This paper studies the communication complexity of 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 that communication complexities of both DRAO and DRAO-S are not improvable. Numerical experiments are conducted to demonstrate the encouraging empirical performance of the DRAO-S method.
翻译:本文研究了网络上风险反向优化的通信复杂性。 问题概括了人们深思熟虑的风险中中性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性问题的通信复杂性方面存在差距。 我们提出两种分布式算法,即分散式风险反优化方法(DRAO)和分散式风险反优化方法(DRAO-S),以缩小这一差距。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问题,实现了最佳的通信复杂性。DRAO-S方法通过引入一个新的滑动亚轨法(只需要对设定的模糊度作预测P美元),消除了强烈的假设。我们观察到DRAO-S的美元预测数是最佳的。此外,我们开发了更低的复杂度界限,以显示DRAO和DRAO-S的通信复杂性是不可能实现的。 Numeral-S实验是令人鼓舞的。