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Cooperative Convex Optimization in Networked Systems: Augmented Lagrangian Algorithms with Directed Gossip Communication

Dusan Jakovetic, Joao Xavier, Jose M. F. Moura

arXiv:1007.3706v2cs.ITmath.IT

TL;DR

Distributed networked systems need to optimize a common quantity from private costs and constraints despite communication failures. This paper proposes augmented-Lagrangian gossip algorithms for that setting, proves convergence, and demonstrates them in two application simulations.

  • Problem

    The paper addresses distributed optimization of a common quantity from private convex costs and constraints when networked systems may lack a central unit and communication may fail.

    Source: §I · p. 2

  • Method

    AL–G uses augmented-Lagrangian dual updates and randomized Gauss-Seidel-type primal updates, with AL–MG and AL–BG as variants.

    Source: §I · pp. 2–3

  • Results

    The paper proves convergence of AL–G and its variants and demonstrates their applicability in simulations of logistic regression and cooperative spectrum sensing.

    Source: §VII · pp. 25–26

  • Takeaways & Limitations

    The proposed algorithms are presented as generic tools for networked optimization with private constraints, random networks, and asymmetric link failures.

    Source: §VII · p. 25

Abstract

from arXiv · show

We study distributed optimization in networked systems, where nodes cooperate to find the optimal quantity of common interest, x=x^\star. The objective function of the corresponding optimization problem is the sum of private (known only by a node,) convex, nodes' objectives and each node imposes a private convex constraint on the allowed values of x. We solve this problem for generic connected network topologies with asymmetric random link failures with a novel distributed, decentralized algorithm. We refer to this algorithm as AL-G (augmented Lagrangian gossiping,) and to its variants as AL-MG (augmented Lagrangian multi neighbor gossiping) and AL-BG (augmented Lagrangian broadcast gossiping.) The AL-G algorithm is based on the augmented Lagrangian dual function. Dual variables are updated by the standard method of multipliers, at a slow time scale. To update the primal variables, we propose a novel, Gauss-Seidel type, randomized algorithm, at a fast time scale. AL-G uses unidirectional gossip communication, only between immediate neighbors in the network and is resilient to random link failures. For networks with reliable communication (i.e., no failures,) the simplified, AL-BG (augmented Lagrangian broadcast gossiping) algorithm reduces communication, computation and data storage cost. We prove convergence for all proposed algorithms and demonstrate by simulations the effectiveness on two applications: l_1-regularized logistic regression for classification and cooperative spectrum sensing for cognitive radio networks.

I. INTRODUCTION

The paper addresses distributed optimization with private convex costs and constraints in networked systems where communication may fail. It develops augmented-Lagrangian gossip algorithms for this setting and proves convergence of their primal updates.

I. INTRODUCTION

  • Nodes seek a common scalar or vector minimizing summed private costs subject to private constraints, in systems that may lack a central unit and experience communication failures.The model is motivated by networked systems including wireless ad-hoc and sensor networks.Source: §I · p. 2
  • The dual update uses the standard method of multipliers, while a randomized Gauss-Seidel-type algorithm updates primal variables asynchronously through gossip.The dual and primal updates run at slow and fast time scales, respectively.Source: §I · p. 2
  • The paper proves convergence of the primal update under private constraints, generic network topologies, random link failures, and gossip communication.The proof is adapted to the AL–MG and AL–BG variants.Source: §I · p. 3
  • Simulations cover l1-regularized logistic regression and cooperative spectrum sensing, and compare communication and computation costs with existing algorithms.The examples are presented as corroborating convergence and comparing performance.Source: §I · p. 3
  • AL–G addresses private costs and constraints, random networks, asymmetric link failures, and asynchronous gossip communication.The paper contrasts this scope with prior primal-dual methods and develops AL–MG and AL–BG variants.Source: §I · pp. 2, 4

II. PROBLEM MODEL

The model represents a connected network through local and edge variables, with asynchronous gossip transmissions subject to random link failures. The algorithm uses separate time scales for primal and dual updates.

II. PROBLEM MODEL

  • The supergraph is connected and bidirectional, while actual network realizations are directed subgraphs that may include unreliable communication links.Each node knows in advance which nodes it can communicate with.Source: §II · pp. 4–5
  • The optimization model assumes convex, coercive local costs and closed convex private constraint sets with a nonempty intersection.A relative-interior regularity condition is also stated to support strong duality.Source: §II · p. 5
  • The reformulation gives each node a local copy xi and each directed edge variables yij, coupled to enforce agreement across copies.The reformulated and original problems are equivalent because the supergraph is connected.Source: §II · pp. 5–6
  • Asynchronous communication is modeled with independent Poisson clocks, and transmissions across physical links may fail while local virtual links remain reliable.Link availability is temporally independent, with positive probability for each directed edge.Source: §II · pp. 6–7

III. AL–G ALGORITHM (AUGMENTED LAGRANGIAN GOSSIPING)

  • The communication model assumes a static supergraph and does not cover scenarios with moving nodes that make the supergraph time-varying.At each slow time-scale slot, dual variables update without communication; primal updates use fast asynchronous slots.Source: §II · pp. 4, 6, 7

A. Dualization

The dualization assigns multipliers to the reformulation’s agreement constraints and forms an augmented-Lagrangian dual. Under the stated assumptions, strong duality holds.

A. Dualization

  • The reformulation’s local-copy and edge-agreement constraints receive multipliers μ(i,j), μ(j,i), and λ{i,j}, with each endpoint maintaining a copy of λ.The endpoint copies are constrained to agree.Source: A. Dualization · p. 7
  • The augmented-Lagrangian dual function and optimization problem are defined by the paper’s equations (3)–(5), with positive parameter ρ.The stated strong-duality argument uses a Slater point satisfying the reformulated equality constraints.Source: A. Dualization · p. 8
  • Under Assumption 1, strong duality holds between primal problem (2) and dual problem (5), so d⋆ = f⋆; the optimal dual-solution set D⋆ is independent of ρ for ρ ≥ 0.Source: A. Dualization · p. 8

B. Solving the dual: D–AL–G (dual augmented Lagrangian gossiping) algorithm

D–AL–G updates dual variables through the method of multipliers, while P–AL–G solves the fixed-dual primal problem through randomized block-coordinate updates. Its updates use neighborhood information and are implemented with asynchronous gossip communication.

B. Solving the dual: D–AL–G (dual augmented Lagrangian gossiping) algorithm

  • D–AL–G applies the method of multipliers to the unconstrained maximization of L′, with nodes maintaining separate copies of shared dual variables.Finite-time termination of P–AL–G means the update does not guarantee equality of the neighboring copies.Source: B. Solving the dual: D–AL–G (dual… · pp. 9–10
  • Under Assumption 1, limit points of the dual-variable sequence solve (2), and the corresponding limit points of x_i solve (1).The implementable dual update uses inexact primal solutions obtained when P–AL–G terminates.Source: B. Solving the dual: D–AL–G (dual… · pp. 9–10

C. Solving for (9): P–AL–G algorithm

1) Simplified notation and an abstract model of the P–AL–G:

The abstract model treats primal variables as blocks and expresses P–AL–G as randomized single-block minimization. The overall algorithm alternates this primal procedure with dual updates, and the penalty parameters can be adjusted locally.

1) Simplified notation and an abstract model of the P–AL–G:

  • P–AL–G is modeled as minimizing L(z) over one block coordinate at a time, with each block selected with positive probability.The unified vector z contains the x_i and y_ij primal variables.Source: 1) Simplified notation and an abstract… · p. 11
  • AL–G repeatedly runs P–AL–G to obtain primal values and then updates dual variables; primal updates are asynchronous gossip at a fast time scale, while dual updates are synchronous and require no communication.Nodes synchronize timers and separate communication phases from idle intervals for dual updates.Source: 1) Simplified notation and an abstract… · pp. 11–12
  • The method of multipliers remains convergent with different penalty parameters, and nodes can adapt them locally by increasing a penalty when constraint violation has not decreased sufficiently.If the violation ratio is at most the local threshold κ<1, the penalty stays unchanged; otherwise it is multiplied by σ>1.Source: 1) Simplified notation and an abstract… · p. 12

IV. CONVERGENCE ANALYSIS OF THE AL–G ALGORITHM

The convergence analysis establishes almost-sure convergence of the P–AL–G primal iteration under stated assumptions and explains why finite-time inexact primal solves do not invalidate convergence of the overall AL–G scheme. It also gives an expected-time stopping guarantee for entering and remaining near the optimal set.

IV. CONVERGENCE ANALYSIS OF THE AL–G ALGORITHM

  • Finite-time P–AL–G solves produce inexact primal solutions, but this does not violate convergence of the overall primal-dual AL–G scheme.The paper notes that simulations corroborate this conclusion.Source: §IV · p. 13
  • Under Assumptions 1 and 3, P–AL–G converges almost surely to the optimal solution set B, with Assumption 3 requiring every block-optimal point to be globally optimal.The stated convergence result is that dist(z(k),B) tends to zero with probability one.Source: §IV · pp. 13, 16
  • Assumption 1 ensures the solution set B is nonempty and compact, while the objective and its constrained extension are coercive.These properties support the analysis of the optimal set and convergence neighborhoods.Source: §IV · pp. 13–14
  • After entering U_ϵ(B), the P–AL–G iterates remain there, and the expected entry time K_ϵ is finite.The paper presents an upper bound on E[K_ϵ] as a stopping criterion.Source: §IV · pp. 16–17

V. VARIANTS TO AL–G: AL–MG (AUGMENTED LAGRANGIAN MULTI NEIGHBOR GOSSIPING) AND AL–BG (AUGMENTED LAGRANGIAN BROADCAST GOSSIPING) ALGORITHMS

The paper introduces multi-neighbor and broadcast variants of AL–G, with convergence proofs for both stated in the appendix. The supplied method details focus on AL–MG, which retains AL–G’s reformulation and dual update while changing its primal communication pattern.

V. VARIANTS TO AL–G: AL–MG (AUGMENTED LAGRANGIAN MULTI NEIGHBOR GOSSIPING) AND AL–BG (AUGMENTED LAGRANGIAN BROADCAST GOSSIPING) ALGORITHMS

  • AL–MG and AL–BG are introduced as variants of AL–G, and convergence proofs for both primal algorithms are given in the appendix.The section identifies separate subsections for the multi-neighbor and broadcast variants.Source: §V · p. 17

A. AL–MG algorithm

  • AL–MG uses the same reformulation as AL–G and the same dual-variable update, with its difference lying in the primal algorithm.The supplied passage does not further specify that primal difference.Source: A. AL–MG algorithm · p. 17
  • In AL–MG, a node transmits its y_ij values to all neighbors at once; successful receptions trigger updates, while link failures may prevent reception.The P–AL–MG algorithm is stated to converge, and overall AL–MG convergence is addressed in the section discussion.Source: A. AL–MG algorithm · pp. 17–18

B. AL–BG algorithm: An algorithm for static networks

For reliable transmissions, AL-BG removes the y_ij variables and solves an equivalent edge-consensus formulation. It uses multiplier updates with distributed randomized primal minimization and broadcasts, while requiring each node to maintain one m-dimensional dual variable.

B. AL–BG algorithm: An algorithm for static networks

  • AL-BG reformulates the problem without y_ij variables, using an equivalent formulation because the supergraph is connected.The reformulation dualizes x_i = x_j constraints on the network edges.Source: B. AL–BG algorithm: An algorithm for… · p. 18
  • The dual is solved by the method of multipliers, while a randomized block-coordinate algorithm minimizes the primal objective over each x_i.At each clock tick, a node updates x_i and broadcasts it to its neighbors.Source: B. AL–BG algorithm: An algorithm for… · pp. 18–19
  • Each node maintains only one m-dimensional dual variable, and convergence of the overall AL-BG scheme follows from the P-AL-BG convergence proof.The overall scheme substitutes P-AL-BG for P-AL-G within the primal-dual algorithm.Source: B. AL–BG algorithm: An algorithm for… · pp. 18–19

VI. SIMULATION EXAMPLES

The simulations test convergence and communication-versus-computation tradeoffs on logistic-regression and spectrum-sensing applications. They report lower communication costs for the proposed algorithms in the stated network settings, alongside greater computation or slower convergence in some comparisons.

VI. SIMULATION EXAMPLES

  • Across both simulation examples, on static networks, AL-BG outperforms existing algorithms in communication cost, while incurring larger computational cost; communication cost is dominant in networked systems supported by wireless communication.Source: §VI · p. 19
  • The two simulation examples corroborate convergence of AL-G and AL-MG on random networks and AL-BG on static networks.The applications are l1-regularized logistic regression and cooperative spectrum sensing.Source: §VI · p. 19
  • On random networks, AL-G and AL-MG outperform and in communication cost for logistic regression, but have larger computational cost.For cooperative spectrum sensing on random networks, AL-G and AL-MG converge more slowly than and.Source: §VI · p. 20

A. l1–regularized logistic regression for classification

This subsection evaluates distributed l1-regularized logistic regression for classification, comparing communication and computational costs across static and random network settings. The reported results show communication savings for the proposed algorithms, with computation tradeoffs.

A. l1–regularized logistic regression for classification

  • The classification task learns a sparse weight vector and offset from labeled feature samples, with l1 regularization and private quadratic constraints.The penalty parameter enforces sparsity, and the private constraints represent node-specific prior knowledge.Source: A. l1–regularized logistic regression… · p. 20
  • The simulations use a 20-node supergraph with 37 undirected edges, five samples per node, and 20-dimensional feature vectors.For random networks, arc failures are independent across arcs at a time slot and across time for a given arc.Source: A. l1–regularized logistic regression… · pp. 20–21
  • The comparison uses PS and MCS baselines alongside AL variants, with fixed-step baselines tuned to match or exceed the AL methods' plateau error.The AL penalty schedule affects convergence speed, and its optimal tuning is left for future study.Source: A. l1–regularized logistic regression… · pp. 21–22
  • Communication cost counts successful and unsuccessful radio transmissions, while computational cost counts floating-point operations across nodes.The paper notes that communication is the dominant cost in wireless-supported networked systems and that operation counts depend on implementation.Source: A. l1–regularized logistic regression… · p. 21
  • AL-BG reached errf below 0.001 in 0.3 · 10^5 transmissions versus MCS's 4 · 10^5 and PS's 18 · 10^5, but used 45 · 10^8 operations versus PS's 13 · 10^8 and MCS's 2 · 10^8.The comparison is for the static network; Fig. 2 plots transmission and floating-point-operation totals.Source: A. l1–regularized logistic regression… · p. 22, Figure 2 · p. 23
  • In random networks, AL-G and AL-MG require fewer transmissions than PS to reach errf below 5·10^-4, while PS requires fewer floating-point operations.The compared algorithms' estimates were feasible along time slots.Source: Figure 2 · p. 23, A. l1–regularized logistic regression… · p. 23

B. Cooperative spectrum sensing for cognitive radio networks

The cognitive-radio simulations apply the algorithms to cooperative estimation of primary-user locations and power spectra, then compare their costs on a static network. AL–BG needs the fewest transmissions to reach the stated error threshold, while AL–BG, MCS, and DCD-Lasso have similar floating-point-operation performance.

B. Cooperative spectrum sensing for cognitive radio networks

VII. CONCLUSION

The paper proposes AL–G for cooperative optimization with private convex costs and constraints over networks with random, potentially asymmetric link failures. It proves convergence for AL–G and its variants, and reports applicability in two simulation examples.

VII. CONCLUSION

  • AL–G addresses private convex costs and constraints over random networks with asymmetric link failures using gossip communication.The objective is the sum of private convex objectives, and each node has a private convex constraint.Source: §VII · p. 25
  • Unlike earlier augmented-Lagrangian primal-dual methods limited to static synchronous networks and primal-subgradient methods handling essentially symmetric failures, AL–G handles random networks and asymmetric failures.Source: §VII · p. 25
  • AL–G updates dual variables synchronously by the method of multipliers and primal variables through P–AL–G, a randomized nonlinear Gauss-Seidel gossip algorithm.The convergence proof uses expected objective decrease after a Gauss-Seidel step; AL–MG and AL–BG convergence is also proved.Source: §VII · p. 25
  • Simulations of l1-regularized logistic regression and cooperative spectrum sensing demonstrate AL–G’s applicability and effectiveness in those applications.Source: §VII · p. 26

APPENDIX

The appendix extends convergence arguments from P–AL–G to P–AL–MG and P–AL–BG. For P–AL–MG, the proof uses positive update probabilities for each variable block and continuity over a compact set.

APPENDIX

  • Lemma 8’s continuity step uses compact coordinate projections and Lemma 12 to show each coordinate-minimized L_i is continuous; the invoked topological result concerns infima over compact sets.Source: APPENDIX · p. 26
  • For P–AL–MG, spatially independent link failures ensure each variable block has positive update probability, allowing the P–AL–G convergence results to carry over with modified expected-decrease and continuity arguments.The random update selections are described as i.i.d.Source: APPENDIX · pp. 26–27
  • The P–AL–MG proof establishes positive expected decrease away from the feasible optimum by showing that zero decrease would make the point block-optimal and therefore optimal, a contradiction.Source: APPENDIX · p. 27
  • The P–AL–BG convergence analysis follows from its equivalence to P–AL–G in the optimization view, with N primal variables and no void updates.Source: APPENDIX · p. 27
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