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A Fast Distributed Proximal-Gradient Method

Annie I. Chen, Asuman Ozdaglar

arXiv:1210.2289v1cs.DCcs.LGstat.ML

TL;DR

The paper addresses distributed optimization of averaged private convex objectives with distinct differentiable components and a shared nondifferentiable component over time-varying networks. It combines proximal-gradient updates, multi-step consensus, and Nesterov acceleration, obtaining a 1/k convergence rate in communication rounds and verifying the method numerically.

  • Problem

    The paper studies how to optimize averages of private local objectives over networked agents when local differentiable terms differ but a nondifferentiable component is shared.

  • Method

    The method combines local gradient steps, multiple consensus communications, a proximal step for the shared nondifferentiable term, and Nesterov-type acceleration.

  • Results

    The method achieves convergence at rate 1/k in the number of communication rounds, faster than most existing distributed methods.

  • Takeaways & Limitations

    Multi-step consensus enables an inexact centralized proximal-gradient analysis with controlled errors and supports the reported faster convergence rate.

Abstract

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We present a distributed proximal-gradient method for optimizing the average of convex functions, each of which is the private local objective of an agent in a network with time-varying topology. The local objectives have distinct differentiable components, but they share a common nondifferentiable component, which has a favorable structure suitable for effective computation of the proximal operator. In our method, each agent iteratively updates its estimate of the global minimum by optimizing its local objective function, and exchanging estimates with others via communication in the network. Using Nesterov-type acceleration techniques and multiple communication steps per iteration, we show that this method converges at the rate 1/k (where k is the number of communication rounds between the agents), which is faster than the convergence rate of the existing distributed methods for solving this problem. The superior convergence rate of our method is also verified by numerical experiments.

I. INTRODUCTION

The paper studies distributed optimization of an average of private local objectives over networked agents, focusing on objectives with differentiable local terms and a shared nondifferentiable term. It proposes an accelerated proximal-gradient method using multi-step consensus to achieve a faster communication-round convergence rate than most existing distributed methods.

  • I. INTRODUCTION: The global objective averages private local functions, and agents cooperatively optimize it using local information and neighbor communication.The formulation covers distributed sensing and machine learning settings.
  • I. INTRODUCTION: Existing first-order distributed methods typically converge at rate 1/√n in the number of communication steps.The cited exception is a distributed gradient method with a diminishing step size under certain network conditions.
  • I. INTRODUCTION: The proposed method targets objectives fi(x) = gi(x) + h(x), with distinct differentiable gi and a common nondifferentiable h, over time-varying networks.Each iteration combines a gradient step, k-step consensus, a proximal step, and Nesterov-type acceleration.
  • I. INTRODUCTION: Multi-step consensus brings agent estimates close enough to reinterpret the distributed procedure as an inexact centralized proximal-gradient method with controlled error.This reformulation supports the convergence analysis using accelerated inexact proximal-gradient results.

II. PRELIMINARIES

The preliminaries introduce the concepts and convergence results needed for the later analysis, including proximal-operator properties and inexact centralized proximal-gradient rates.

  • II. PRELIMINARIES: The section prepares the analysis by reviewing proximal-operator properties and convergence results for inexact centralized proximal-gradient methods.The inexact method is characterized through errors introduced during optimization.

A. Properties of the Proximal Operator

The proximal operator is defined for a closed proper convex function and provides a single-valued transformation with subgradient and nonexpansiveness properties.

  • A. Properties of the Proximal Operator: For closed proper convex h and α > 0, the proximal operator maps x to the unique minimizer of a regularized optimization problem.The resulting map is single-valued.
  • A. Properties of the Proximal Operator: The section establishes basic proximal-operator properties, including nonexpansiveness.These properties support the later convergence analysis.
  • A. Properties of the Proximal Operator: The proximal output y satisfies α^-1(x − y) ∈ ∂h(y), linking the proximal step to a subgradient of h.The passage also expresses y as x − αz for an appropriate subgradient z.

B. Inexact Proximal-Gradient Method

The inexact proximal-gradient analysis connects summable distributed errors to accelerated convergence, using polynomial-geometric bounds generated by multi-step consensus.

  • B. Inexact Proximal-Gradient Method: The inexact recursion incorporates error vectors and ε-optimal proximal solutions into the accelerated proximal-gradient updates.The ε-optimal solutions characterize approximate proximal computations.
  • B. Inexact Proximal-Gradient Method: If the inexact accelerated proximal-gradient method’s error sequences are summable, it achieves the optimal convergence rate O(1/n^2).The result applies under the stated convexity, Lipschitz-gradient, lower-semicontinuity, and minimizer assumptions.
  • B. Inexact Proximal-Gradient Method: The analysis also notes that the accelerated inexact result holds for a constant step size α ≤ 1/L.Here L is the Lipschitz constant of the differentiable component’s gradient.
  • B. Inexact Proximal-Gradient Method: The distributed method’s errors can be bounded by polynomial-geometric sequences of the form p(k)γ^k, with γ ∈ (0, 1).These bounds arise from the distributed formulation and multi-step consensus.
  • B. Inexact Proximal-Gradient Method: Polynomial-geometric sequences are summable, allowing their error bounds to satisfy the summability condition required for accelerated inexact proximal-gradient convergence.The supporting proposition considers polynomials multiplied by a geometric factor with γ < 1.

III. MODEL AND METHOD

The paper studies distributed optimization with local objectives combining differentiable private terms and a common convex nondifferentiable term. Its method combines gradient, multi-step consensus, proximal, and Nesterov acceleration steps under time-varying network conditions.

  • III. MODEL AND METHOD: The model uses local objectives fi(x) = gi(x) + h(x), where gi is differentiable and h is common and nondifferentiable.The global objective is formed from private local objectives, and agents cooperatively seek its minimizer.
  • III. MODEL AND METHOD: The analysis assumes convex smooth differentiable components with Lipschitz gradients and a convex common nonsmooth component.The differentiable gradients and the nonsmooth term also have boundedness conditions used in the method’s analysis.
  • III. MODEL AND METHOD: The multi-step consensus stage repeatedly exchanges estimates using time-varying weight matrices to bring agents’ estimates toward agreement.Communication weights are organized through products of matrices A(t), with connectivity assumed over bounded time windows.
  • III. MODEL AND METHOD: Each iteration applies a local gradient step, multi-step consensus, a proximal step, and Nesterov-type acceleration.The consensus stage precedes the proximal step and helps bring agents’ estimates close together.
  • III. MODEL AND METHOD: Under the network assumptions, products of communication matrices converge geometrically and uniformly toward equal influence across agents.This geometric convergence controls the distance between each agent’s consensus output and the average estimate.

A. Formulation as an Inexact Method

The distributed algorithm is reformulated as an inexact centralized proximal-gradient method. Consensus errors and proximal-step discrepancies become explicit error sequences controlled by the multi-step communication stage.

  • A. Formulation as an Inexact Method: The distributed iterates can be represented as an inexact centralized proximal-gradient method for the average iterate.The formulation compares the distributed average with the exact centralized proximal step.
  • A. Formulation as an Inexact Method: The two error sequences are bounded using discrepancies between individual agent estimates, the consensus average, and the exact proximal update.The bounds rely on gradient Lipschitz continuity, convexity of h, bounded subgradients, and proximal-operator nonexpansiveness.
  • A. Formulation as an Inexact Method: Because consensus controls these errors, summability of both sequences yields the exact inexact-method convergence rate O(1/n^2).The distributed analysis therefore reduces the rate question to establishing summability of the communication-induced errors.

B. Convergence Rate Analysis

The convergence analysis proves that the communication-induced error sequences are summable and then translates iteration-wise convergence into a communication-round bound. Since iteration k uses k communication steps, the method achieves O(1/t) convergence in total communication steps.

  • B. Convergence Rate Analysis: The recursive iterate expressions provide the bounds needed to show that the distributed error terms have polynomial growth multiplied by geometric decay.These bounds are established through recursive expressions and a second-order polynomial estimate.
  • B. Convergence Rate Analysis: The proof establishes summability of the error sequences by bounding them with polynomial-geometric sequences.Polynomial-geometric summability is then used to invoke the inexact proximal-gradient convergence result.
  • B. Convergence Rate Analysis: Iteration k requires k communication steps, so completing n iterations requires a quadratic number of total communication steps.The resulting relation between iterations and communication rounds converts the iteration-wise bound into O(1/t).
  • B. Convergence Rate Analysis: f(x(t))−f(x∗) ≤ 2D/(4t + 1 − √(1 + 8t)) = O(1/t) after t communication steps.The theorem assumes a constant stepsize α ≤ 1/L and measures the objective gap at the average iterate.

C. Beyond O(1/t)

The section examines reducing communication steps below the default s_k=k while preserving convergence, and reports improved iteration and communication complexity under suitable choices.

  • Communication-step choices: O(log k) communication steps per iteration can be sufficient, smaller than the previous O(k) choice, under a condition involving the network parameter γ.The sufficient condition is derived from summability of γ^{s_k} weighted by polynomial factors.
  • Iteration complexity: O(1/n^2) convergence in iteration number is guaranteed when γ is known and the improved choice of s_k is used.Here n denotes the iteration number rather than total communication steps.
  • Communication complexity: The resulting rate is better than O(1/t) in total communication steps because the total time T(n) is bounded above by O(n^2).The comparison follows from relating iteration count to total communication steps.
  • Practical boundary: The improved schedule requires knowledge of γ, which may be unavailable when communication-network information or guarantees are unknown.The method can still be implemented with s_k=k in that case.

IV. NUMERICAL EXPERIMENTS

Numerical experiments evaluate distributed optimization methods on L1-regularized logistic regression using a 20 Newsgroups networked data split. The accelerated multi-step method achieves exact O(1/t) convergence and outperforms the alternatives shown.

  • Task and dataset: The experiment uses L1-regularized logistic regression on about 20,000 news articles evenly distributed across 20 topics.The task learns a classification model for a chosen topic label.
  • Distributed setup: Training data are distributed across 10 data centers with 1129 samples each, and communication uses randomly selected weight matrices generated from connected random graphs.All weight matrices satisfy the stated network assumption.
  • Results: The accelerated multi-step method attains exact O(1/t) convergence, outperforming all other evaluated methods.The comparison is shown in Figure 1.
  • Method comparison: The post-proximal consensus variant reaches only an O(1/t) error neighborhood, while the proposed pre-proximal consensus method achieves exact convergence.The comparison highlights the placement of the consensus stage relative to the proximal step.

V. CONCLUSION AND FUTURE WORK

The paper concludes that its distributed proximal-gradient method achieves O(1/t) convergence using multiple communication steps and Nesterov acceleration. It identifies extensions to distinct nondifferentiable components, single-step acceleration, and lower bounds as open problems.

  • Conclusion: The method solves averages of convex functions with distinct differentiable components and a common nondifferentiable component at O(1/t) communication complexity.The rate is measured in total communication steps and is reported as superior to most existing distributed methods.
  • Future work: Future work includes generalizing the result to distinct nondifferentiable functions, analyzing accelerated single-step proximal gradient, and deriving lower bounds.These are stated as open questions under the paper’s current framework.

APPENDIX

The appendix develops technical bounds for the distributed algorithm using induction, consensus-matrix properties, convexity, and proximal-operator nonexpansiveness. It assembles these estimates into the convergence proof.

  • Parameterization: The appendix uses β_k=(k−1)/(k+2) and notes that bounded subgradients imply a corresponding bound on x(k).The parameter choice simplifies the proof’s recursive expressions.
  • Technical bounds: The proof bounds update quantities by combining the algorithm’s component relations with gradient bounds and recursive estimates.Several steps substitute earlier inequalities into the main recursions before shifting indices.
  • Consensus and proximal bounds: Doubly stochastic communication matrices and convexity of the norm are used to control consensus-related terms.The appendix also invokes nonexpansiveness of the proximal operator.
  • Induction: An induction argument establishes the required bounds for k=2 and propagates them from k to k+1 by comparing coefficients.The proof selects finite constants satisfying the resulting coefficient inequalities.
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