究了网络上凸风险厌恶优化的通信复杂度。该问题推广了已被广泛研究的风险中立有限和分布式优化问题，其重要性源于在不确定环境中处理风险的需求。针对现有文献中的算法，解决风险厌恶问题与风险中立问题之间存在通信复杂度差距。我们提出了两种分布式算法，即分布式风险厌恶优化（DRAO）方法和带滑动机制的分布式风险厌恶优化（DRAO-S）方法，以弥合这一差距。具体而言，DRAO方法通过假设服务器节点中可轻松求解特定鞍点子问题，实现了最优通信复杂度。DRAO-S方法通过引入一种新颖的鞍点滑动子程序，消除了这一强假设，该子程序仅需要对模糊集$P$进行投影。我们观察到DRAO-S执行的$P$-投影次数是最优的。此外，我们推导了匹配的下界复杂度，以证明DRAO和DRAO-S的通信复杂度无法进一步改进。数值实验展示了DRAO-S方法令人鼓舞的实证性能。