Asynchronous Saddle Point Algorithm for Stochastic Optimization in Heterogeneous Networks

Abstract

We consider expected risk minimization in multi-agent systems comprised of distinct subsets of agents operating without a common time-scale. Each individual in the network is charged with minimizing the global objective function, which is an average of sum of the statistical average loss function of each agent in the network. Since agents are not assumed to observe data from identical distributions, the hypothesis that all agents seek a common action is violated, and thus the hypothesis upon which consensus constraints are formulated is violated. Thus, we consider nonlinear network proximity constraints which incentivize nearby nodes to make decisions which are close to one another but not necessarily coincide. Moreover, agents are not assumed to receive their sequentially arriving observations on a common time index, and thus seek to learn in an asynchronous manner. An asynchronous stochastic variant of the Arrow-Hurwicz saddle point method is proposed to solve this problem which operates by alternating primal stochastic descent steps and Lagrange multiplier updates which penalize the discrepancies between agents. This tool leads to an implementation that allows for each agent to operate asynchronously with local information only and message passing with neighbors. Our main result establishes that the proposed method yields convergence in expectation both in terms of the primal sub-optimality and constraint violation to radii of sizes O( √ T ) and O(T 3/4 ), respectively. Empirical evaluation on an asynchronously operating wireless network that manages user channel interference through an adaptive communications pricing mechanism demonstrates that our theoretical results translates well to practice.

Decentralized Multi-Task Stochastic Optimization With Compressed Communications

“…Using the property of the double sum similar to the updates from ( 12) to (13) in the last term of the above equation,…”

“…Such a scenario arises in, for example, distributed multitask adaptive signal processing, where the weight vectors at neighboring nodes are not the same [10,11]. One of the first papers that has analyzed such departure from consensus optimization is [12] where the formulation included proximity constraints between neighboring nodes, which were handled through construction of Lagrangians and using saddle-point algorithms, and extended to the asynchronous setting in [13].…”

“…Before further discussing the contents and contributions of this paper, let us point out that saddle-point algorithms (a.k.a. primal-dual algorithms) have been extensively used in the literature on constrained optimization, such as deterministic centralized optimization [33,34], decentralized optimization [35], stochastic optimization [12,13,36], and online optimization [37,38].…”

Communication Efficient Framework for Decentralized Machine Learning

“…There are a variety of distributed optimization algorithms proposed in the literature, such as primal methods [9], [10], [11], [12] and primal-dual methods [13], [14], [15]. The performance of distributed optimization algorithms is commonly characterized by their computation time and communication cost.…”

Decentralised federated learning with adaptive partial gradient aggregation

“…Many distributed optimisation methods have been proposed to overcome the challenges in large scale distributed machine learning, such as primal methods [13][14][15] and primal-dual methods [16][17][18], while, in federated learning settings, the communication cost often becomes dominant compared to the computation cost.…”