[Paper Review] AB/Push-Pull Method for Distributed Optimization in Time-Varying Directed Networks

Angelia Nedich, Duong Thuy Anh Nguyen+1 more|arXiv (Cornell University)|Sep 14, 2022

Distributed Control Multi-Agent Systems4 citations

TL;DR

This paper proposes the AB/Push-Pull method for distributed optimization in time-varying directed networks, using both row- and column-stochastic mixing matrices to track the optimal decision and average gradient. It establishes linear convergence with explicit stepsize bounds derived from graph diameter, cost function properties, and matrix connectivity, improving upon prior work by linking contraction parameters directly to network structure.

ABSTRACT

In this paper, we study the distributed optimization problem for a system of agents embedded in time-varying directed communication networks. Each agent has its own cost function and agents cooperate to determine the global decision that minimizes the summation of all individual cost functions. We consider the so-called push-pull gradient-based algorithm (termed as AB/Push-Pull) which employs both row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost while ensuring consensus and optimality. We show that the algorithm converges linearly to the optimal solution over a time-varying directed network for a constant stepsize when the agent's cost function is smooth and strongly convex. The linear convergence of the method has been shown in Saadatniaki et al. (2020), where the multi-step consensus contraction parameters for row- and column-stochastic mixing matrices are not directly related to the underlying graph structure, and the explicit range for the stepsize value is not provided. With respect to Saadatniaki et al. (2020), the novelty of this work is twofold: (1) we establish the one-step consensus contraction for both row- and column-stochastic mixing matrices with the contraction parameters given explicitly in terms of the graph diameter and other graph properties; and (2) we provide explicit upper bounds for the stepsize value in terms of the properties of the cost functions, the mixing matrices, and the graph connectivity structure.

Motivation & Objective

• To address distributed optimization in time-varying directed networks where agents have local cost functions and must collaboratively minimize the sum.
• To develop a gradient-based method that ensures linear convergence under smooth and strongly convex cost functions.
• To provide explicit, graph-structure-dependent upper bounds for the stepsize in the AB/Push-Pull algorithm.
• To establish one-step consensus contraction for both row- and column-stochastic matrices using network diameter and connectivity metrics.
• To extend prior AB/Push-Pull convergence results from static to time-varying directed graphs with stronger analytical guarantees.

Proposed method

• The method uses two separate mixing matrices: a row-stochastic matrix for pushing decision estimates and a column-stochastic matrix for pulling gradient information.
• Each agent maintains two estimates: one for the optimal decision variable and one for the average gradient of local cost functions.
• The algorithm operates in a push-pull fashion: decisions are pushed via the row-stochastic matrix, gradients are pulled via the column-stochastic matrix.
• Contraction properties are derived using the graph diameter and connectivity structure, ensuring one-step consensus for both estimates.
• The convergence analysis relies on bounding the spectral radius of a system matrix derived from the algorithm dynamics.
• Explicit stepsize upper bounds are derived by solving a system of inequalities involving Lipschitz constants, strong convexity parameters, and mixing matrix properties.

Experimental results

Research questions

• RQ1 Can the AB/Push-Pull method achieve linear convergence in time-varying directed networks with smooth and strongly convex cost functions?
• RQ2 How can the stepsize be bounded explicitly in terms of network structure and function properties to ensure convergence?
• RQ3 What is the relationship between the contraction parameters of the mixing matrices and the underlying graph’s diameter and connectivity?
• RQ4 How does the method’s performance compare to Push-DIGing in ill-conditioned and unbalanced networks?
• RQ5 Can the method be extended to networks with periodic strong connectivity instead of uniform strong connectivity?

Key findings

• The AB/Push-Pull method achieves linear convergence to the global minimizer for smooth and strongly convex cost functions in time-varying directed networks.
• The contraction parameters for both row- and column-stochastic matrices are explicitly linked to the graph diameter and other structural properties.
• An explicit upper bound for the stepsize is derived, depending on the Lipschitz constant, strong convexity parameter, and mixing matrix properties.
• The method outperforms Push-DIGing in convergence speed, especially in ill-conditioned problems and unbalanced networks, due to a larger allowable stepsize.
• The convergence analysis holds under a C-strongly-connected graph sequence, extending applicability to periodic network connectivity.
• The spectral radius of the system matrix is bounded below 1 when the stepsize is chosen within the derived explicit range, ensuring linear convergence.