In this work, we investigate stochastic quasi-Newton methods for minimizing a finite sum of cost functions over a decentralized network. In Part I, we develop a general algorithmic framework that incorporates stochastic quasi-Newton approximations with variance reduction so as to achieve fast convergence. At each time each node constructs a local, inexact quasi-Newton direction that asymptotically approaches the global, exact one. To be specific, (i) A local gradient approximation is constructed by using dynamic average consensus to track the average of variance-reduced local stochastic gradients over the entire network; (ii) A local Hessian inverse approximation is assumed to be positive definite with bounded eigenvalues, and how to construct it to satisfy these assumptions will be given in Part II. Compared to the existing decentralized stochastic first-order methods, the proposed general framework introduces the second-order curvature information without incurring extra sampling or communication. With a fixed step size, we establish the conditions under which the proposed general framework linearly converges to an exact optimal solution.
翻译:在这项工作中,我们调查在分散的网络中将成本功能的有限总和最小化的准牛顿方法。在第一部分,我们开发了一个总体算法框架,其中纳入随机准牛顿近似值,并减少差异,以便实现快速趋同。每次每个节点都构建一个局部的、不精确的准牛顿方向,而该方向与全局的、准确的一阶方法相近。具体来说,(一) 本地梯度近似值是通过使用动态平均共识构建的,以跟踪整个网络中差异减少的本地蒸气梯度的平均值。 (二) 假定本地的赫森反向近似值与封闭的eigen值是肯定的,如何构建满足这些假设将在第二部分中给出。 与现有的分散的随机第一阶方法相比,拟议的总框架引入了第二阶曲线信息,而无需额外的取样或通信。我们设定了固定的步数,我们设定了拟议总框架线性走向最佳解决方案的条件。