A General Framework of Exact Primal-Dual First-Order Algorithms for Distributed Optimization

Abstract

In this paper, we study the problem of minimizing a sum of convex objective functions, each of which is locally available to an agent in the network. Distributed optimization algorithms make it possible for the agents to cooperatively solve the problem through local computations and communications with neighbors. Lagrangian-based distributed optimization algorithms have received significant attention in recent years, due to their exact convergence property. However, many of these algorithms have slow convergence or are expensive to execute. In this paper, we develop a flexible framework of first-order primal-dual algorithms (FlexPD), which allows for multiple primal steps per iteration and can be customized for various applications with different computation and communication limitations. For strongly convex and Lipschitz gradient objective functions, we establish linear convergence of our proposed algorithms to the optimal solution. Simulation results confirm the superior performance of our algorithm compared to the existing methods.

Primal-dual methods for large-scale and distributed convex optimization and data analytics

“…Reference [38] considers several deterministic and randomized variants to solve (16) inexactly in an iterative fashion, including gradient-like and Jacobi-like primal updates. Reference [66] considers gradient-like primal updates and shows by simulation that performing a few inner iterations (2-4, more precisely) usually improves performance over the single inner iteration round-methods like EXTRA [34].…”

A stochastic averaging gradient algorithm with multi‐step communication for distributed optimization

“…However, the minimization of a local objective function at each iteration is required for distributed ADMM methods, which leads to computational complexity and might break the balance between computation costs and performance. This bottleneck is overcome by References 25,26 through the use of a multi‐step communication strategy that performs multiple gradient descent steps and a dual step per iteration. By adjusting the number of gradient descent steps during each iteration, the algorithms in References 25,26 achieve a balance for computation or communication costs and performance tradeoffs.…”

“…This bottleneck is overcome by References 25,26 through the use of a multi‐step communication strategy that performs multiple gradient descent steps and a dual step per iteration. By adjusting the number of gradient descent steps during each iteration, the algorithms in References 25,26 achieve a balance for computation or communication costs and performance tradeoffs. It is worth noting that the multi‐step communication strategy proposed in the literature 25,26 is different from multi‐step distributed online learning 27 .…”

“…By adjusting the number of gradient descent steps during each iteration, the algorithms in References 25,26 achieve a balance for computation or communication costs and performance tradeoffs. It is worth noting that the multi‐step communication strategy proposed in the literature 25,26 is different from multi‐step distributed online learning 27 . Multi‐step distributed online learning uses the past multi‐step performance metrics as input for predicting the future multi‐step performance metrics online or offline, while multi‐step communication uses the past one‐step information as input to update the variable for the current step.…”

A Unified Contraction Analysis of a Class of Distributed Algorithms for Composite Optimization

“…Our focus is on the design of distributed algorithms for Problem (P) that provably converge at a linear rate. When G = 0, several distributed schemes have been proposed in the literature enjoying such a property; examples include EXTRA [1], AugDGM [2], NEXT [3], SONATA [4], [5], DIGing [6], NIDS [7], Exact Diffusion [8], MSDA [9], and the distributed algorithms in [10], [11], and [12]. When G = 0 results are scarce; to our knowledge, the only two schemes available in the literature achieving linear rate for (P) are SONATA [5] and the distributed proximal gradient algorithm [13].…”