An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This however leads to a slower convergence. A second alternative is to use a fixed step-size and run independent replicas of the algorithm and average these. A third option is to run replicas of the algorithm and allow them to interact. It is unclear which of these options works best. To address this question, we reduce the problem of the computation of an exact minimizer with noisy gradient information to the study of stochastic mirror descent with interacting particles. We study the convergence of stochastic mirror descent and make explicit the tradeoffs between communication and variance reduction. We provide theoretical and numerical evidence to suggest that interaction helps to improve convergence and reduce the variance of the estimate.

带有噪声信息的优化中的一个开放问题是与噪声量无关的精确最小化器的计算。随机逼近算法的标准做法是使用逐渐减小的步长。但是，这导致收敛变慢。第二种替代方法是使用固定的步长并运行算法的独立副本并将其平均。第三种选择是运行算法的副本并允许它们进行交互。尚不清楚这些选项中哪个最合适。为了解决这个问题，我们减少了带有噪声梯度信息的精确最小化器的计算问题，以研究具有相互作用粒子的随机镜下降。我们研究了随机镜像下降的收敛性，并明确了通信与方差减少之间的权衡。