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Newton-Raphson Consensus for Distributed Convex Optimization

Damiano Varagnolo, Filippo Zanella, Angelo Cenedese, Gianluigi Pillonetto, Luca Schenato

arXiv:1511.01509v1math.OC

TL;DR

The paper addresses distributed minimization of sums of private smooth convex costs without centralized coordination. It proposes Newton-Raphson Consensus, which combines average consensus and separation of time scales to distribute approximate Newton directions. Under suitable assumptions, the method converges exponentially to the global minimizer, while simulations report faster communication convergence than standard ADMM implementations.

  • Problem

    The problem is to compute the minimizer of a sum of private multidimensional convex costs across communicating agents without centralized coordination.

  • Method

    Newton-Raphson Consensus combines average-consensus protocols, local Taylor expansions, and separation of time scales to update distributed approximate Newton directions.

  • Results

    The algorithm is proved to converge exponentially to the global minimizer under suitable smoothness assumptions, and simulations report faster communication convergence than standard ADMM implementations.

  • Takeaways & Limitations

    The strategy provides a distributed second-order optimization procedure that avoids Lagrangian formalisms for smooth, unconstrained, multidimensional, separable convex problems.

  • Takeaways & Limitations

    The convergence proof requires sufficiently slow local updates, and the reported speed advantage over ADMM is based on simulations without a theoretical comparison proof.

Abstract

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We address the problem of distributed uncon- strained convex optimization under separability assumptions, i.e., the framework where each agent of a network is endowed with a local private multidimensional convex cost, is subject to communication constraints, and wants to collaborate to compute the minimizer of the sum of the local costs. We propose a design methodology that combines average consensus algorithms and separation of time-scales ideas. This strategy is proved, under suitable hypotheses, to be globally convergent to the true minimizer. Intuitively, the procedure lets the agents distributedly compute and sequentially update an approximated Newton- Raphson direction by means of suitable average consensus ratios. We show with numerical simulations that the speed of convergence of this strategy is comparable with alternative optimization strategies such as the Alternating Direction Method of Multipliers. Finally, we propose some alternative strategies which trade-off communication and computational requirements with convergence speed.

I. INTRODUCTION

The paper studies distributed unconstrained convex optimization without centralized coordination, where agents cooperate over communication networks to minimize a sum of private local costs. It introduces Newton-Raphson Consensus, combining average consensus with time-scale separation, and reports exponential convergence under suitable assumptions plus faster simulated communication convergence than standard ADMM implementations.

  • Distributed optimization targets networked systems without a centralized coordination unit, including complex systems requiring autonomous management with limited human intervention.
  • Newton-Raphson Consensus (NRC) uses average consensus to maintain an approximated Newton direction while agents update local guesses toward it.
  • The approach does not use Lagrangian formalisms, while alternative strategies trade communication and computational requirements against convergence speed.
  • Singular perturbation analysis establishes exponential convergence of the distributed Newton-Raphson scheme under suitable assumptions.
  • Numerical simulations on real-world databases report faster convergence than standard distributed ADMM implementations in communication iterations, without a theoretical proof of this comparison.
  • The paper focuses on unconstrained convex problems in which agents cooperatively compute the minimizer of a global cost formed from private local costs.

C. Stability of discretized dynamics

This section establishes that suitable forward-Euler discretizations preserve global exponential stability of continuous-time dynamics. The result is then applied to generalized Newton-Raphson dynamics and several descent-dynamics correspondences.

  • Under suitable Lyapunov conditions, a globally exponentially stable continuous-time system remains globally exponentially stable after sufficiently small forward-Euler discretization.
  • The stability result applies to continuous Newton-Raphson dynamics and supports analyzing their discrete-time implementations.
  • The generalized dynamics includes continuous versions of gradient descent and Jacobi descent through suitable choices of h′(x).
  • Under Assumption 1, both continuous and discrete generalized Newton-Raphson dynamics are globally exponentially stable.
  • For Newton-Raphson descent, the linearized continuous dynamics is ˙x = −x, with unitary and direction-independent convergence rate.

E. Stability of multi-agent NR dynamics

The paper generalizes single-agent Newton-Raphson dynamics to coupled multi-agent systems that reproduce the centralized behavior under identical initialization. Under additional global assumptions, the generalization and its discrete counterpart are globally exponentially stable, including under modeled perturbations.

  • The multi-agent construction couples N systems so that identical initial conditions produce trajectories matching N decoupled centralized generalized Newton-Raphson systems.
  • The coupled dynamics can be written as N independent systems driven by a common forcing term ψ(x), enabling its use as the algorithm’s slow dynamics.
  • The corresponding discrete-time dynamics is also globally exponentially stable under the same theorem combination used for the continuous system.
  • The analysis introduces perturbation terms and derives interconnection bounds to establish stability for dynamics with non-vanishing perturbations.

G. Multi-agent NR dynamics under non-vanishing perturbations

Under suitable assumptions, the perturbed multi-agent Newton-Raphson dynamics admit a smoothly parameterized equilibrium and are globally exponentially stable, including their discrete-time counterparts.

  • Perturbation model: A translated perturbed system places the perturbation-dependent equilibrium at the origin, enabling stability analysis of the shifted dynamics.The translated flow has the origin as an equilibrium for perturbations within the specified radius.
  • Stability: Under the stated assumptions, the origin is a globally exponentially stable equilibrium for the perturbed dynamics.A Lyapunov function and positive bounding scalars support the stability proof.
  • Stability: The continuous-time perturbed dynamics and their discrete-time counterparts are globally exponentially stable by combining the perturbation theorem with the discretization result.The result applies when the perturbation flow satisfies the required global Lipschitz condition.
  • Quadratic costs: Quadratic costs satisfying the specified conditions meet Assumptions 1, 5, and 9 for the selected h′ structure.This establishes a concrete cost class covered by the perturbation analysis.

IV. NEWTON-RAPHSON CONSENSUS

Newton-Raphson Consensus distributively minimizes the sum of private convex costs by combining consensus-based estimates with a slow Newton update. Under suitable assumptions it converges exponentially to the global minimizer, while perturbations produce a smooth nearby limit and alternative parameter choices trade communication, computation, and speed.

  • Algorithm and scope: Newton-Raphson Consensus computes the minimizer of a separable convex sum, including cases where the global minimizer is nonzero.The proof translates the cost-function argument to reduce analysis to the zero-minimizer case.
  • Algorithm and scope: The algorithm uses slow x_i updates driven by consensus variables y_i and z_i, whose ratio approximates a common Newton descent direction.When consensus is sufficiently faster, agents receive nearly identical forcing terms and converge toward the optimum.
  • Convergence: With zero perturbations, local estimates converge exponentially to the global minimizer for sufficiently small ε and arbitrary initial conditions.The discrete algorithm is treated as a forward-Euler discretization of globally exponentially stable continuous dynamics.
  • Convergence: Finite-precision perturbations can shift the limit away from x∗, but the computed solution varies smoothly with the initial perturbations.The perturbed limit is Ψ(ξy, ξz), and small initialization errors do not produce dramatic errors.
  • Parameter selection: The critical ε ensuring convergence is not quantified; a rule of thumb is to make consensus 2-10 times faster than the slow dynamics.The fast rate depends on the spectral gap of P, whereas the slow rate is governed by nonlinear Newton dynamics.
  • Parameter selection: For quadratic costs with ε = 1, the error satisfies ∥x_i(k) − x∗∥≤α(ρ(P))^k, while near-quadratic costs can be limited by consensus convergence.Admissible ε values may also exceed those suggested by the conservative rule of thumb.

A. On the selection of the structure of h(x)

The structure chosen for h(x) determines the distributed method’s convergence behavior and resource requirements. Newton-Raphson uses full Hessians, Jacobi reduces communication through diagonalization, and gradient descent avoids second-order information.

  • Trade-offs: Different h_i structures provide trade-offs among convergence properties, computational requirements, and communication costs.The paper summarizes the per-unit, per-step costs of NRC, JC, and GDC in Table I.
  • Method choices: Choosing h_i(x)=∇²f_i(x) yields Newton-Raphson Consensus, which approximates continuous-time Newton-Raphson dynamics for sufficiently small ε.This choice uses second-order information to construct the distributed update.
  • Method choices: Choosing a diagonal Hessian approximation yields Jacobi Consensus with less exchanged information but generally slower convergence than Newton-Raphson on skewed global costs.Diagonalization reduces communication relative to exchanging full Hessian information.
  • Method choices: Choosing h_i(x)=I yields Gradient Descent Consensus, a distributed gradient-descent procedure motivated when second derivatives are unavailable or unsuitable.This option avoids reliance on continuous second derivatives.
  • Limitations: The allowable ε depends on h_i, and quasi-Newton extensions may require modifications when Hessian approximations lose symmetry, positive definiteness, or conditioning.The paper identifies such extensions as future directions.
  • Trade-offs: In scalar scenarios, JC and NRC are equivalent, whereas GDC requires only first-derivative information.The algorithms therefore coincide or differ depending on problem dimension and derivative requirements.
  • Evaluation: The numerical study compares NRC, JC, and GDC with other distributed optimization algorithms on random geometric graphs, including costs that may violate global-stability assumptions.The simulations examine convergence performance and numerical stability beyond the theorem assumptions.

A. Effects of the choice of ε

The experiments examine how ε, initialization, local Hessian approximations, and algorithmic choices affect distributed optimization convergence. Smaller ε improves tracking of the slow Newton-Raphson dynamics, while NRC and FNRC outperform slower alternatives in communication iterations.

  • Effects of ε: As ε decreases, local trajectories more closely track the slow Newton-Raphson dynamics, which converges to the global optimum x∗.Smaller ε increases separation between fast consensus and slow optimization dynamics.
  • Initial conditions: With ε = 0.01 and N = 30, Algorithm 1 converges to x∗ from arbitrary local initial conditions when α = β = 0.Perturbed auxiliary initial conditions were also evaluated through 300 Monte Carlo runs.
  • Optimization problems: The simulations use distributed spam classification and housing regression tasks over 30-user communication graphs.The classification experiment uses 5,000 emails; the regression experiment uses 506 examples and four features.
  • Algorithm variants: The experiments compare NRC, JC, and GDC using different choices of h_i(x) for classification and regression problems.NRC uses analytical Hessians, while JC uses a Jacobi approximation and GDC corresponds to a gradient-based choice.
  • Algorithm variants: JC performs close to NRC because its Jacobi Hessian approximation is accurate for these problems, whereas GDC converges more slowly.The reported difference between NRC and JC is evident but not pronounced.
  • Comparisons: NRC and FNRC converge faster than ADMM in communication iterations, while DSM and DCM are much slower.The authors describe NRC as a possible ADMM alternative but state that further comparisons are needed.

VI. CONCLUSION

The paper presents a distributed Newton-Raphson strategy for smooth, separable, unconstrained convex optimization. It combines local second-order models with average consensus and proves convergence under a sufficiently slow step size, while simulations report faster communication convergence than ADMM.

  • VI. CONCLUSION: The proposed strategy lets agents combine local second-order Taylor expansions through average consensus to infer distributed Newton directions.It targets convex, unconstrained, multidimensional, smooth, separable cost functions without relying on Lagrangian formalisms.
  • VI. CONCLUSION: Convergence requires distinct time scales, with consensus operating faster than sufficiently slow local update steps.The algorithm is proved convergent only when the step size is sufficiently slow.
  • VI. CONCLUSION: Numerical simulations on real-world databases show faster communication-iteration convergence than ADMM when the proposed method is suitably tuned.The conclusion explicitly notes that this comparison lacks a theoretical proof.
  • VI. CONCLUSION: Open directions include local step-size tuning, asynchronous communication, convergence-rate analysis, theoretical ADMM comparison, and nonsmooth extensions.Larger ε may accelerate convergence but can lead to instability.

APPENDIX

The appendix establishes stability and convergence properties through Lyapunov arguments, singular-perturbation analysis, and local equilibrium analysis. It also connects the discrete algorithm to continuous-time dynamics and consensus-preserving reductions.

  • Global bounds: The appendix combines local cone inequalities with a whole-domain inequality by separately analyzing behavior near and beyond radius r.Inequality (38) describes the local proper cone, while inequality (39) describes the global improper cone.
  • Equilibrium analysis: For sufficiently small perturbations, the equilibrium x_eq(ξ) exists uniquely and is continuously differentiable by the Implicit Function Theorem.Invertibility follows from the positive-definite Hessian condition near x∗.
  • Stability analysis: The proof of Theorem 12 treats Algorithm 1 as an autonomous singularly perturbed system and combines boundary-layer and reduced-system Lyapunov functions.The proof proceeds by establishing the equilibrium, changing variables, analyzing both subsystems, and joining their Lyapunov functions.
  • Stability analysis: Algorithm 1 is proved exponentially stable under the hypotheses of Theorem 2.
  • Distributed dynamics: The algorithm can be represented as a forward-Euler discretization of continuous-time dynamics driven by an average-consensus matrix.The consensus matrix is doubly stochastic and is lifted across the local-cost dimension.
  • Distributed dynamics: The transformed disagreement dynamics preserve zero averages for y′ and z′, enabling elimination of average dynamics during reduction.The appendix states that 1T y′(t) and z′(t) remain zero for all t under the chosen initial conditions.
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