Linear Convergence Rate of a Class of Distributed Augmented Lagrangian Algorithms

A. Motivation

• Distributed augmented-Lagrangian methods lacked convergence-rate analysis despite their use in signal-processing and learning applications.Nodes communicate with neighbors while minimizing the sum of their locally known costs.Source: A. Motivation · p. 1

B. Contributions

• The framework gives globally linear rates for inexact methods under smoothness, bounded-Hessian, and network-dependent conditions.The stated conditions relate the inexactness level, Hessian bounds, and network spectral gap.Source: B. Contributions · p. 2
• Four analyzed variants are deterministic Jacobi, deterministic gradient, randomized Jacobi, and randomized gradient AL methods.Their rates are characterized in per-node communications, and the gradient variants also in per-node gradient evaluations.Source: B. Contributions · p. 2
• The framework establishes linear rates for a wide class of AL methods, including randomized primal updates, while specialized ADMM analyses can give tighter bounds.The paper distinguishes its general analysis from existing work focused on specific methods.Source: B. Contributions · p. 2
• With additional knowledge of system parameters, gradient AL variants achieve faster rates than standard distributed gradient methods at the same communication cost.The comparison also notes similar per-iteration computational cost and additional system-parameter knowledge.Source: B. Contributions · p. 2

Distributed multi-agent optimization

• Prior distributed ADMM analyses give tighter bounds for the specific methods they study, while several similar distributed primal-dual works do not analyze convergence rates.The paper also notes tighter bounds for some related methods and specialized problems.Source: C. Related work · pp. 2–3

Centralized optimization

• Unlike much centralized inexact-AL work, this analysis initializes inner primal algorithms from the previous primal variable and measures primal suboptimality through inner iterations.The cited centralized studies often use arbitrary initialization and establish rates at the outer-iteration level.Source: C. Related work · p. 3

Saddle point algorithms

• The distributed AL method is an inexact Arrow-Hurwitz-Uzawa method on a specific saddle-point system, with an inexactness measure and inner-iteration accounting that differ from cited analyses.The paper says it could not find equivalent results in the discussed literature.Source: C. Related work · p. 3

Jacobi/Gauss-Seidel algorithms

• The analysis treats Jacobi and Gauss-Seidel as inner solvers for the AL problem, deriving factors that determine the inner-iteration count needed for overall linear convergence.The paper distinguishes this focus from convergence of those algorithms in general settings.Source: C. Related work · p. 4

• The section is organized around the network and optimization models, deterministic AL methods, and randomized methods.These topics appear in Subsections II-A, II-B, and II-C, respectively.Source: §II · p. 4

• Each node knows its own convex cost, which is twice continuously differentiable with Hessian bounded between hmin and hmax.Under these assumptions, the optimization problem has a unique solution.Source: A. Optimization and network models · p. 4
• The communication graph is connected and undirected, with a symmetric stochastic weight matrix and spectral gap λ2(L) measuring connectivity.The stated network assumptions include positive definiteness of W and λN−1(W) < 1.Source: A. Optimization and network models · p. 4
• Dual variables start at zero, primal variables start equal across nodes, and convergence requires appropriately chosen shared algorithm parameters.Parameter setting requires bounds on hmin, hmax, and λ2(L); randomized methods also require knowledge of N.Source: A. Optimization and network models · p. 5

B. Deterministic Methods

• Both deterministic variants perform dual gradient-ascent updates at outer iterations while the primal variables are fixed.The Jacobi variant initializes inner primal variables from the previous outer iterate and sets the outer iterate to the result after τ inner steps.Source: B. Deterministic Methods · p. 5
• Each Jacobi inner iteration updates local primal variables using neighbor information and broadcasts the updated variables; each outer iteration updates the dual variables.The method uses τ inner iterations per outer iteration, each requiring one broadcast transmission per node; outer dual updates require no communication.Source: B. Deterministic Methods · p. 5
• The gradient variant replaces the Jacobi primal update with gradient descent on the augmented-Lagrangian function and adds step-size parameter β.The described algorithm also uses the weight matrix W, τ, penalty parameter ρ, and dual step-size α.Source: B. Deterministic Methods · p. 5

C. Randomized Methods

• Independent rate-1 Poisson clocks trigger primal updates; at each tick, one uniformly selected node updates its variable and broadcasts to neighbors.Across all nodes, the tick process has rate N, yielding an average of τN primal updates during an outer interval of length τ.Source: C. Randomized Methods · pp. 5–6
• During each outer interval, the number and sequence of inner updates are random, determined by the clock ticks and the node whose clock ticks.The variables after these inner updates become the primal variables for the next outer iteration; the randomized gradient update uses step-size β.Source: C. Randomized Methods · p. 6
• The two randomized variants differ in their primal updates: one uses Gauss-Seidel updates, while the other substitutes gradient updates.The dual variables are updated at the synchronized outer-iteration times.Source: C. Randomized Methods · p. 6

III. ANALYSIS OF INEXACT AUGMENTED LAGRANGIAN

• In the randomized Gauss-Seidel procedure, nodes perform a random number of inner updates during each outer iteration, updating the selected node and relevant neighbor variables.The selected node is determined by the tick sequence; the inner variables are carried forward as the next outer iterate.Source: §III · p. 6
• After the inner updates, all nodes update their dual variables using the primal variables at the end of the outer iteration.The section introduces an analysis framework, states its result, and applies it to four distributed algorithms.Source: §III · p. 6

A. Inexact AL algorithm: Convergence rate

• Each inexact primal update must reduce its distance to the exact augmented-Lagrangian minimizer by a factor ξ relative to the previous iterate.The iterative subproblem solver starts from the preceding primal variable and stops once this accuracy condition is met.Source: A. Inexact AL algorithm: Convergence… · pp. 6–7
• Under Assumptions 1 and 2 and suitable algorithm and network parameters, the inexact method converges linearly at every node to the solution of the global problem.The theorem quantifies the convergence factor and how small ξ must be, though the supplied passages do not give the expressions.Source: A. Inexact AL algorithm: Convergence… · p. 7
• Dµ reflects problem difficulty for distributed solution: it is zero when local costs share a minimizer, while differing local minimizers can require inter-node communication.The passage relates Dµ to the dual optimum but does not provide that relationship’s expression.Source: A. Inexact AL algorithm: Convergence… · p. 7

B. Auxiliary results and proof of Theorem 1

• The nonlinear saddle-point system has a unique solution, with primal variables equal to the consensus optimizer and dual variables equal to the negative cost gradient there.This identifies the limit point to which the convergence proof applies.Source: B. Auxiliary results and proof of… · p. 7
• The primal-error lemma decomposes the error into inner-solve residual and distance from the AL minimizer to the solution, using the inner-solve accuracy condition.The proof bounds the residual using the contraction factor ξ for the inner iterations.Source: B. Auxiliary results and proof of… · p. 8
• Rather than bound raw dual error, the proof uses a Laplacian-eigenbasis transformation; spectral bounds then control its contraction through λ2 and the Hessian and penalty parameters.The transformation projects onto the nonzero Laplacian eigenspace and scales by inverse square roots of its eigenvalues.Source: B. Auxiliary results and proof of… · pp. 8–9
• The maximum of primal error and scaled dual error contracts by factor r, and the proof establishes r < 1 under Theorem 1’s conditions.Unwinding this recursion yields the stated linear convergence bound.Source: B. Auxiliary results and proof of… · p. 9

IV. ANALYSIS OF DISTRIBUTED AUGMENTED LAGRANGIAN METHODS

• Jacobi updates converge linearly in outer and inner iterations, hence communications; choosing ρ=hmax and α=hmin+ρ gives outer factor r=1−Ω(λ2), shifting cost to inner iterations.The paper notes this choice eliminates the condition-number effect at the outer-iteration level, at a cost in the inner-iteration factor.Source: §IV · p. 9
• For reasonable α, β, and ρ (e.g., α=ρ=hmin, β=1/(ρ+hmax)), the results guarantee neither convergence nor linear rates at τ=1; whether some parameters ensure globally linear rates remains open.Prior work establishes global linear convergence for the Jacobi variant at τ=1 for any α=ρ>0.Source: §IV · p. 10
• For deterministic gradient updates, ξ=(1−β hmin)^τ, yielding linear convergence in inner iterations and per-node communications under Theorem 1’s conditions.The inner-solve error bound assumes β≤1/(hmax+ρ).Source: §IV · p. 10
• Randomized Gauss–Seidel and randomized gradient updates yield expected inner-solve contractions e^−ητ and e^−η′τ, respectively, supporting global linear convergence under the adapted framework.The expected number of inner iterations is τ for both variants.Source: §IV · pp. 10, 12

V. SIMULATION EXAMPLE

• The example uses l2-regularized logistic losses on a geometric network of 10 nodes with 28 links.Source: §V · p. 12
• The chosen algorithm parameters satisfy the stated conditions guaranteeing linear convergence, but the simulations also test τ = 1, for which the theory gives no such guarantee.Theoretical inner-iteration counts are large for the randomized gradient method.Source: §V · p. 12
• The methods show linear convergence; Jacobi has lower communication cost but higher computational cost, and randomized methods behave similarly to deterministic variants.Gradient variants with the theoretical inner-iteration count converge slowly, particularly the randomized gradient method.Source: §V · pp. 12–13, Figure 1 · p. 13

VI. CONCLUSION

• Four deterministic and randomized AL methods—Jacobi, gradient descent, nonlinear Gauss-Seidel, and gradient descent updates—have globally linear convergence rates under twice continuously differentiable costs with bounded Hessians.Source: §VI · p. 13
• The convergence rates are explicitly characterized in terms of underlying network parameters.Source: §VI · p. 13
• Simulation examples demonstrate linear convergence of the proposed methods.Source: §VI · p. 13