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Distributed Basis Pursuit

João F. C. Mota, João M. F. Xavier, Pedro M. Q. Aguiar, Markus Püschel

arXiv:1009.1128v3math.OCcs.ITeess.SY

TL;DR

The paper addresses how to solve Basis Pursuit when matrix data are distributed across network nodes and communication is costly. It proposes decentralized ADMM for row- or column-partitioned matrices, and simulations report lower communication requirements than competing methods, under connected static networks.

  • Problem

    The problem is to solve Basis Pursuit in a distributed network where no central processor exists and each node accesses only one matrix block, while communications are costly.

  • Method

    The paper proposes D-ADMM, a decentralized alternating direction method of multipliers for solving BP with either row or column partitions of A.

  • Results

    D-ADMM always required fewer communications than competing algorithms in the reported simulations; in type II experiments, it used on average 42% of D-Lasso's communications.

  • Takeaways & Limitations

    The algorithms apply to connected static networks ranging from sparse sensor networks to dense supercomputing platforms, with communication savings particularly relevant to energy-constrained settings.

  • Takeaways & Limitations

    The network topology is assumed to be connected and static, and execution-time comparisons are outside the paper's scope because wireless MAC protocols affect timing.

Abstract

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We propose a distributed algorithm for solving the optimization problem Basis Pursuit (BP). BP finds the least L1-norm solution of the underdetermined linear system Ax = b and is used, for example, in compressed sensing for reconstruction. Our algorithm solves BP on a distributed platform such as a sensor network, and is designed to minimize the communication between nodes. The algorithm only requires the network to be connected, has no notion of a central processing node, and no node has access to the entire matrix A at any time. We consider two scenarios in which either the columns or the rows of A are distributed among the compute nodes. Our algorithm, named D-ADMM, is a decentralized implementation of the alternating direction method of multipliers. We show through numerical simulation that our algorithm requires considerably less communications between the nodes than the state-of-the-art algorithms.

I. INTRODUCTION

The paper develops a distributed approach to Basis Pursuit for connected networks, where nodes retain only matrix blocks and communicate without a central processor. It targets row- and column-partitioned matrices and reduces communications relative to prior distributed methods.

  • Motivation: Basis Pursuit minimizes the ℓ1 norm subject to Ax = b, seeking a small-norm solution of an underdetermined linear system.It is a convex relaxation of sparsity-seeking ℓ0 minimization and supports compressed-sensing reconstruction.
  • Problem statement and contribution: The paper solves Basis Pursuit across P compute nodes without a central processor, with each node accessing only its local block Ap.The distributed setting is motivated by applications such as sensor networks, where communication is a major performance concern.
  • Problem statement and contribution: The matrix A is distributed either by rows or by columns, yielding two frameworks for solving BP over the network.In row partitioning, blocks contain rows; in column partitioning, blocks contain columns.
  • Problem statement and contribution: Simulations evaluate the distributed algorithm across applications and network settings, with the objective of reducing communications needed to reach a target accuracy.The paper contrasts this communication focus with prior approaches that converge slowly or require restrictive network structures.
  • Problem statement and contribution: The proposed algorithm uses ADMM in a decentralized setting and is designed to operate on any connected network topology.The network is assumed connected and static, while the approach avoids the central-node requirement of related ADMM work.

II. ROW PARTITION

The row-partition formulation gives each node local rows of A and measurements, then enforces agreement among local copies of x across network edges. D-ADMM updates color classes in parallel and converges for bipartite graphs, while the general-graph extension lacks a convergence guarantee.

  • Problem reformulation: Each node stores a block of rows Ap and the corresponding measurements bp, with the blocks collectively partitioning A and b.The local blocks satisfy m1 + ··· + mP = m, and Ap and bp are available only at node p.
  • Problem reformulation: Replacing x with local copies xp decouples the objective, while edge constraints xi = xj enforce global consistency on a connected network.The reformulation minimizes the sum of local L1 norms subject to local measurement constraints and agreement across every network edge.
  • D-ADMM: D-ADMM applies ADMM to the reformulated problem, allowing nodes in the same color class to solve their optimization problems in parallel.For bipartite graphs, the two color classes alternate; each node communicates its updated local variable to its neighbors at each iteration.
  • Convergence: For bipartite graphs, every sequence {x_p^(k)} produced by D-ADMM converges to a solution of BP, and every node then knows a BP solution.The convergence theorem applies under the stated assumptions, including a connected static network and the problem’s solvability conditions.
  • General graphs: For arbitrary connected graphs, the colored D-ADMM generalization uses C sequential color classes, but the paper provides only experimental evidence of convergence.One outer iteration takes C units when one local optimization in step 4 is counted as one unit.

III. COLUMN PARTITION

The column-partition algorithm adapts D-ADMM through a regularized problem and its dual, allowing each node to use only its local matrix block. For bipartite graphs, it converges to an optimal regularized solution and, for sufficiently small δ, to a BP solution.

  • Each node stores only its column block Ap, while b is known by all nodes in the connected network.
  • Regularizing BP: Regularized BP adds δ/2∥x∥2 to the ℓ1 objective, making the solution unique and selecting BP’s least-ℓ2-norm solution when δ is sufficiently small.
  • Regularizing BP: There exists a threshold ¯δ such that solving the regularized problem recovers a BP solution for every 0 < δ < ¯δ.
  • Dual problem: The column-partition method uses the dual because A⊤ is partitioned by rows, enabling reuse of the row-partition algorithm after minor modifications.
  • Adapting the algorithm: D-ADMM clones the dual variable across nodes and enforces agreement over network edges, with each node computing its local dual objective term from Ap.
  • Convergence and implementation: For bipartite graphs, Algorithm 3 converges to an optimal solution of the regularized problem and, with sufficiently small δ, to a BP solution.
  • Convergence and implementation: A very small δ can cause ill-conditioning in the local update, although δ = 10^-3 produced high-precision BP solutions without numerical problems in the simulations.

IV. OTHER ALGORITHMS

The paper compares D-ADMM with single- and double-looped distributed BP algorithms using communication steps as the primary performance measure. These alternatives differ in synchronization, graph-convergence guarantees, loop structure, and per-iteration computation.

  • Single-looped algorithms: D-ADMM is single-looped and transmits a vector of size n to neighboring nodes at each iteration.
  • Performance measure: Communication steps measure performance: single-looped methods use one step per iteration, while double-looped methods use one per inner iteration.
  • Single-looped algorithms: The subgradient method combines consensus and subgradient updates for convex problems with private node constraints.
  • Single-looped algorithms: D-Lasso uses ADMM after introducing edge variables, with closed-form updates for the auxiliary variables and local updates for node variables.
  • Comparison with D-Lasso: D-ADMM is proven to converge only for bipartite graphs, whereas D-Lasso is proven to converge for any connected graph.
  • Double-looped algorithms: Double-looped methods use an outer loop for the dual problem and an inner loop for its distributed optimization subproblem.
  • Double-looped algorithms: MM/NGS and MM/DQA use method-of-multipliers outer loops, while DN uses Nesterov’s method; their inner methods optimize node blocks iteratively.
  • Communication-time caveat: Although D-ADMM requires fewer communication steps, execution-time comparisons are outside the paper’s scope because step durations and wireless MAC protocols differ.

V. EXPERIMENTAL RESULTS

The experiments evaluate D-ADMM and competing distributed BP algorithms across multiple data scenarios, network models, parameter settings, and accuracy targets. D-ADMM generally requires fewer communication steps, although execution-time and parameter-tuning trade-offs remain relevant.

  • Experimental setup: Experiments cover five data scenarios and seven network configurations, using networks with 50 or 64 nodes.The networks are generated from several models, including Erdős-Rényi, Watts-Strogatz, Barabási-Albert, geometric, and lattice graphs.
  • Experimental setup: The study evaluates both fixed-parameter and parameter-search settings: type I reflects practical fixed tuning, whereas type II assesses algorithm capabilities more thoroughly.Type I uses fixed ρ values across scenarios and networks; type II tests several ρ values for each setting.
  • Type I experiments: In scenario 3, D-ADMM uses fewer communications than ρ-dependent algorithms, while Subgradient is better only for 1% accuracy on networks 1, 2, and 6.Subgradient reaches the maximum communication limit before achieving 10−3% accuracy, and the other algorithms show roughly similar relative behavior at both accuracies.
  • Scaling and execution time: Communication counts scale sublinearly with network size in the fitted models: C ≃11.7 · P^0.8 for D-ADMM and C ≃29.5 · P^0.77 for D-Lasso.The average number of colors was 4.6, implying that in a collision-free network D-ADMM would be 1.8 times slower than D-Lasso.
  • Type II experiments: D-ADMM uses an average of 42% of D-Lasso's communications in type II experiments, with a 10% standard deviation.The largest difference is 28% in scenario 2, network 4; the smallest is 72% in scenario 5, network 5.

VI. FINAL REMARKS AND CONCLUSIONS

The paper presents D-ADMM for solving BP with either row- or column-partitioned matrices over connected, static networks. Simulations find fewer communications than competing algorithms, especially for energy-constrained sensor-network settings.

  • Contribution: D-ADMM solves BP when matrix rows or columns are distributed across compute nodes.The two frameworks support row and column partitions of A.
  • Scope: The communication network need only be connected and static, allowing application to sparse sensor networks and dense supercomputing platforms.No central processing node is required by the distributed framework.
  • Conclusion: Across simulations with several data and network types, the proposed algorithms require fewer communications than competing algorithms.The paper identifies this reduction as particularly important in energy-constrained sensor networks.

APPENDIX A ALTERNATING DIRECTION METHOD OF MULTIPLIERS

ADMM solves a two-block convex optimization problem by alternating minimization steps followed by a dual-variable update. Under stated convexity, polyhedral-set, rank, and solvability assumptions, the iterates converge, while the generalized multi-block extension lacks an established corresponding theorem here.

  • ADMM formulation: The augmented Lagrangian combines f(x), g(y), the constraint term λ⊤(Ax + By), and a quadratic penalty controlled by ρ > 0.
  • ADMM formulation: ADMM alternates minimization over x and y, then updates the dual variable λ using a gradient-based step.The x minimization precedes the y minimization, so the two primal updates are not simultaneous.
  • Convergence: Under convexity, polyhedral-set, full-column-rank, and solvability assumptions, {(x(k), y(k))} converges to a solution and {λ(k)} converges to a dual solution.
  • Convergence: ADMM has a convergence rate O(1/k), including when its quadratic term is linearized.Linearization can simplify the associated optimization problem.
  • Generalized ADMM: The generalized ADMM extends the formulation to I > 2 convex blocks with polyhedral sets and full-column-rank matrices.The paper applies it despite the stated lack of knowledge about whether Theorem 4 extends to this case; its simulations never failed to converge.

APPENDIX B PROBLEM FOR EACH NODE: ROW PARTITION

Each node solves its local optimization problem through a differentiable dual formulation. Unique coordinate minimizers provide the primal solution after the dual optimum is found, while BB iterations and warm starts reduce local computational effort.

  • Local problem: Each node solves its local problem by formulating and solving its dual problem.The dual variable is λ ∈ R^m, with u(λ) = v − A⊤λ.
  • Dual solution: The dual objective is differentiable with gradient b − Ax(λ), because each coordinate minimizer x_i(λ) is unique.Strict convexity of each coordinate function follows from c > 0.
  • Recovering the primal solution: Once λ⋆ solves the dual problem, the primal solution is x(λ⋆).
  • Numerical solution: The Barzilai-Borwein algorithm uses O(n) flops per iteration plus gradient-computation costs and is known to converge R-superlinearly for generic unconstrained optimization.
  • Numerical solution: Warm starts reduce iterations by initializing the iteration k + 1 solve with the BB solution from iteration k.Consecutive local problems are expected to have nearby solutions when v and c change only slightly.
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