An Analysis Tool for Push-Sum Based Distributed Optimization

The push-sum algorithm is probably the most important distributed averaging approach over directed graphs, which has been applied to various problems including distributed optimization. This paper establishes the explicit absolute probability sequence for the push-sum algorithm, and based on which, constructs quadratic Lyapunov functions for push-sum based distributed optimization algorithms. As illustrative examples, the proposed novel analysis tool can improve the convergence rates of the subgradient-push and stochastic gradient-push, two important algorithms for distributed convex optimization over unbalanced directed graphs. Specifically, the paper proves that the subgradient-push algorithm converges at a rate of $O(1/\sqrt{t})$ for general convex functions and stochastic gradient-push algorithm converges at a rate of $O(1/t)$ for strongly convex functions, over time-varying unbalanced directed graphs. Both rates are respectively the same as the state-of-the-art rates of their single-agent counterparts and thus optimal, which closes the theoretical gap between the centralized and push-sum based (sub)gradient methods. The paper further proposes a heterogeneous push-sum based subgradient algorithm in which each agent can arbitrarily switch between subgradient-push and push-subgradient. The heterogeneous algorithm thus subsumes both subgradient-push and push-subgradient as special cases, and still converges to an optimal point at an optimal rate. The proposed tool can also be extended to analyze distributed weighted averaging.Comment: arXiv admin note: substantial text overlap with arXiv:2203.16623, arXiv:2303.1706

FROST -- Fast row-stochastic optimization with uncoordinated step-sizes

In this paper, we discuss distributed optimization over directed graphs, where doubly-stochastic weights cannot be constructed. Most of the existing algorithms overcome this issue by applying push-sum consensus, which utilizes column-stochastic weights. The formulation of column-stochastic weights requires each agent to know (at least) its out-degree, which may be impractical in e.g., broadcast-based communication protocols. In contrast, we describe FROST (Fast Row-stochastic-Optimization with uncoordinated STep-sizes), an optimization algorithm applicable to directed graphs that does not require the knowledge of out-degrees; the implementation of which is straightforward as each agent locally assigns weights to the incoming information and locally chooses a suitable step-size. We show that FROST converges linearly to the optimal solution for smooth and strongly-convex functions given that the largest step-size is positive and sufficiently small.Comment: Submitted for journal publication, currently under revie

Distributed Nonconvex Multiagent Optimization Over Time-Varying Networks

We study nonconvex distributed optimization in multiagent networks where the communications between nodes is modeled as a time-varying sequence of arbitrary digraphs. We introduce a novel broadcast-based distributed algorithmic framework for the (constrained) minimization of the sum of a smooth (possibly nonconvex and nonseparable) function, i.e., the agents' sum-utility, plus a convex (possibly nonsmooth and nonseparable) regularizer. The latter is usually employed to enforce some structure in the solution, typically sparsity. The proposed method hinges on Successive Convex Approximation (SCA) techniques coupled with i) a tracking mechanism instrumental to locally estimate the gradients of agents' cost functions; and ii) a novel broadcast protocol to disseminate information and distribute the computation among the agents. Asymptotic convergence to stationary solutions is established. A key feature of the proposed algorithm is that it neither requires the double-stochasticity of the consensus matrices (but only column stochasticity) nor the knowledge of the graph sequence to implement. To the best of our knowledge, the proposed framework is the first broadcast-based distributed algorithm for convex and nonconvex constrained optimization over arbitrary, time-varying digraphs. Numerical results show that our algorithm outperforms current schemes on both convex and nonconvex problems.Comment: Copyright 2001 SS&C. Published in the Proceedings of the 50th annual Asilomar conference on signals, systems, and computers, Nov. 6-9, 2016, CA, US