Learning Over Multitask Graphs—Part I: Stability Analysis

Abstract

This paper formulates a multitask optimization problem where agents in the network have individual objectives to meet, or individual parameter vectors to estimate, subject to a smoothness condition over the graph. The smoothness condition softens the transition in the tasks among adjacent nodes and allows incorporating information about the graph structure into the solution of the inference problem. A diffusion strategy is devised that responds to streaming data and employs stochastic approximations in place of actual gradient vectors, which are generally unavailable.The approach relies on minimizing a global cost consisting of the aggregate sum of individual costs regularized by a term that promotes smoothness. We show in this Part I of the work, under conditions on the step-size parameter, that the adaptive strategy induces a contraction mapping and leads to small estimation errors on the order of the small step-size. The results in the accompanying Part II will reveal explicitly the influence of the network topology and the regularization strength on the network performance and will provide insights into the design of effective multitask strategies for distributed inference over networks.

Learning Over Multitask Graphs—Part II: Performance Analysis

“…We denote the unique minimizer of J k (w k ) by w o k . Let us recall the assumption on the risks {J k (w k )} used in Part I [2]. Assumption 1.…”

“…Let w k,i denote the estimate of w o k,η at iteration i and node k. In order to solve (4) in a fully distributed and adaptive manner, we proposed in Part I [2] the following diffusion-type algorithm:…”

“…In this section, we recall the assumptions used in Part I [2] to establish the network mean-square error stability (9). Assumption 2.…”

“…For any values of η in the range 0 < η < ∞, the network behaves in a multitask mode where agents seek their individual models while at the same time ensuring that these models satisfy certain smoothness and closeness conditions dictated by the value of η. We carried out in Part I [2] a detailed stability analysis of the proposed strategy. We showed, under conditions on the step-size learning parameter µ, that the adaptive strategy induces a contraction mapping and that despite gradient noise, it is able to converge in the mean-square-error sense within O(µ) from the solution of the regularized problem, for sufficiently small µ.…”

“…Based on the results established in Part I [2], we shall derive in this paper a closed-form expression for the steady-state network mean-square-error relative to the minimizer of the regularized cost. This closed form expression will reveal explicitly the influence of the regularization strength, network topology (through the eigenvalues and eigenvectors of the Laplacian matrix), gradient noise, and data characteristics, on the network performance.…”