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Harnessing Smoothness to Accelerate Distributed Optimization

Guannan Qu, Na Li

arXiv:1605.07112v2math.OC

TL;DR

For smooth convex objectives, the paper asks how consensus-based distributed optimization can harness smoothness to converge faster than existing DGD methods. It introduces a history-based gradient estimation scheme and reports convergence rates matching centralized gradient methods, up to constants.

  • Problem

    For smooth convex objectives, existing DGD methods do not fully exploit smoothness to match centralized gradient descent rates.

    Source: §I · p. 2

  • Method

    The algorithm combines gradient descent with a history-based scheme for estimating the average gradient.

    Source: C. Harnessing Smoothness via History… · p. 10, §VI · p. 24

  • Results

    The method achieves O(1/t) convergence for convex smooth objectives and linear convergence for strongly convex smooth objectives, with rates comparable to centralized gradient methods except for constants.

    Source: §VI · p. 24

  • Takeaways & Limitations

    The reported rates show that distributed optimization can harness smoothness with convergence comparable to centralized gradient methods, apart from constants.

    Source: §VI · p. 24

Abstract

from arXiv · show

There has been a growing effort in studying the distributed optimization problem over a network. The objective is to optimize a global function formed by a sum of local functions, using only local computation and communication. Literature has developed consensus-based distributed (sub)gradient descent (DGD) methods and has shown that they have the same convergence rate $O(\frac{\log t}{\sqrt{t}})$ as the centralized (sub)gradient methods (CGD) when the function is convex but possibly nonsmooth. However, when the function is convex and smooth, under the framework of DGD, it is unclear how to harness the smoothness to obtain a faster convergence rate comparable to CGD's convergence rate. In this paper, we propose a distributed algorithm that, despite using the same amount of communication per iteration as DGD, can effectively harnesses the function smoothness and converge to the optimum with a rate of $O(\frac{1}{t})$. If the objective function is further strongly convex, our algorithm has a linear convergence rate. Both rates match the convergence rate of CGD. The key step in our algorithm is a novel gradient estimation scheme that uses history information to achieve fast and accurate estimation of the average gradient. To motivate the necessity of history information, we also show that it is impossible for a class of distributed algorithms like DGD to achieve a linear convergence rate without using history information even if the objective function is strongly convex and smooth.

I. INTRODUCTION

The paper targets faster consensus-based distributed optimization for smooth convex objectives without increasing communication per gradient evaluation. Its history-based gradient estimator supports convergence rates matching centralized gradient descent, while a lower-bound result highlights the role of history information.

I. INTRODUCTION

  • Existing DGD methods do not fully exploit smoothness: their smooth-objective convergence is slower than centralized gradient descent.The introduction reports at most Ω(1/t^(2/3)) with a diminishing step size and notes that constant step sizes need not guarantee exact convergence.Source: §I · p. 2
  • The algorithm achieves O(1/t) convergence for smooth convex objectives and linear convergence for smooth strongly convex objectives, matching centralized rates apart from worse constants.The linear rate is O(γ^t) for some γ in (0, 1).Source: §I · p. 2
  • A history-based gradient estimation scheme estimates the average gradient quickly and accurately within a gradient-descent algorithm.The proposed method uses one communication step per gradient evaluation.Source: §I · p. 2
  • Without history information, a class of DGD-like algorithms cannot attain linear convergence even for strongly convex smooth objectives.Source: §I · p. 2
  • The estimation scheme can be cast as a general method for distributed versions of first-order algorithms, including Nesterov gradient descent.The authors expect the resulting distributed algorithms to have convergence rates similar to their centralized counterparts.Source: §I · p. 2
  • The paper focuses on consensus-based first-order distributed algorithms, not other distributed optimization approaches such as second-order methods or ADMM.Source: §I · p. 3

II. PROBLEM AND ALGORITHM

The paper formulates distributed optimization as minimizing the average of agents’ local functions using local computation and communication. Its consensus-based algorithm has each agent maintain a solution estimate and an estimate of the average gradient.

A. Problem Formulation

  • The objective is to minimize the average of n agents’ convex local functions, with agents communicating over a connected graph and making decisions from local functions and neighbors’ information.The analysis considers convex, smooth functions and also the case where each local function is strongly convex; it assumes the global objective has a minimizer.Source: A. Problem Formulation · pp. 3–4

B. Algorithm

  • Each agent maintains a solution estimate xi(t) and a vector si(t) designated to estimate the average gradient.The algorithm is consensus-based, and its update uses a gradient-estimation rule for si(t).Source: B. Algorithm · pp. 4–5
  • The algorithm can operate in a fully distributed fashion, with each node sending its solution and gradient-estimate vectors to its neighbors.The consensus weights are zero for non-neighbors; the two consensus weight matrices may be chosen differently, although the analysis uses the same matrix.Source: B. Algorithm · p. 5
  • The key contribution is the gradient-estimation scheme, which the authors say can also be used to obtain distributed versions of other gradient-based algorithms.The proposed extension to a broad range of centralized algorithms is presented as a conjecture.Source: B. Algorithm · p. 5

C. Convergence of the Algorithm

The convergence results establish O(1/t) convergence for smooth convex objectives and linear convergence under strong convexity. The rates match centralized gradient descent, although the distributed method has worse constants and step-size rules tied to the consensus matrix.

C. Convergence of the Algorithm

  • O(rho(G(eta))^t) bounds the decay of optimizer distance, consensus error, and gradient-estimation error; objective error decays as O(rho(G(eta))^(2t)).These bounds are stated under the smooth and strongly convex assumptions and the step-size condition of Theorem 1.Source: C. Convergence of the Algorithm · pp. 5–6
  • The convergence-rate types match CGD, but the distributed algorithm has worse constants in both cases.The linear convergence-rate constant is also described as usually worse than CGD’s.Source: C. Convergence of the Algorithm · pp. 6–7
  • O(1/t) is the stated convergence rate when the local functions are smooth, while strongly convex smooth functions yield a linear rate.The paper says both rates preserve the convergence-rate types of centralized gradient descent.Source: C. Convergence of the Algorithm · pp. 5–6
  • The step-size rules depend on the consensus matrix W, and the convergence rate depends on the network size through W’s second-largest singular value sigma.The paper notes that the rate does not explicitly depend on n under the stated result’s initial-condition qualification.Source: C. Convergence of the Algorithm · p. 7
  • With lazy Metropolis weights and an upper bound U on the number of agents, the paper gives relaxed step-size conditions under which its strongly convex and smooth-case results hold.For the smooth case, the supplied corollary states 0 < eta <= 1/(160 beta (71 U^2)^2).Source: C. Convergence of the Algorithm · p. 7

III. ALGORITHM DEVELOPMENT: MOTIVATION

The section reviews why standard distributed gradient methods do not exploit smoothness to match centralized rates with limited communication. It motivates an alternative that uses only one or a constant number of communication steps per gradient evaluation.

A. Review of Distributed First-Order Optimization Algorithms

  • Consensus-based DGD has rate O(log t/sqrt(t)) for the running-best objective under the cited result, matching centralized subgradient descent up to a log t factor.The supplied passage identifies this as a result in.Source: A. Review of Distributed First-Order… · p. 8
  • Method (5), with a vanishing step size, cannot converge faster than Omega(1/t^(2/3)); CGD with a sufficiently small constant step size achieves O(1/t), and has a linear rate when f is strongly convex.Source: A. Review of Distributed First-Order… · p. 8
  • Using multiple consensus steps after each gradient step addresses DGD’s non-convergence problem, but the required number of steps grows as the algorithm proceeds.The cited examples use ct = Theta(log t) in and ct = Theta(t) in.Source: A. Review of Distributed First-Order… · p. 8

III. ALGORITHM DEVELOPMENT: MOTIVATION

  • The review motivates distributed algorithms that harness smoothness while using only one or a constant number of communication steps per gradient evaluation.This is the stated motivation for the paper’s algorithm-development discussion.Source: §III · p. 7, A. Review of Distributed First-Order… · p. 9

B. An Impossibility Result

The paper proves that a broad class of distributed first-order algorithms without history information cannot guarantee linear convergence, even for strongly convex, smooth objectives. It then describes a history-based gradient estimator and how the resulting algorithm differs from related work.

B. An Impossibility Result

  • The impossibility result covers updates that combine neighbors’ information through a continuous consensus function with a local scaled-gradient update whose step size has a limit.The update function is assumed Lipschitz continuous in its gradient input.Source: B. An Impossibility Result · p. 9
  • For two agents, some strongly convex, smooth objectives yield arbitrarily late steps whose error ratio is at least any chosen δ∈(0,1), precluding linear convergence guarantees for this class.The theorem assumes convergence to the minimizer for all objectives and initial points in the class.Source: B. An Impossibility Result · p. 9

C. Harnessing Smoothness via History Information

  • The proposed algorithm uses history information to estimate the average gradient, with its estimator separated from the gradient-descent update.The estimated average gradient serves as a possible stopping criterion, and the separation supports extension to other centralized methods.Source: C. Harnessing Smoothness via History… · pp. 10–11
  • Unlike, the proposed algorithm has looser consensus-matrix requirements, but its step-size bounds also depend on W.The paper notes this dependence as a downside of its current results.Source: C. Harnessing Smoothness via History… · p. 11

IV. CONVERGENCE ANALYSIS

The convergence analysis introduces average-sequence update equations and auxiliary bounds, then gives an intuition for the strongly convex linear-rate result. Its key intuition is that optimization error and gradient-estimation error can decay together.

A. Analysis Setup

  • The analysis introduces update equations for the average iterate and estimated gradient, with auxiliary inequalities derived from the doubly stochastic consensus matrix and β-smoothness.The lemmas govern the average sequences and supply bounds used in the convergence proofs.Source: A. Analysis Setup · p. 11

B. Why the Algorithm Works: An Intuitive Explanation

  • Under strong convexity and smoothness, linear decay of the gradient-estimation error implies linear decay of distance to the optimizer, and vice versa.The authors describe this mutual dependence as intuition, not by itself a proof of linear convergence.Source: B. Why the Algorithm Works: An… · p. 12
  • The gradient-descent and gradient-estimation steps facilitate each other, unlike approaches that first bound consensus error and then use it to bound objective error.The paper says the formal convergence proof follows in subsequent subsections.Source: B. Why the Algorithm Works: An… · p. 12

C. Convergence Analysis: Strongly Convex

For strongly convex, smooth objectives, the proof bounds gradient-estimation error, consensus error, and distance to the optimizer through a contracting linear system. Smoothness avoids additive constants that would otherwise require a vanishing step size.

C. Convergence Analysis: Strongly Convex

  • The proof bounds the three errors using linear combinations of their past values, then combines them into a linear system inequality.The resulting system contracts to establish the linear convergence rate.Source: C. Convergence Analysis: Strongly Convex · pp. 12, 14, 16
  • The gradient-estimation error, consensus error, and optimizer distance converge at O(ρ(G(η))^k), yielding objective error O(ρ(G(η))^(2k)).The proof also derives the same squared-rate objective-error bound for each agent’s iterate.Source: C. Convergence Analysis: Strongly Convex · p. 15
  • Using β-smoothness to bound differences of gradients avoids additive constants, whereas DGD-like analyses typically require vanishing step sizes to control such constants.The paper links this distinction to the slower convergence associated with vanishing step sizes.Source: C. Convergence Analysis: Strongly Convex · p. 16

D. Convergence Analysis: Non-strongly Convex Case

For the non-strongly-convex case, the proof bounds consensus error, treats the average gradient as an inexact gradient, and applies a centralized gradient-descent analysis to the average iterate. The resulting O(1/t) convergence rate is preserved despite the inexactness.

D. Convergence Analysis: Non-strongly Convex Case

A. Experiments with different objective functions

The experiments compare the proposed algorithm with distributed and centralized gradient methods across three objective-function cases. The reported figures include objective-error plots, with a rescaled error plot for case III to examine sublinear convergence.

A. Experiments with different objective functions

  • The setup uses 100 agents and an Erdős–Rényi graph with connectivity probability 0.3.The text states this setup in the case-II description.Source: A. Experiments with different… · p. 21
  • The comparisons include vanishing- and fixed-step DGD, the algorithm in [22], and centralized gradient descent (CGD).The simulations use a common randomly initialized starting point for the agents.Source: A. Experiments with different… · p. 21
  • Case I uses square-loss linear regression, case II logistic-regression loss, and case III smooth convex functions whose Hessian is zero at the optimum.For case III, the functions are specified as f_i(x) = u(x) + b_i x.Source: A. Experiments with different… · p. 22
  • Figures 1–3 present simulation results for three objective-function cases, with case III also plotting t times objective error to examine O(1/t) decay.The text says cases I and II satisfy Assumptions 1 and 2, while case III satisfies only Assumption 1.Source: A. Experiments with different… · p. 22, Figure 3 · p. 23

B. Experiments with different graph sizes

The graph-size experiments test whether convergence behavior remains similar as the number of agents grows while the graph parameter σ stays similar. The conclusion states the method's rates for smooth convex and strongly convex smooth objectives.

B. Experiments with different graph sizes

VI. CONCLUSION

  • The conclusion states O(1/t) convergence for convex smooth objectives and linear convergence for strongly convex smooth objectives, comparable to centralized gradient methods except for constants.The method features a gradient-estimation scheme.Source: §VI · p. 24
  • The authors identify applying the gradient-estimation scheme to other first-order methods, including Nesterov gradient methods, as future work.Source: §VI · p. 24

APPENDIX

The appendix establishes properties of the gradient estimator and consensus error, proves a connection between estimation error and linear convergence, and derives a bound used in the main analysis.

A. Proof of Lemma 7 and Lemma 8

  • The proof uses double stochasticity of W to obtain a recursive identity for the average gradient estimate, initialized by setting s(0) equal to the initial gradient.It derives the average estimate at time t + 1 from its initial value and the change in the average gradient.Source: A. Proof of Lemma 7 and Lemma 8 · p. 26

B. Proof of Proposition 9

  • Proposition 9's proof shows that linear decay of gradient-estimation error yields linear decay of distance to the optimizer, and conversely.The forward direction also establishes linear decay of consensus error.Source: B. Proof of Proposition 9 · pp. 27–28

C. Proof of Lemma 10

D. Derivation of (23)

  • In deriving (23), the proof bounds the quantities θ1 and θ2 by θ and identifies the second row of z̃(k) with the consensus error.It then combines (34), (35), and (24) to obtain (23).Source: D. Derivation of (23) · pp. 29–30

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