Source-linked AI summary
Linear convergence in optimization over directed graphs with row-stochastic matrices
Chenguang Xi, Van Sy Mai, Ran Xin, Eyad H. Abed, Usman A. Khan
TL;DR
Directed-graph distributed optimization needs fast convergence without requiring agents to know neighbors’ out-degrees. The paper introduces a row-stochastic algorithm with an auxiliary eigenvector-tracking variable, achieving O(μ^k) convergence under strongly-convex objectives with Lipschitz-continuous gradients.
Problem
Existing directed-graph algorithms require neighbors’ out-degree knowledge because they use column-stochastic matrices, while the paper seeks distributed optimization over directed networks.
Method
The algorithm uses row-stochastic weights and an additional variable whose asymptotic left-eigenvector estimate scales gradients to cancel imbalance.
Results
O(μ^k) convergence is achieved for 0 < μ < 1 when local objectives are strongly-convex with Lipschitz-continuous gradients.
Takeaways & Limitations
The proposed method provides a fully distributed directed-graph optimization algorithm without requiring agents’ out-degree information.
Abstract
from arXiv · showhide
This paper considers a distributed optimization problem over a multi-agent network, in which the objective function is a sum of individual cost functions at the agents. We focus on the case when communication between the agents is described by a \emph{directed} graph. Existing distributed optimization algorithms for directed graphs require at least the knowledge of the neighbors' out-degree at each agent (due to the requirement of column-stochastic matrices). In contrast, our algorithm requires no such knowledge. Moreover, the proposed algorithm achieves the best known rate of convergence for this class of problems, $O(μ^k)$ for $0<μ<1$, where $k$ is the number of iterations, given that the objective functions are strongly-convex and have Lipschitz-continuous gradients. Numerical experiments are also provided to illustrate the theoretical findings.
I. INTRODUCTION
Distributed optimization minimizes a sum of local objectives over agent networks, but directed communication creates challenges for fast, fully distributed methods. The paper addresses these challenges with a row-stochastic algorithm that avoids out-degree knowledge and achieves linear convergence under strong convexity.
- Motivation: Distributed optimization asks agents to cooperatively minimize the sum of local objective functions using local information exchange.The framework includes strongly-connected networks in which each agent knows only its own objective function.
- Motivation: Directed communication is practically relevant because agents may communicate asymmetrically or lose communication links.Such networks do not generally satisfy the undirected or balanced assumptions used by earlier methods.
- Prior work: Earlier directed-graph methods commonly require column-stochastic weights, which in turn require agents to know neighbors’ out-degrees.This requirement is identified as impractical in many situations.
- Prior work: Row-stochastic matrices are easier to implement because each agent can locally choose its weights.Prior row-stochastic work cited here achieved O(√(ln k/k)) for arbitrary convex functions.
- Contribution: The proposed algorithm uses only row-stochastic matrices, requires agent identifiers but no out-degree information, and achieves O(μ^k) convergence for strongly-convex functions.Here 0 < μ < 1 and k denotes the iteration count.
II. PROBLEM, ASSUMPTIONS, AND ALGORITHM
The paper formulates distributed minimization over strongly-connected directed networks and develops an algorithm using row-stochastic weights. An auxiliary variable estimates the row-stochastic matrix’s left eigenvector so gradient scaling cancels the resulting imbalance.
- Problem formulation: Agents cooperatively minimize the sum of differentiable local objectives over a strongly-connected directed graph.Each local objective is known only to its corresponding agent.
- Assumptions: The analysis assumes strongly-connected communication, known unique agent identifiers, and local objectives that are differentiable, strongly-convex, and have Lipschitz-continuous gradients.These assumptions support the stated optimization and convergence results.
- Algorithm: Each agent maintains xk,i, zk,i ∈ R^p and yk,i ∈ R^n, updating three vector states at every iteration.The initialization uses arbitrary x0,i, z0,i = ∇fi(x0,i), and y0,i = ei.
- Algorithm: The weight matrix is row-stochastic, and the implementation does not require any agent’s out-degree.Agents use locally assigned weights rather than column-stochastic weights requiring out-degree information.
- Algorithm: The yk,i updates converge to the left eigenvector of the row-stochastic matrix, and gradient scaling by these iterates cancels communication imbalance.This modification enables convergence to the optimal solution using only row-stochastic weights.
III. MAIN RESULTS
The paper establishes linear convergence by analyzing an iterative error relation whose dynamics are governed by matrices G and H_k. For sufficiently small step-sizes, the spectral radius of G is below one, H_k decays linearly, and the algorithm converges linearly to the optimum.
- Error analysis: The analysis reduces linear convergence to showing that the error sequence t_k converges linearly to zero.Because convergence of ||t_k||_2 implies convergence of x_k to x*, this sequence is the central object of the proof.
- Error analysis: Theorem 1 expresses the error recursion through matrices G and H_k, so convergence follows from ρ(G) < 1 and linear decay of H_k.The decay of H_k follows from 0 < γ_1 < 1, while the spectral-radius condition is established separately.
- Spectral-radius condition: For step-size α ∈ (0, ᾱ), Lemma 4 proves ρ(G_α) < 1 by analyzing how the eigenvalues change from α = 0.The eigenvalue equal to 1 decreases as α increases slightly, while the other two eigenvalues remain below 1; continuity then yields the result over the admissible interval.
- Geometric decay: Lemmas 4 and 5 show that G^k and H_k decay linearly for sufficiently small step-size α.The proof combines the spectral-radius bound for G with geometric bounds characterized by constants γ_1, γ_2, and γ.
- Main convergence result: Theorem 2 establishes that the iterates x_k converge linearly to x* when α ∈ (0, ᾱ).The proof obtains ||t_k||_2 = O(μ^k) and then bounds ||x_k − x*||_2 by a constant multiple of (γ + ξ)^k.
- Step-size requirement: The admissible upper bound ᾱ depends on network parameters and cannot be computed locally, so agents must use a common sufficiently small step-size.The common value may need to be pre-programmed to avoid an additional agreement protocol.
IV. PROOF OF THEOREM 1
The section introduces auxiliary relations used in the proof.
- The proof begins by establishing relevant auxiliary relations.
- These relations support the subsequent convergence analysis.
- The section prepares the technical steps needed for the theorem proof.
A. Auxiliary Relations
The auxiliary results characterize the key recursions and contraction behavior used in the convergence proof. They include iterative relations for transformed sequences and a fixed-step gradient-descent contraction result.
- Lemma 7 derives iterative equations governing the sequences bzk and bxk.
- The updates include a transformed gradient sequence and an optimization iterate with step-size α.
- The proof uses Y∞A = Y∞ to establish one of the sequence relations.
- The analysis requires showing that the series P∞ k=0 skγk is finite.
- For a strongly-convex function with Lipschitz-continuous gradient, fixed-step gradient descent contracts distance to the optimizer when 0 < α < 2/nl.
B. Proof of Theorem 1
The proof bounds consensus, optimization, and gradient-tracking errors through recursive inequalities. It then combines these bounds into a linear convergence argument.
- The proof bounds three errors: ∥xk+1−bxk+1∥, ∥bxk+1−x∗∥2, and ∥zk+1−bzk+1∥.
- Step 1 bounds the disagreement ∥xk+1−bxk+1∥ using the algorithm update and Lemma 7(b).
- Step 2 bounds ∥bxk+1−x∗∥2 by relating the transformed update to centralized gradient descent.
- The proof bounds additional terms by splitting the right-hand side and applying the relations from Lemma 7 and Lemma 3.
- The final step combines the bounds from the preceding steps to complete the proof.
V. NUMERICAL EXPERIMENTS
The experiments compare the proposed method with existing distributed optimization algorithms on logistic regression over a directed network. The comparison emphasizes convergence behavior, step-size requirements, and information requirements.
- The experiments compare the proposed algorithm with DEXTRA, ADD-OPT, Push-DIGing, Subgradient-Push, Directed-Distributed Subgradient Descent, and Weight-Balancing Distributed Subgradient Descent.
- Fig. 1 depicts the directed graph used in the logistic-regression experiment.
- The proposed algorithm, DEXTRA, and ADD-OPT use constant step-sizes α = 0.008, α = 0.1, and α = 0.03, respectively.
- The proposed algorithm has linear convergence under strong convexity, unlike the sublinear convergence of the compared methods that support general convex functions.
- Compared with ADD-OPT, Push-DIGing, and DEXTRA, the proposed algorithm requires no knowledge of agents’ out-degree but requires unique identifiers.
VI. CONCLUSIONS
The paper addresses distributed optimization over directed graphs without requiring agents to know neighbors’ out-degrees. Its algorithm achieves O(µk), 0 < µ < 1, for the stated strongly-convex objective setting.
- O(µk), 0 < µ < 1, is achieved as the algorithm’s convergence rate for the stated objective-function conditions.The passage identifies this as the best rate for this class of problems.
- The algorithm requires no knowledge of neighbors’ out-degrees.Existing directed-graph algorithms require such knowledge because they use column-stochastic matrices.
- The setting is distributed minimization of summed local objectives over a multi-agent network with directed communication.