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Fast Distributed Gradient Methods

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

arXiv:1112.2972v4cs.IT

TL;DR

The paper asks whether distributed optimization can converge faster for convex costs with Lipschitz continuous and bounded gradients. It proposes two Nesterov-based distributed gradient algorithms and reports affirmative faster-convergence results, with D–NC requiring knowledge of L and μ(W).

  • Problem

    The paper asks whether distributed gradient methods can achieve faster convergence for convex costs with Lipschitz continuous and bounded gradients.

  • Method

    The paper proposes two distributed gradient algorithms built from the centralized Nesterov gradient method, while D–NC operates with knowledge of L and μ(W).

  • Results

    The proposed methods achieve faster convergence rates than those established for the considered class of distributed gradient methods.

  • Takeaways & Limitations

    Nesterov acceleration can provide fast distributed gradient algorithms for networks minimizing sums of individual cost functions.

  • Takeaways & Limitations

    Without Assumption 1(a), the disagreement vectors may fail to converge to zero and the objective values may fail to converge to f⋆.

Abstract

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We study distributed optimization problems when $N$ nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant $L$), and bounded gradient. We propose two fast distributed gradient algorithms based on the centralized Nesterov gradient algorithm and establish their convergence rates in terms of the per-node communications $\mathcal{K}$ and the per-node gradient evaluations $k$. Our first method, Distributed Nesterov Gradient, achieves rates $O\left({\log \mathcal{K}}/{\mathcal{K}}\right)$ and $O\left({\log k}/{k}\right)$. Our second method, Distributed Nesterov gradient with Consensus iterations, assumes at all nodes knowledge of $L$ and $μ(W)$ -- the second largest singular value of the $N \times N$ doubly stochastic weight matrix $W$. It achieves rates $O\left({1}/{\mathcal{K}^{2-ξ}}\right)$ and $O\left({1}/{k^2}\right)$ ($ξ>0$ arbitrarily small). Further, we give with both methods explicit dependence of the convergence constants on $N$ and $W$. Simulation examples illustrate our findings.

I. INTRODUCTION

The paper studies distributed minimization of a sum of convex, smooth, bounded-gradient costs and develops two Nesterov-based methods with improved convergence rates. The rates depend on communications, gradient evaluations, network parameters, and assumptions such as global knowledge and bounded gradients.

  • Problem setting: N nodes minimize a sum of convex functions of a common vector variable, with each function known only to its node.The network is generic and connected.
  • Assumptions: Convex functions with Lipschitz continuous and bounded gradients define the function class studied by the paper.The Lipschitz constant is L.
  • Contributions: O(log K/K) communications and O(log k/k) gradient evaluations are achieved by Distributed Nesterov Gradient (D–NG).D–NG uses one communication per iteration, so k = K.
  • Comparison: Both methods are established as strictly better than distributed subgradient and dual-averaging methods for the considered function class.The comparison includes at least the O(log k/k) rate established for the proposed methods.
  • Limitations: Dropping bounded gradients can make the worst-case convergence rates of both proposed methods arbitrarily slow.The paper notes that a weaker replacement assumption may be possible.

III. DISTRIBUTED NESTEROV BASED ALGORITHMS

The paper presents D–NG and D–NC as distributed implementations of Nesterov-style updates. D–NG uses implicit consensus with a diminishing step-size, whereas D–NC uses explicit, increasing consensus rounds and requires network and smoothness knowledge.

  • A. Distributed Nesterov gradient algorithm (D–NG): D–NG generates local solution estimates x_i(k) and auxiliary variables y_i(k) from common initialization.The auxiliary sequence supports the accelerated update.
  • A. Distributed Nesterov gradient algorithm (D–NG): D–NG uses α_k = c/(k+1) and a Nesterov sequence β_k, because a constant step-size generally converges only to a solution neighborhood.The constant c > 0 can be arbitrary.
  • A. Distributed Nesterov gradient algorithm (D–NG): Each D–NG iteration averages neighboring auxiliary variables, takes a local negative-gradient step, and updates the auxiliary variable.The communication is performed with the averaging weights in W.
  • B. Distributed Nesterov gradient with Consensus iterations (D–NC): D–NC uses a constant step-size α ≤ 1/(2L) and operates across outer optimization iterations and inner consensus iterations.Nodes perform consensus around the solution and auxiliary updates.
  • B. Distributed Nesterov gradient with Consensus iterations (D–NC): D–NC increases the number of inner consensus iterations as log k and makes this count depend on μ(W).Unlike D–NG, D–NC uses explicit multistep consensus at each outer iteration.

IV. INTERMEDIATE RESULTS: INEXACT NESTEROV GRADIENT METHOD

The convergence analysis reduces both distributed methods to an inexact centralized Nesterov method applied to global averages. Network disagreement creates oracle error, so bounding disagreement is the central step linking consensus behavior to optimization progress.

  • Framework: The global averages of the nodes’ estimates are analyzed through an inexact centralized Nesterov gradient framework.This framework supplies the per-iteration progress relation used for both algorithms.
  • A. Inexact Nesterov gradient method: The inexact method has the same per-iteration progress as exact Nesterov gradient except for deterioration proportional to (k+1)^2δ_(k−1).Lemma 2 identifies δ_(k−1) as the gradient-direction inexactness term.
  • B. Algorithms D–NG and D–NC in the inexact oracle framework: For D–NG, node disagreement is represented by deviations of local auxiliary variables from their global average.Analogous disagreement vectors are defined for the solution estimates.
  • B. Algorithms D–NG and D–NC in the inexact oracle framework: The distributed gradient construction forms an inexact oracle for the sum objective with L_k = 2NL and δ_k = L∥e_y(k)∥^2.The oracle error is determined by the squared auxiliary-variable disagreement.
  • B. Algorithms D–NG and D–NC in the inexact oracle framework: For D–NC, consensus powers are implemented through τ_x(k) and τ_y(k) distributed consensus steps.The first and second consensus rounds correspond to the solution and auxiliary updates, respectively.
  • Analysis strategy: Convergence-rate proofs for both methods reduce to bounding disagreement and applying the inexact Nesterov progress lemma.This is the analysis strategy stated in the paper.

V. ALGORITHM D–NG: CONVERGENCE ANALYSIS

D–NG analysis first establishes asymptotic consensus, then uses inexact Nesterov analysis to derive convergence rates and network dependence. The method achieves O(log k/k) convergence under the stated assumptions, with constants depending on step size, network parameters, and problem quantities.

  • A. Disagreement estimate: O(1/k) disagreement bounds for ex(k) and ey(k) establish asymptotic consensus among node estimates.The differences between the nodes’ solution and auxiliary estimates converge to zero.
  • A. Disagreement estimate: Theorem 4 bounds D–NG disagreement by modeling the augmented state as a linear time-varying system with gradient-driven inputs.The resulting upper bound is identified as an important technical contribution relative to systems with a single disagreement state.
  • B. Convergence rate and network scaling: O(log k/k) convergence holds for D–NG when c ≤ 1/(2L), and remains valid for c > 1/(2L) with a larger convergence constant.The latter statement assumes xi(0) = yi(0) = 0 for all nodes.
  • B. Convergence rate and network scaling: Network scaling distinguishes arbitrary constant c, yielding p = 3, from c = (1 − μ(W))/(2L), yielding p = 1.The analysis gives explicit dependence of the convergence bound on N and W through the network disagreement constants.

VI. ALGORITHM D–NC: CONVERGENCE ANALYSIS

This section presents the convergence analysis of D–NC. It covers disagreement estimation and then derives convergence rates together with their dependence on the underlying network.

  • VI. Algorithm D–NC: Convergence Analysis: D–NC analysis is divided into disagreement estimation and convergence-rate analysis with network scaling.The section contains separate subsections for these two analytical components.
  • VI. Algorithm D–NC: Convergence Analysis: The first analytical component estimates the disagreements ex(k) and ey(k) under D–NC.
  • VI. Algorithm D–NC: Convergence Analysis: The second analytical component derives the convergence rate and its dependence on the underlying network.

A. Disagreement estimate

D–NC uses inner consensus iterations to control disagreement between node estimates while outer iterations perform accelerated optimization. The analysis establishes explicit disagreement and convergence bounds under network-dependent communication schedules.

  • A. Disagreement estimate: D–NC bounds ex(k) and ey(k) by analyzing the consensus dynamics through auxiliary disagreement recursions.The proof tracks the maximum of the two disagreement norms and uses the contraction factor μ(W).
  • B. Convergence rate and network scaling: O(k log k) communication rounds suffice after k outer iterations, with the schedule determined by μ(W).The communication count is the sum of the inner consensus rounds τx(t) and τy(t).
  • B. Convergence rate and network scaling: O(N^1/2/[(1 − μ(W))K]^(2−ξ)) and O(N^1/2/k^2) scaling follows for D–NC under known L and μ(W).The network-scaling result assumes α ≤ 1/(2L), and ξ is an arbitrarily small positive constant.

VII. COMPARISONS WITH THE LITERATURE AND DISCUSSION OF THE ASSUMPTIONS

The paper compares D–NG and D–NC with prior distributed methods and examines how network knowledge and function assumptions affect convergence. It also discusses implementation requirements, network-weight modifications, and limitations when Lipschitz constants or bounded gradients are unavailable.

  • Network scaling: D–NG’s modified weight matrix changes the spectral gap by only a constant factor, so its convergence rate is unaffected up to a numerical constant.The rate can therefore be expressed using the original matrix W rather than the modified matrix W′.
  • Comparisons with the literature: Table I compares D–NG, D–NC, and prior algorithms by implementation requirements and communications needed to reach ε-accuracy.The comparison distinguishes knowledge required for convergence from knowledge needed for stopping and step-size optimization.
  • Assumptions and limitations: Without local Lipschitz constants, D–NC can use α_k = 1/(k + 1)^p for p ∈ (0, 1], but only with a deteriorated convergence rate.A distributed line-search variant is proposed as a future research direction rather than established here.
  • Assumptions and limitations: If bounded gradients are removed, the established convergence rates do not hold and the worst-case rates of both D–NG and D–NC can become arbitrarily slow.The paper notes that a weaker replacement for bounded gradients may be possible.
  • Comparisons with the literature: D–NG achieves O(log k/k) in the considered worst-case analysis, while the compared method in has an Ω(1/k^2/3) lower bound.The comparison is made for a two-node connected network and functions satisfying the paper’s assumptions.

B. Discussion on Assumptions

The discussion tests which assumptions support convergence and shows that relaxing network, smoothness, or bounded-gradient conditions can prevent convergence or make worst-case behavior arbitrarily poor. Simulations also compare D–NG and D–NC across losses and problem difficulties.

  • Assumption 1(a): Without Assumption 1(a), the disagreement state may not converge to zero, and node optimality gaps may fail to converge to f⋆.The example uses N = 2 and W = I.
  • Assumption 1(b): Relaxing Assumption 1(b) can make D–NG disagreement and node optimality gaps grow unbounded in simulation.The example uses a two-node network with W11 = W22 = 1/10 and W12 = W21 = 9/10.
  • Assumption 2: Relaxing Lipschitz continuity of gradients causes D–NG gaps and iterates to grow unbounded, while D–NC gaps can remain bounded away from zero.These outcomes are reported for the same two-node weight matrix in simulations.
  • Assumption 3: Dropping bounded gradients invalidates the established rates for both methods and can make their worst-case convergence arbitrarily slow.The hard instances produce arbitrarily large disagreement inputs even near the solution.
  • Simulation results: D–NG is faster than the compared methods over moderate accuracies, but (second variant) becomes faster at accuracies of about 2·10^-5 and finer.At ϵ = 10^-2, D–NG requires about 10^4 transmissions, versus approximately 3.16·10^4 for (second variant).
  • Simulation results: As θ decreases, D–NC becomes faster and its performance relative to D–NG improves; for large θ, D–NC deteriorates relative to D–NG.For θ = 0.01, the curves intersect near 4·10^-3; for θ = 1000, D–NG remains better to at least 10^-7.
  • Interpretation: D–NC resembles centralized Nesterov with a constant step-size, whereas D–NG resembles it with a diminishing 1/k step-size.Multiple consensus rounds can be unnecessary for easy problems, while disagreement dynamics matter for difficult problems.

IX. CONCLUSION

The conclusion proposes two distributed gradient algorithms for sums of convex costs with Lipschitz continuous and bounded gradients, extending centralized Nesterov acceleration to networks. Their rates and network-dependent constants are established, and simulations illustrate their behavior.

  • Contribution: The paper proposes two distributed gradient algorithms for networked minimization of sums of individual node costs.Both methods are built from the centralized Nesterov gradient method.
  • Contribution: D–NC requires all nodes to know L and μ(W), while its convergence rates and those of D–NG depend explicitly on network parameters.The conclusion states that simulations illustrate the proposed methods.

APPENDIX

The appendix analyzes the time-varying linear dynamics underlying the distributed method by decomposing system matrices into network eigenmodes and bounding their products.

  • Proof strategy: The proof models the augmented state as a linear time-varying system driven by disagreement-gradient inputs.It first unwinds the recursion, then bounds system-matrix norms and completes the theorem with summation arguments.
  • Matrix decomposition: An orthonormal eigenbasis transforms the transition matrix into block-diagonal 2×2 mode dynamics without changing its singular values or spectral norm.The decomposition uses the weight matrix eigenstructure and a permutation matrix.
  • Mode analysis: The consensus mode has eigenvalue λ1(fW) = 0, making its transition powers vanish after two or more steps.For this mode, Σ1 is constant with norm 1 and Σ1^s = 0 for s ≥ 2.
  • Mode analysis: For non-consensus modes, the proof represents each time-varying block as a perturbation of a diagonalizable matrix and bounds products using norm inequalities.The argument separates the fixed spectral component from the step-size-dependent perturbation.

3) Summation:

The summation analysis bounds accumulated disagreement and gradient effects using harmonic and geometric-series inequalities, while hard instances explain when distributed methods can become arbitrarily difficult.

  • 3) Summation:: The proof bounds the accumulated transition effects by splitting the sum into early and late time ranges.Harmonic and geometric-series bounds control the two resulting sums.
  • 3) Summation:: Logarithmic bounds on harmonic terms and geometric decay yield the convergence estimate used to establish Theorem 4.The argument applies sub-multiplicative and sub-additive norm properties before invoking the consensus constant.
  • Hard instances: The hard-instance construction uses two-node functions whose separated minimizers create large disagreement-driving gradients near the global solution.Increasing θ makes the problem increasingly difficult because the disagreement-dynamics inputs become arbitrarily large.
  • Hard instances: Eigenvalue decompositions of W reduce the hard-instance recursions to scalar products along consensus and disagreement directions.The analysis uses W = QΛQ^⊤ and tracks factors involving λ1 and 1.
  • Hard instances: Choosing θ based on the iteration horizon produces a lower-bound construction for the competing distributed gradient method.The construction sets θ = 1/(s_k(τ)) ≤ 1 after bounding the relevant products.

Step 3: Upper bounding ∥x(k)∥:

The proof upper-bounds the optimality-gap expression by analyzing e_k(τ) over three τ-regimes. These regime-wise bounds, together with (57), complete the proof of (32).

  • Step 4: Upper bounding the optimality gap: The right-hand side is upper-bounded by taking the infimum of e_k(τ) over τ ∈ [0, ∞).The interval is split into [0, 3/4], [3/4, 1], and [1, ∞).
  • Step 4: Upper bounding the optimality gap: inf_[0,3/4) e_k(τ) = Ω(1/k^2/3).
  • Step 4: Upper bounding the optimality gap: For τ ∈ [3/4, 1], s_k(τ) ≤ 3(log k)(k + 1)^(1−τ) for k ≥ 3, yielding a corresponding lower bound on inf_[3/4,1] e_k(τ).
  • Step 4: Upper bounding the optimality gap: Combining the regime-wise bounds with (57) completes the proof of (32).

C. Relaxing bounded gradients: Proof of (34) for D–NC

The proof of (34) for D–NC constructs a two-node example, diagonalizes the weight matrix, and tracks the transformed coordinates. The resulting lower bound shows that the per-node objective gap can scale as α²θ²/(16k^4).

  • Proof setup: The proof fixes θ > 0 and selects the functions f_i according to (35) to analyze D–NC.The proof of the corresponding D–NG statement is described as similar and referenced elsewhere.
  • Proof setup: The D–NC recursion updates x(k) using (1−α)W^τx(k)y(k−1) and an αθW^τx(k)(1, −1)^⊤ term.
  • Spectral transformation: The initialization is x(0) = y(0) = (0, 0)^⊤, and the weight matrix is decomposed as W = QΛQ^⊤.The transformed variables are z(k) = Q^⊤x(k) and w(k) = Q^⊤y(k).
  • Spectral transformation: The first transformed coordinate is identified with the scalar recursion used earlier, with μ := λ1 and z^(1)(k), w^(1)(k) corresponding to the transformed dynamics.
  • Resulting lower bound: α²θ²/(16k^4) lower-bounds max_i=1,2(f(x_i(k))−f⋆) for k ≥ 10.With α = 1/(2L) = 1/2, the proof uses this bound to select θ as a function of k and M.
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