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Zero-Gradient-Sum Algorithms for Distributed Convex Optimization: The Continuous-Time Case
Jie Lu, Choon Yik Tang
TL;DR
The paper addresses distributed convex optimization over networks and develops continuous-time ZGS algorithms for this setting. It establishes asymptotic and exponential convergence, derives convergence-rate bounds, and relates these algorithms to basic distributed consensus algorithms.
Problem
The paper addresses solving an unconstrained, separable, convex optimization problem over an N-node multi-hop network, with each node observing a convex function.
Method
The paper systematically constructs continuous-time ZGS algorithms and analyzes trajectories using a Lyapunov-based approach to establish convergence.
Results
The algorithms achieve asymptotic and, for a subset, exponential convergence, with lower and upper bounds derived for their convergence rates.
Takeaways & Limitations
ZGS algorithms for distributed convex optimization are closely related to basic distributed consensus algorithms, with consensus-rate characterization appearing as a special case.
Takeaways & Limitations
The convergence-rate results generally depend on the initial state, except that for quadratic functions the relevant parameters can be chosen from their smallest and largest values.
Abstract
from arXiv · showhide
This paper presents a set of continuous-time distributed algorithms that solve unconstrained, separable, convex optimization problems over undirected networks with fixed topologies. The algorithms are developed using a Lyapunov function candidate that exploits convexity, and are called Zero-Gradient-Sum (ZGS) algorithms as they yield nonlinear networked dynamical systems that evolve invariantly on a zero-gradient-sum manifold and converge asymptotically to the unknown optimizer. We also describe a systematic way to construct ZGS algorithms, show that a subset of them actually converge exponentially, and obtain lower and upper bounds on their convergence rates in terms of the network topologies, problem characteristics, and algorithm parameters, including the algebraic connectivity, Laplacian spectral radius, and function curvatures. The findings of this paper may be regarded as a natural generalization of several well-known algorithms and results for distributed consensus, to distributed convex optimization.
1 Introduction
The paper extends distributed convex optimization from prior discrete-time scalar settings to continuous-time, multidimensional algorithms over networks. Its ZGS framework provides asymptotic and, for a subset, exponential convergence with topology- and curvature-dependent rate bounds.
- Motivation: The problem requires networked nodes, each observing a convex function, to jointly determine an optimizer minimizing their summed objective.The setting is unconstrained, separable, and distributed over a multi-hop network.
- Prior work: Earlier work mainly studied discrete-time asynchronous algorithms and only the scalar version of the optimization problem.Prior approaches included incremental and non-incremental subgradient methods, as well as gossip-style algorithms.
- Contributions: The paper develops Zero-Gradient-Sum (ZGS) algorithms whose nonlinear trajectories remain on an invariant zero-gradient-sum manifold and converge asymptotically to the unknown minimizer.The construction assumes twice continuously differentiable, strongly convex local functions and uses a convexity-inspired Lyapunov function.
- Contributions: A subset of ZGS algorithms converges exponentially, with lower and upper rate bounds depending on network topology, problem characteristics, and algorithm parameters.The bounds involve algebraic connectivity, Laplacian spectral radius, and local-function curvatures.
- Connections to consensus: Existing continuous-time distributed consensus algorithms are special cases of ZGS algorithms and require only slight modification to address the optimization problem.The algebraic-connectivity characterization of linear consensus convergence is included as a special case of Theorem 2.
2 Preliminaries
The preliminaries define strong convexity for twice continuously differentiable functions through equivalent conditions involving function values, gradients, and Hessians. They also introduce global strong convexity through a positive convexity parameter.
- Strong convexity: A twice continuously differentiable function is locally strongly convex when the equivalent strong-convexity conditions hold on every convex compact set.The conditions are parameterized by a positive constant θ.
- Notation: The preliminaries identify the gradient, Hessian, Euclidean norm, identity matrix, and positive-semidefinite matrix ordering used in these conditions.Matrix inequality A ≥ B means A − B is positive semidefinite.
- Strong convexity: A function is strongly convex on R^n when a single constant θ > 0 satisfies the equivalent conditions globally.This θ is called the convexity parameter.
- Curvature conditions: The preliminaries also consider twice continuously differentiable functions on convex sets under conditions involving an arbitrary constant Θ > 0.These conditions extend the curvature framework used later in the analysis.
3 Problem Formulation
The paper formulates distributed optimization on a connected, undirected, fixed-topology network where strongly convex local functions are summed into a globally minimized objective. Each node uses local state dynamics and neighbor communication to drive its estimate toward the common optimizer.
- Network model: The network is a connected, undirected graph with N ≥ 2 nodes, bidirectional links, one-hop neighbor sets, and fixed topology.Communication is assumed delay-free, error-free, and unquantized.
- Function assumptions: Each node observes a twice continuously differentiable, strongly convex function fi with convexity parameter θi > 0 and locally Lipschitz Hessian.These assumptions support the optimization and convergence analysis.
- Optimization problem: The nodes seek the unique optimizer x∗ of the unconstrained separable objective F formed by summing the local functions.At x∗, the summed gradient satisfies ∇F(x∗) = 0.
- Algorithm formulation: The distributed algorithm assigns each node a state xi(t) representing its estimate of x∗ and evolves it through locally Lipschitz dynamics depending on its state and neighbor states.The goal is asymptotic, or preferably exponential, convergence of all estimates to x∗.
- Communication: Nodes need to exchange neighbor states continuously when the dynamics depend on them, while function exchange is unnecessary for the ZGS constructions described later.This contrasts with earlier PE and CHE algorithms, which require exchanging local functions.
4 Zero-Gradient-Sum Algorithms
The paper constructs Zero-Gradient-Sum algorithms by designing trajectories that remain on a zero-gradient-sum manifold while a convexity-based Lyapunov function decreases toward the optimizer. Under the stated assumptions, every ZGS algorithm has a unique global solution and converges asymptotically to the optimizer.
- Convergence guarantee: The construction avoids requiring local or global asymptotic stability of the full system, relying instead on one suitable trajectory with nonincreasing Lyapunov value.The paper identifies this trajectory-based condition as sufficient for achieving the optimization goal.
- Geometric construction: The approach uses agreement and zero-gradient-sum sets whose intersection contains only the optimizer.The agreement set requires all node states to coincide, while the zero-gradient-sum manifold requires the sum of local gradients to vanish.
- Lyapunov-based design: The Lyapunov design removes dependence on the unknown optimizer from the derivative and makes the derivative nonpositive, with equality only on the agreement set.The construction imposes conditions on the local dynamics so that the unknown optimizer does not determine the sign of the Lyapunov derivative.
- Lyapunov-based design: The required initial conditions maintain a zero gradient sum for all time, enabling convergence on the invariant manifold.The gradient sum remains constant along trajectories and must be initialized at zero to achieve convergence to the optimizer.
- Geometric construction: The algorithm conditions make the zero-gradient-sum manifold positively invariant, so trajectories begin on and remain within it.The initial-state constraints place the trajectory on the manifold, and the dynamics preserve it over time.
- Convergence guarantee: Every ZGS algorithm has a unique solution, preserves the manifold, decreases the Lyapunov function, and converges asymptotically to the optimizer.Theorem 1 establishes existence and uniqueness, invariance, strict Lyapunov decrease away from the optimizer, convergence of the Lyapunov value to zero, and state convergence.
5 Convergence Rate Analysis
The paper establishes exponential convergence for a subset of ZGS algorithms and derives lower and upper rate bounds linked to network, function, and algorithm parameters. These bounds specialize to algebraic-connectivity and Laplacian-spectral-radius expressions and recover known consensus rates.
- Scope and specialization: The rate bounds depend generally on the initial state through curvature constants, except that quadratic node and edge functions permit state-independent Hessian-eigenvalue bounds.The analysis also frames the interplay among topology, objective curvature, and algorithm parameters for further study.
- Lower convergence-rate bound: Theorem 2 establishes exponential convergence for ZGS algorithms (32) and provides a lower convergence-rate bound ρ.The bound is defined through positive semidefinite matrices P and Q and satisfies ρP ≤ Q.
- Lower convergence-rate bound: ρ is the smallest eigenvalue of ¯P^-1/2 ¯Q ¯P^-1/2, while an explicit but looser bound uses the graph’s algebraic connectivity λ2.The algebraic-connectivity bound follows from comparing Laplacians of the complete graph and the underlying graph.
- Scope and specialization: For scalar quadratic consensus, the bounds recover the known algebraic-connectivity rate ∥x(t)−x∗∥≤∥x(0)−x∗∥e−λ2t and generalize consensus analysis to distributed convex optimization.The specialization sets relevant curvature parameters to 1 and connects Theorem 2 and Corollary 1 to a standard linear consensus result.
- Upper convergence-rate bound: ˜ρ is the largest eigenvalue of ˜P^-1/2 ˜Q ˜P^-1/2, while a looser explicit upper bound depends on the Laplacian spectral radius λN.In the scalar quadratic consensus case, the resulting estimate is ∥x(t)−x∗∥≥∥x(0)−x∗∥e−λN t.
6 Conclusion
Using a convexity-based Lyapunov function, the paper develops continuous-time ZGS algorithms for distributed convex optimization. These algorithms converge asymptotically or exponentially, with bounded convergence rates, and relate closely to distributed-consensus algorithms.
- The paper develops a set of continuous-time ZGS algorithms for distributed convex optimization over networks.
- The algorithms achieve asymptotic convergence, while a subset also achieves exponential convergence.
- The paper derives lower and upper bounds on the algorithms’ convergence rates.
- ZGS algorithms are closely related to basic distributed-consensus algorithms.
- This relationship suggests possible extensions of ZGS algorithms paralleling extensions of distributed-consensus algorithms.