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:本文研究了网络风险反向优化在通信方面的复杂程度。 这个问题概括了经过充分研究的风险中和有限和分布式优化问题及其重要性,因为需要在不确定的环境中处理风险。 关于文献中的算法,在解决风险反常和风险中不偏重问题的通信复杂性方面存在差距。 我们提出两种分散式算法,即分散式风险反优化(DRAO)方法和分散式风险以滑动(DRAO-S)法来缩小差距。 具体地说,DRAO方法通过假设在服务器节点上可以很容易地解决某个垫点子子问题而实现了最佳的通信复杂性。 DRAO-S方法通过引入一个新的挂点滑动亚路径来消除强烈的假设,只需要对设定的模糊度作预测$P$。 我们观察到DRAO-S所执行的美元预测数是最佳的。 此外,我们开发了更低的复杂度约束,以显示DRAO和DRAO-S的通信复杂性,从而无法进行实验。</s>