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Balancing Communication and Computation in Distributed Optimization

Albert S. Berahas, Raghu Bollapragada, Nitish Shirish Keskar, Ermin Wei

arXiv:1709.02999v4math.OC

TL;DR

Iteration counts alone overlook the differing costs of communication and computation across applications. The paper proposes an adaptive weighted-cost framework and decomposes DGD steps to create customizable methods. Its reported comparisons show diminishing returns from additional consensus rounds, and the cost-based comparison depends on the chosen cost structure.

  • Problem

    Iteration-count measures overlook per-iteration complexity, while applications can assign different costs to communication and computation.

    Source: §I · p. 1, §II · p. 3

  • Method

    The paper weights communication and computation counts by application-dependent costs and decomposes DGD steps to create customizable variants.

    Source: §I · p. 2, §II · p. 3, §IV · p. 4

  • Results

    Additional consensus rounds improve the convergence neighborhood with diminishing returns; at cost budget 104 and cc = cg = 1, only the two-consensus-step method is reported competitive with and better than baseline DGD.

    Source: B. Numerical Results for DGDt · p. 8

  • Takeaways & Limitations

    NEAR-DGD+ with an increasing number of communication steps has an error neighborhood that shrinks to zero for any increasing sequence t(k) tending to infinity.

    Source: A. Convergence Analysis of NEAR-DGDt… · p. 12

    The reported cost-based comparison is highly dependent on the chosen cost structure, here cc = cg = 1.

    Source: B. Numerical Results for DGDt · p. 8

Abstract

from arXiv · show

Methods for distributed optimization have received significant attention in recent years owing to their wide applicability in various domains. A distributed optimization method typically consists of two key components: communication and computation. More specifically, at every iteration (or every several iterations) of a distributed algorithm, each node in the network requires some form of information exchange with its neighboring nodes (communication) and the computation step related to a (sub)-gradient (computation). The standard way of judging an algorithm via only the number of iterations overlooks the complexity associated with each iteration. Moreover, various applications deploying distributed methods may prefer a different composition of communication and computation. Motivated by this discrepancy, in this work we propose an adaptive cost framework which adjusts the cost measure depending on the features of various applications. We present a flexible algorithmic framework, where communication and computation steps are explicitly decomposed to enable algorithm customization for various applications. We apply this framework to the well-known distributed gradient descent (DGD) method, and show that the resulting customized algorithms, which we call DGD$^t$, NEAR-DGD$^t$ and NEAR-DGD$^+$, compare favorably to their base algorithms, both theoretically and empirically. The proposed NEAR-DGD$^+$ algorithm is an exact first-order method where the communication and computation steps are nested, and when the number of communication steps is adaptively increased, the method converges to the optimal solution. We test the performance and illustrate the flexibility of the methods, as well as practical variants, on quadratic functions and classification problems that arise in machine learning, in terms of iterations, gradient evaluations, communications and the proposed cost framework.

I. INTRODUCTION

The paper argues that iteration counts miss application-dependent communication and computation costs, and proposes a weighted cost framework with flexible DGD-based methods. It reviews related approaches, introduces the framework, and previews the methods and their theoretical and empirical evaluation.

I. INTRODUCTION

  • Distributed optimization methods are commonly judged by iteration count, which overlooks per-iteration complexity and differing application constraints.The paper motivates a measure that accounts for communication and computation separately.Source: §I · p. 1

A. Literature Review

  • The review contrasts iteration-counted distributed methods with communication-aware approaches, noting that prior designs often lack environment-specific flexibility.The closest work adjusts communication frequency to reduce runtime, while overlooking other aspects such as energy consumption.Source: A. Literature Review · p. 2

B. Contributions

  • The contributions are an application-adaptive weighted cost metric and tunable DGD-based classes that decompose communication and computation steps.NEAR-DGD+ nests the steps and, with communication increased at rate k, achieves exact linear convergence; the paper also evaluates variants on quadratic and logistic-regression problems.Source: B. Contributions · pp. 2–3

II. ADAPTIVE COST FRAMEWORK

  • The proposed cost adds communication and computation counts weighted by application-dependent parameters cc and cg.The parameters can represent energy, runtime, or a combination of both, and can reflect different costs in different environments.Source: §II · p. 3

III. PRELIMINARIES

  • The preliminaries introduce a compact reformulation and review DGD as the building block for the paper’s flexible algorithms.The paper organizes its methods around solving the reformulated distributed optimization problem.Source: §III · p. 3

A. Equivalent Reformulations

The section expresses the consensus optimization problem in compact form using a consensus matrix. Its properties characterize agreement across the connected network and quantify the contraction of disagreement.

A. Equivalent Reformulations

  • The reformulation concatenates all local variables into x and uses W to represent network weights.W is an n × n matrix whose entries encode connections between agents.Source: A. Equivalent Reformulations · p. 3
  • W is symmetric and doubly stochastic, with positive diagonal entries and positive off-diagonal entries exactly for neighboring agents.With connected network consensus, (W ⊗ I_p)x = x holds exactly when all local copies agree, making the compact and consensus formulations equivalent.Source: A. Equivalent Reformulations · p. 3
  • The second-largest eigenvalue magnitude β of W is strictly below 1 and is used to characterize network consensus behavior.W has one eigenvalue equal to 1 and all other eigenvalues have magnitude less than 1.Source: A. Equivalent Reformulations · pp. 3–4

B. Distributed Gradient Descent

DGD updates each agent’s estimate using its local gradient and information from neighboring estimates. Its convergence behavior depends on the stepsize and convexity assumptions.

B. Distributed Gradient Descent

  • DGD is a first-order method in which each agent combines a local gradient step with information exchanged with its neighbors.The update uses the local estimate and a weighted average of neighboring estimates before stepping in the negative local-gradient direction.Source: B. Distributed Gradient Descent · p. 4
  • The standard convergence analysis assumes Lipschitz-continuous local gradients and strong convexity of the global or local objectives.The local strong-convexity assumption implies strong convexity of the global objective.Source: B. Distributed Gradient Descent · p. 4
  • With diminishing stepsizes, DGD converges sublinearly to a feasible optimal solution; with an appropriate constant stepsize, it converges linearly to a neighborhood instead.The constant-stepsize limit may violate the equality constraint in the consensus formulation.Source: B. Distributed Gradient Descent · p. 4

IV. DGDt: A DISTRIBUTED GRADIENT DESCENT VARIANT

DGDt generalizes DGD by separating its communication and computation steps and performing multiple consensus rounds per iteration. The added communication can improve feasibility, with the method suited to settings where communication is relatively cheap.

IV. DGDt: A DISTRIBUTED GRADIENT DESCENT VARIANT

  • DGDt decomposes and rearranges DGD’s communication and computation steps to create methods with different step counts.The section focuses on constant-stepsize implementations.Source: §IV · p. 4
  • DGDt performs t consensus steps per iteration to improve solution quality without sacrificing convergence speed.The method is motivated by DGD’s constant-stepsize infeasibility with respect to the consensus equality constraint.Source: §IV · p. 4
  • More consensus rounds bring the solution closer to feasibility, but require sequential communications that cost more per iteration.The method may be desirable when communication costs much less than computation.Source: §IV · p. 5

A. Convergence Analysis of DGDt

With a constant stepsize, DGDt converges to a neighborhood of the optimum that is smaller than standard DGD’s, but multiple consensus steps alone do not guarantee exact convergence. Under the stated assumptions and stepsize condition, its local iterates converge from zero initialization.

A. Convergence Analysis of DGDt

  • Under the stated assumptions and stepsize condition, DGDt’s local iterates converge when initialized at zero.Source: A. Convergence Analysis of DGDt · pp. 5–6
  • The analysis bounds both the deviation of local iterates from their mean and the resulting deviation in their gradients.Source: A. Convergence Analysis of DGDt · p. 6
  • DGDt converges to a smaller neighborhood of the optimum than standard DGD with a constant stepsize.The neighborhood depends on the stepsize, the second-largest eigenvalue of W, and the number of consensus steps.Source: A. Convergence Analysis of DGDt · pp. 7–8
  • Multiple consensus steps improve DGDt’s convergence neighborhood but cannot eliminate its error term or guarantee convergence to the solution.The analysis states that the error term remains nonzero even as the number of consensus steps tends to infinity.Source: A. Convergence Analysis of DGDt · p. 8

B. Numerical Results for DGDt

On a quadratic problem, multiple consensus steps improved the convergence neighborhood, with diminishing gains as the number of steps increased. Under the tested equal communication and computation costs, DGD2 was the only variant reported as competitive with and better than baseline DGD at a fixed cost budget.

B. Numerical Results for DGDt

  • The experiment compared DGD variants with 1, 2, 5, or 10 consensus steps by relative error across iterations, cost, gradient evaluations, and communications.The quadratic setup used n = 10 agents, p = 10 dimensions, and condition number 10^2.Source: B. Numerical Results for DGDt · p. 8
  • Figure 1 illustrates that multiple communication rounds improve the neighborhood of convergence, with diminishing returns: the neighborhood improves substantially when increasing from 1 consensus step to 2 consensus steps.Source: B. Numerical Results for DGDt · p. 8
  • At a fixed cost budget with cc = cg = 1, DGD2 was the only tested variant reported as competitive with and better than baseline DGD.The comparison is shown alongside relative error in Figure 1.Source: B. Numerical Results for DGDt · p. 8
  • The relative performance pattern remained similar across tested quadratic problems with different characteristics and graph sizes under cc = cg = 1.Source: B. Numerical Results for DGDt · p. 8

V. NEAR-DGD

NEAR-DGD decomposes gradient and consensus operations and nests them in an alternating sequence, aiming to use newer information for gradient computation. Its variants allow multiple consensus steps per gradient step, including an adaptive version that increases the number of consensus steps over iterations.

V. NEAR-DGD

  • The framework separates communication and computation operations, enabling customization to balance their costs for different applications.Source: §V · p. 8
  • NEAR-DGD nests and alternates the gradient and consensus operations, computing the gradient after the consensus step.Unlike standard DGD, the two steps are inherently sequential; each NEAR-DGD iteration uses the same amount of communication and computation as DGD.Source: §V · pp. 8–9
  • NEAR-DGDt extends NEAR-DGD by performing multiple consensus steps per gradient step.Source: §V · p. 9
  • NEAR-DGD+ allows the number of nested consensus steps to vary and increase across iterations.The method uses t(k) consensus operations at iteration k.Source: §V · p. 9

A. Convergence Analysis of NEAR-DGDt and NEAR-DGD+

With fixed consensus steps, NEAR-DGD^t converges linearly only to a neighborhood of the optimum; increasing consensus steps in NEAR-DGD+ removes that error and can yield exact linear convergence.

A. Convergence Analysis of NEAR-DGDt and NEAR-DGD+

  • For NEAR-DGD^t, the average iterates converge linearly to an optimal-solution neighborhood whose size depends on the step length, the second-largest eigenvalue of W, and the consensus-step count.The local agent estimates also converge to a neighborhood under the stated assumptions and step-length conditions.Source: A. Convergence Analysis of NEAR-DGDt… · p. 11
  • For NEAR-DGD+, any strictly increasing, unbounded consensus-step schedule makes the error neighborhood shrink to zero and the iterates converge to the optimum.The neighborhood size is O(β^(2t(k))).Source: A. Convergence Analysis of NEAR-DGDt… · p. 12
  • With t(k) = k, NEAR-DGD+ converges to the solution at an R-linear rate, and its local x iterates converge while its local y iterates do not.The result assumes the stated assumptions and step-length conditions.Source: A. Convergence Analysis of NEAR-DGDt… · p. 12
  • The NEAR-DGD+ work-complexity analysis counts gradient evaluations and communication rounds needed for an ε-accurate average iterate, but only local x iterates are covered because local y iterates do not converge.The stated work complexity is under Theorem V.6's conditions.Source: A. Convergence Analysis of NEAR-DGDt… · p. 12

B. Numerical Results for NEAR-DGD and NEAR-DGD+

The numerical study tests NEAR-DGD^t and NEAR-DGD+ on quadratic and machine-learning classification problems to assess whether their theoretical convergence behavior appears in practice. It compares six NEAR-DGD variants with DGD, including practical NEAR-DGD+ schedules.

B. Numerical Results for NEAR-DGD and NEAR-DGD+

  • The experiments assess whether NEAR-DGD^t converges to a solution neighborhood and NEAR-DGD+ to the solution on strongly convex objectives.Tests include quadratic and classification problems arising in machine learning.Source: B. Numerical Results for NEAR-DGD and… · p. 13
  • Six NEAR-DGD variants are compared with DGD, including schedules that increase consensus steps each iteration or double them every 500 iterations.NEAR-DGD (10,1,−), with multiple gradient steps per iteration, is included for comparison although the paper's theory does not cover it.Source: B. Numerical Results for NEAR-DGD and… · p. 13

1) Quadratic Problems:

On the quadratic problem, increasing-consensus NEAR-DGD+ variants converge to the solution, while fixed-consensus methods reach only a neighborhood. Practical NEAR-DGD+ variants perform better when measured by cost or communications, and cost rankings depend on the application's communication-to-computation prices.

1) Quadratic Problems:

  • Methods that increase consensus steps converge to the solution, whereas methods with fixed consensus steps converge only to a neighborhood.NEAR-DGD reaches a better neighborhood than DGD on this problem, which the authors say is not predicted by theory and is probably problem-specific.Source: 1) Quadratic Problems · p. 13
  • The practical NEAR-DGD+ variants perform significantly better than other methods in cost or communications, although NEAR-DGD+ (1,1,k) is fastest by iterations.The iteration advantage reflects its higher work per iteration; the cost comparison assumes cc = cg = 1.Source: 1) Quadratic Problems · p. 13, Figure 2 · p. 13
  • The practical NEAR-DGD+ variants are best by cost across the tested cost structures, but the benefit of multiple consensus steps varies with the structure.Figure 3 compares cc = 1, cg = 10; cc = 1, cg = 1; and cc = 10, cg = 1.Source: 1) Quadratic Problems · pp. 13–14, Figure 3 · p. 14

2) Binary Classification Logistic Regression Problems:

On the mushroom logistic-regression problem, increasing-consensus variants converge to the optimum while other methods reach only a neighborhood. NEAR-DGD (1,1,−) appears competitive by cost at a fixed budget, but its error stagnates as NEAR-DGD+ continues improving.

2) Binary Classification Logistic Regression Problems:

  • On the mushrooms dataset, variants that increase the number of consensus steps converge to the optimal solution; the reported setup uses 10 nodes, 114 dimensions, and 812 local data points per node.Source: 2) Binary Classification Logistic… · p. 14
  • NEAR-DGD (1,1,−) reaches a significantly better neighborhood than DGD and appears cost-competitive with NEAR-DGD+ at a fixed budget, but its error stagnates while NEAR-DGD+ continues improving.The cost plot ends when DGD terminates, so it does not show the full experiment.Source: 2) Binary Classification Logistic… · p. 14

VI. FINAL REMARKS AND FUTURE WORK

The paper applies its adaptive cost framework to DGD, decomposing communication and computation to construct three flexible methods. NEAR-DGD+ converges to the optimum with constant step length, while broader applications and cost factors remain future work.

VI. FINAL REMARKS AND FUTURE WORK

  • The framework decomposes DGD’s communication and computation steps to construct DGD^t, NEAR-DGD^t, and NEAR-DGD+.The methods are evaluated on quadratic and binary classification logistic regression problems.Source: §VI · p. 14
  • NEAR-DGD+ converges to the optimal solution with constant steplength and, for strongly convex functions, at a linear rate.Standard DGD with constant steplength converges only to a neighborhood of the optimum.Source: §VI · p. 14
  • Future work includes applying the decomposition to other distributed optimization methods and adding memory access, quantization, and dynamic environments to the cost framework.Automatically adjusting communication and computation steps to the environment is also under investigation.Source: §VI · p. 14

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