Source-linked AI summary
A linear algorithm for optimization over directed graphs with geometric convergence
Ran Xin, Usman A. Khan
TL;DR
The paper tackles distributed minimization of private convex objectives over directed graphs, where existing push-sum approaches add computation, communication, and nonlinearity. It proposes a linear algorithm based on inexact gradients and gradient estimation, and proves geometric convergence to the global optimum under strong convexity, Lipschitz-continuous gradients, and a sufficiently small step-size.
Problem
Distributed optimization over directed graphs is challenging because constructing doubly-stochastic weights may be impossible, while common push-sum methods require extra computation, communication, and nonlinearity.
Method
The paper proposes a linear distributed algorithm based on an inexact gradient method and a gradient estimation technique.
Results
Under strongly-convex objectives with Lipschitz-continuous gradients and a sufficiently small step-size, the proposed algorithm converges geometrically to the global optimum.
Takeaways & Limitations
The method provides geometric-convergence distributed optimization over directed graphs without relying on the push-sum strategy described in the paper.
Abstract
from arXiv · showhide
In this letter, we study distributed optimization, where a network of agents, abstracted as a directed graph, collaborates to minimize the average of locally-known convex functions. Most of the existing approaches over directed graphs are based on push-sum (type) techniques, which use an independent algorithm to asymptotically learn either the left or right eigenvector of the underlying weight matrices. This strategy causes additional computation, communication, and nonlinearity in the algorithm. In contrast, we propose a linear algorithm based on an inexact gradient method and a gradient estimation technique. Under the assumptions that each local function is strongly-convex with Lipschitz-continuous gradients, we show that the proposed algorithm geometrically converges to the global minimizer with a sufficiently small step-size. We present simulations to illustrate the theoretical findings.
I. INTRODUCTION
The paper addresses distributed optimization over arbitrary directed graphs, where agents minimize an average of private convex functions despite the difficulty of constructing doubly-stochastic weights. It proposes a linear method that geometrically converges under strong convexity, Lipschitz-continuous gradients, and a sufficiently small step-size.
- Problem: Each agent holds a private convex function, and the network jointly minimizes their average through information exchange.The optimization is distributed across a multi-agent network, with each agent accessing only its own function.
- Problem: Directed graphs may lack doubly-stochastic weight matrices because their adjacency matrices are generally only row-stochastic or column-stochastic.This makes methods designed for undirected or weight-balanced networks impractical for some one-way communication settings.
- Related work: Existing DGD-based approaches can converge slowly because diminishing step-sizes yield rates such as O(lnk) for general convex functions and O(lnk) for strongly-convex functions.Other methods achieve geometric convergence under stronger conditions, but some require symmetric weights or step-size intervals.
- Related work: Fast directed-graph methods commonly use push-sum techniques that learn a weight-matrix eigenvector through an independent algorithm, making the resulting method nonlinear.The strategy also causes additional computation and communication.
- Contribution: The paper provides a linear distributed optimization algorithm for directed graphs that geometrically converges to the global optimum with a sufficiently small step-size.The guarantee assumes strongly-convex objective functions with Lipschitz-continuous gradients.
II. ALGORITHM DEVELOPMENT
The paper formulates distributed convex optimization over strongly-connected directed graphs and proposes a linear algorithm using row- and column-stochastic weight matrices. It contrasts this approach with push-sum-based methods that require additional computation, communication, or nonlinearity.
- Problem formulation: The network is strongly connected, agents know their out-degrees, and each local objective is strongly convex with globally Lipschitz-continuous gradients.These assumptions ensure the global objective has a unique optimizer.
- Proposed algorithm: Each agent maintains primal and gradient-estimation variables, x_i(k) and y_i(k), and initializes y_i(0) with its local gradient.The primal initialization is arbitrary, while the gradient estimate is initialized as y_i(0) = ∇f_i(x_i(0)).
- Proposed algorithm: The proposed updates use a row-stochastic matrix A for x_i(k) mixing and a column-stochastic matrix B for transmitting gradient-related information.A is locally implementable through incoming-weight selection, while B requires agents to know their out-degrees.
- Proposed algorithm: The gradient-estimation update asymptotically tracks the aggregate local gradients, making the primal update approach centralized gradient descent.The descent direction y_i(k) becomes the gradient of the global objective over time.
- Relation with existing work: Unlike push-sum approaches, the proposed algorithm avoids an independent eigenvector-learning procedure and its associated nonlinearity, computation, and communication costs.The method is related to doubly-stochastic consensus-based updates but does not require doubly-stochastic weights, which are generally unavailable on directed graphs.
- Relation with existing work: Extending doubly-stochastic consensus linearly to directed graphs is non-trivial because simultaneous contractions for row- and column-stochastic matrices require arbitrary norms and norm-equivalence constants.The proposed analysis addresses the loss of the convenient eigenstructure associated with doubly-stochastic weights.
III. CONVERGENCE ANALYSIS
The convergence analysis rewrites the agent-level updates in compact matrix form using stacked local variables and Kronecker-extended mixing matrices. The initialization sets the stacked gradient variable to the aggregate local-gradient vector and leaves the primal state arbitrary.
- Matrix formulation: The stacked vectors x(k) and y(k) collect all agents’ local primal and gradient variables for matrix-form analysis.The matrices are extended as A = A ⊗ I_p and B = B ⊗ I_p.
- Matrix formulation: The compact matrix formulation represents the proposed updates using the Kronecker-extended matrices A and B and the optimizer x∗.The initial condition is y(0) = ∇f(0), while x(0) is arbitrary.
A. Auxiliary relations
The auxiliary analysis establishes contraction and gradient-preservation relations for the row- and column-stochastic processes, then combines them with strongly convex gradient-descent bounds. These relations support a linear contraction argument under a sufficiently small step-size.
- Auxiliary relations: The row- and column-stochastic processes admit contraction bounds in compatible vector norms, with constants σ_A and σ_B satisfying 0 < σ_A < 1 and 0 < σ_B < 1.Norm equivalence is used because the directed-graph analysis does not rely solely on the Euclidean norm.
- Auxiliary relations: The average of the agents’ y_i(k) variables preserves the average of local gradients.This invariant follows from the column-stochasticity of B and the initialization y(0) = ∇f(0).
- Auxiliary relations: For a strongly convex function with Lipschitz-continuous gradient, a gradient step contracts distance to the minimizer by τ = max(|1 − αθ|, |1 − βθ|).Here α and β are the strong-convexity and Lipschitz-continuity constants, respectively.
- Auxiliary relations: The analysis tracks consensus error, optimality error, and gradient-estimation error as separate quantities bounded across iterations.These are respectively measured relative to A∞x(k), 1_n ⊗ x∗, and B∞y(k).
- Auxiliary relations: The convergence theorem is derived under the step-size condition 0 < η < 2/(nβπ⊤_r π_c).The theorem states a linear matrix inequality entry-wise under this condition.
- Auxiliary relations: The proof bounds the gradient-estimation magnitude using consensus, optimality, and gradient-estimation errors, completing the contraction relationship.The resulting inequalities connect the bounds from the preceding auxiliary lemmas.
B. Main results
The main result establishes geometric convergence of the proposed algorithm under a sufficiently small step-size, using a matrix inequality whose associated spectral radius is below one.
- Theorem 1 gives a sufficient step-size condition, 0 < η < 2 nβπ⊤ r πc, under which a linear matrix inequality holds entry-wise.
- The convergence proof constructs t(k) ∈ R3 and J(η) ∈ R3×3 to express the relevant error relations.
- ρ(J(η)) is strictly less than 1, so ||x(k) − 1_n ⊗ x∗||2 converges to zero geometrically at rate O(ρ(J(η))^k).
- Solving the resulting inequalities identifies conditions on the step-size η and a positive vector ε that ensure the required matrix inequality.
IV. NUMERICAL EXPERIMENTS
The simulations study distributed logistic regression on directed graphs, comparing the proposed method with other algorithms and across graphs with different edge counts. They plot average agent residuals and report geometric convergence comparable to fast directed-graph methods.
- The experiment uses distributed logistic regression, with each agent retaining private binary-classification training data.
- The setting uses n = 8 agents and p = 5 features, with Gaussian feature vectors and randomly generated Bernoulli labels.
- The proposed algorithm is compared with ADD-OPT/Push-DIGing, FROST, and subgradient-push over the directed graph G1.
- The proposed algorithm is also evaluated on directed graphs G1, G2, and G3, where graphs to the right have a few more edges.
- The plots show average residuals at each agent, and the proposed method has geometric convergence comparable to other fast directed-graph algorithms.
V. CONCLUSIONS
The paper presents a linear distributed optimization algorithm for directed graphs and establishes geometric convergence under strong convexity and Lipschitz-continuous gradients. Its analysis uses simultaneous contractions and nonnegative matrix theory.
- The paper describes a linear distributed algorithm for optimization over directed graphs, generalizing earlier work over undirected graphs.
- Under strongly-convex objective functions with Lipschitz-continuous gradients, the proposed algorithm achieves geometric convergence to the global optimum.
- The analysis establishes simultaneous contractions of row- and column-stochastic matrices under some arbitrary norms.
- A result from nonnegative matrix theory is used to develop the convergence conditions.