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Asynchronous Distributed ADMM for Large-Scale Optimization- Part I: Algorithm and Convergence Analysis

Tsung-Hui Chang, Mingyi Hong, Wei-Cheng Liao, Xiangfeng Wang

arXiv:1509.02597v2cs.DCcs.LGeess.SY

TL;DR

The paper addresses synchronization bottlenecks in distributed ADMM for large-scale learning over heterogeneous star networks. It proposes asynchronous distributed ADMM and analyzes it under bounded delays, showing convergence to KKT points for general, possibly non-convex problems when parameters are chosen appropriately. It also shows that small implementation changes can substantially alter convergence behavior.

  • Problem

    Synchronous distributed optimization is limited by the slowest worker, while convergence analysis for asynchronous distributed ADMM, especially beyond convex problems, remains limited.

  • Method

    The paper generalizes synchronous distributed ADMM to an asynchronous star-network algorithm in which the master updates using information from a partial set of workers.

  • Results

    For general and possibly non-convex problems, AD-ADMM converges to KKT points when its parameters are chosen according to the maximum network delay.

  • Takeaways & Limitations

    Asynchronous ADMM can support distributed optimization beyond standard convex settings, but its implementation must be handled carefully because small modifications can jeopardize convergence.

Abstract

from arXiv · show

Aiming at solving large-scale learning problems, this paper studies distributed optimization methods based on the alternating direction method of multipliers (ADMM). By formulating the learning problem as a consensus problem, the ADMM can be used to solve the consensus problem in a fully parallel fashion over a computer network with a star topology. However, traditional synchronized computation does not scale well with the problem size, as the speed of the algorithm is limited by the slowest workers. This is particularly true in a heterogeneous network where the computing nodes experience different computation and communication delays. In this paper, we propose an asynchronous distributed ADMM (AD-AMM) which can effectively improve the time efficiency of distributed optimization. Our main interest lies in analyzing the convergence conditions of the AD-ADMM, under the popular partially asynchronous model, which is defined based on a maximum tolerable delay of the network. Specifically, by considering general and possibly non-convex cost functions, we show that the AD-ADMM is guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points as long as the algorithm parameters are chosen appropriately according to the network delay. We further illustrate that the asynchrony of the ADMM has to be handled with care, as slightly modifying the implementation of the AD-ADMM can jeopardize the algorithm convergence, even under a standard convex setting.

I. INTRODUCTION

The paper develops asynchronous distributed ADMM for large-scale learning over star networks, addressing synchronization bottlenecks while analyzing convergence under bounded delays. It extends convergence guarantees to general, possibly non-convex problems and warns that implementation details can alter convergence behavior.

  • Motivation: Large-scale learning with distributed data and computing resources challenges centralized optimization methods and motivates efficient distributed algorithms.The target settings include problems with many training samples or features, including LASSO, logistic regression, SVM, and sparse PCA.
  • Approach: The proposed AD-ADMM lets a master coordinate partial worker updates in a star topology, improving computation efficiency while retaining distributed parallelism.Workers solve partitioned subproblems and send variable information to the master, which summarizes and broadcasts the result.
  • Motivation: Synchronous distributed methods wait for all workers, so heterogeneous computation and communication delays make progress depend on the slowest worker.Asynchronous updates let the master proceed when only a subset of workers has returned information, although this introduces delayed gradients.
  • Efficiency: The paper reports that asynchronous distributed optimization can reduce computation time nearly linearly with the number of workers.This behavior is reported for asynchronous protocols in distributed optimization, particularly when worker delays differ.
  • Convergence: The analysis establishes convergence to KKT points for general and possibly non-convex problems when algorithm parameters are selected according to the maximum network delay.The result uses a bounded-delay model and provides deterministic worst-case convergence conditions rather than statistical assumptions on workers.
  • Caveat: A slight modification of asynchronous ADMM can produce different convergence conditions and even destroy convergence, including for convex problems.The paper supports this warning with an alternative asynchronous implementation and numerical results.

II. APPLICATIONS AND DISTRIBUTED ADMM

The paper formulates large-scale learning and related optimization tasks as consensus problems solvable by distributed ADMM over a star network. Synchronous updates exploit parallelism but remain bottlenecked by the slowest worker in heterogeneous settings.

  • Applications: Regularized empirical risk minimization covers important learning problems including LASSO, logistic regression, SVM, and sparse PCA.The loss depends on training samples, labels, and parameters, while Ω(w) is a convex regularizer.
  • Applications: Splitting training samples across N workers enables distributed optimization, but very large sample counts or parameter dimensions make the learning problem challenging.The paper specifically notes difficulty when the number of samples is very large and when n ≫ m.
  • Distributed ADMM: Distributed ADMM reformulates problem (1) as a consensus problem with variables x_i constrained by x_i = x_0 for all workers.The formulation is equivalent to the original optimization problem and supports distributed solution over a star network.
  • Distributed ADMM: The synchronous distributed ADMM updates the master and workers in parallel, exchanging current variables and dual variables across the star network.The master optimizes x_0, workers optimize their local variables, and the method has established convergence results under several assumptions.
  • Distributed ADMM: Synchronous execution waits for every worker at each iteration, so heterogeneous computation and communication delays force the master and faster workers to idle.Consequently, the optimization pace is determined by the slowest worker and available parallel resources are not fully utilized.

III. ASYNCHRONOUS DISTRIBUTED ADMM

The proposed AD-ADMM removes the synchronization lock by updating from partial worker arrivals while controlling stale information through bounded-delay rules. This can increase update frequency and reduce elapsed time, but delayed information may require more iterations and careful parameter choices.

  • Algorithm Description: AD-ADMM lets the master update when it receives variables from a partial set of workers, rather than waiting for all workers.The master updates only variables associated with arrived workers and sends the current x_0 to those workers.
  • Performance Trade-offs: Removing the synchronization lock allows the master and fast workers to update more frequently, with six asynchronous updates versus two synchronous updates over the illustrated period.The improvement arises because faster nodes need not wait for the slowest worker.
  • Delay Model: The partially asynchronous model imposes a maximum delay τ, requiring every worker to arrive at least once within each τ-iteration window.Worker information used by the master is therefore at most τ iterations old.
  • Algorithm Description: AD-ADMM adds a proximal term with γ > 0 to the master update, introduces delay counters, and requires at least A arrived workers before proceeding.When A = N, the method reduces to synchronous distributed ADMM.
  • Performance Trade-offs: Delayed variable information means asynchronous ADMM may require more iterations to reach the same accuracy, although its improved update frequency can still yield faster convergence in time.The paper says this benefit is particularly relevant when worker computation and communication delays differ.

B. Convergence Analysis

Under stated smoothness, boundedness, and delay assumptions, the analysis establishes deterministic convergence of AD-ADMM to KKT points, including for non-convex worker costs. Convergence requires parameter choices that account for network delay, worker count, and problem curvature.

  • Main convergence result: The convergence result applies without convexity assumptions on the worker functions f_i.The paper explicitly states convergence to KKT points even when the f_i are non-convex.
  • Main convergence result: AD-ADMM converges to the set of KKT points when its penalty parameters are sufficiently large.The iterates generated by the worker and master updates are bounded and have limit points satisfying the KKT conditions.
  • Parameter effects: Longer network delays require more cautious master updates through a larger γ, which acts inversely as the step size for x_0.The worst-case γ increases with the square order of the delay τ, and also increases with the number of workers when τ is fixed.
  • Parameter effects: A large ρ may be essential for proper AD-ADMM convergence, especially on non-convex problems.The paper points to simulations as evidence for this non-convex setting.
  • Comparison with prior analysis: The analysis is deterministic, unlike cited results that apply only to convex problems or establish convergence in expectation.The comparison notes that a realization can theoretically fail to converge under expectation-based conditions, whereas the present results hold deterministically.
  • Proof mechanism: For ρ > L, the worker subproblem becomes strongly convex and is therefore globally solvable even when f_i is non-convex.This follows from the lower Hessian bound implied by the Lipschitz-gradient assumption.

IV. COMPARISON WITH AN ALTERNATIVE SCHEME

The paper compares AD-ADMM with an alternative implementation that assigns dual-variable updates to the master. Although the schemes are equivalent synchronously, asynchronous execution can give them different convergence conditions and practical behavior.

  • Alternative implementation: Algorithm 4 moves the updates of the dual variables {λ_i} to the master, leaving workers to update only {x_i}.The alternative scheme retains the master update of x_0 while changing responsibility for the dual variables.
  • Synchronous comparison: In synchronous networks, Algorithms 2 and 4 are equivalent up to a change in update order and have the same convergence conditions.The equivalence is explicitly limited to the synchronous protocol.
  • Asynchronous comparison: In asynchronous networks, the two algorithms may require distinct convergence conditions and behave very differently in practice.This motivates a separate convergence analysis for Algorithm 4.
  • Algorithm 4 conditions: Because ρ is the dual-gradient ascent step size in Algorithm 4, larger delays require slower updates of the dual variables.The stated interpretation connects the delay-dependent condition to the master’s movement of λ_i.

V. SIMULATION RESULTS

The simulation section examines AD-ADMM’s convergence behavior with respect to master iterations. It studies a non-convex sparse PCA problem and compares the alternative and original schemes on LASSO.

  • Simulation design: The simulations examine AD-ADMM convergence with respect to the master’s iteration number k.The experiments implement Algorithm 3 on a desktop computer.
  • Simulation tasks: The experiments first evaluate AD-ADMM on a non-convex sparse PCA problem.
  • Simulation tasks: The experiments then compare Algorithm 4 with Algorithm 2 on the LASSO problem.

A. Example 1: Sparse PCA

The experiments show that AD-ADMM converges on a non-convex sparse PCA problem and remains effective across delays, whereas a slight asynchronous modification can diverge or slow substantially.

  • A. Example 1: Sparse PCA: AD-ADMM converges properly on the non-convex sparse PCA problem for the tested delay values.The experiment uses 1000 × 500 sparse random matrices, N = 32, and asynchronous worker arrivals.
  • A. Example 1: Sparse PCA: The tested AD-ADMM instance works with γ = 0 across different τ values, although the theorem recommends larger γ in the worst case.This is an empirical observation for the considered example, not a general replacement for the theorem’s parameter guidance.
  • B. Example 2: LASSO: For LASSO, Algorithm 2 with ρ = 500 converges across delay values, while Algorithm 4 diverges asynchronously at τ = 3.Algorithm 4 can be made convergent by reducing ρ, but this requires progressively smaller values as delay increases.
  • B. Example 2: LASSO: Algorithm 4 with ρ = 1 is much slower than Algorithm 2, and it always diverges for the larger n = 1000 setting.The larger setting removes strong convexity of the cost functions; the simulations indicate strong convexity may be practically necessary for Algorithm 4.
  • VI. Concluding Remarks: The simulations conclude that Algorithm 2 significantly outperforms Algorithm 4 asynchronously despite similar synchronous convergence behavior.The paper also observes possible linear convergence for some structured instances.

APPENDIX A

Appendix A derives bounds used in the convergence analysis by exploiting strong convexity, Lipschitz gradients, optimality conditions, and Young’s inequality.

  • APPENDIX A: Optimality conditions and convexity of h provide inequalities used to bound the auxiliary variable update.These inequalities are subsequently combined with earlier relations in the proof.
  • APPENDIX A: The proof bounds differences in the Lyapunov-function analysis using strong-convexity properties of the augmented subproblem.The relevant convexity parameter is expressed through ρ and the lower curvature bound of fi.
  • APPENDIX A: Young’s inequality with selected δ values yields the stated bound in equation (27).The proof applies it to the fourth and fifth terms on the right-hand side.

APPENDIX B

Appendix B completes a bound by accounting for delayed iterates and the bounded number of active workers in the asynchronous analysis.

  • APPENDIX B: The proof uses the delay relation and the assumption that fewer than S workers are active at each iteration.These assumptions control how often historical differences appear in the summation.
  • APPENDIX B: Combining the intermediate inequality with the preceding bound yields equation (32).The appendix states this final substitution as the completion of the argument.

APPENDIX C

Appendix C derives a lower bound for the augmented Lagrangian by applying the descent relation and Lipschitz-gradient assumptions.

  • APPENDIX C: The proof recalls equation (29) and applies it to equation (26) to obtain an intermediate inequality.This step connects the previously established descent relation to the current Lyapunov analysis.
  • APPENDIX C: Lipschitz continuity of ∇fi justifies the descent lemma used in the derivation.The combined inequalities provide the lower bound for Lρ.

APPENDIX D

Appendix D reformulates the asynchronous distributed ADMM updates and derives bounds by combining optimality conditions, delay relations, convexity, and duality arguments. Summing these bounds over iterations yields the stated result.

  • Equivalent formulation: The appendix introduces an equivalent update representation through variable rescaling and a transformed penalty parameter.The changes of variables are λ_i = eλ_i/α and ρ = β/α.
  • Delay control: Delay-dependent terms are bounded using the last message-receipt iterations, Young’s inequality, and the fact that delayed error terms recur at most τ − 1 times.These bounds control terms involving stale worker and dual variables.
  • Optimality analysis: Worker updates are analyzed separately for workers participating in the current iteration and workers outside that set.The proof uses optimality conditions for i ∈ A_k and i ∈ A_c before summing the resulting inequalities.
  • Final bound: Summing the main inequality over k = 0, …, K − 1 and averaging produces the appendix’s bound, which is finalized through duality arguments.The final step combines relations (A.57) and (A.59) to obtain (52).
  • Convergence inequality: Convexity of the worker costs and h is used when substituting the intermediate bounds into the main inequality.The resulting inequality is then combined with parameter choices and the rescaled variables.

Algorithm 2 Asynchronous Distributed ADMM for (4).

The algorithm specifies asynchronous coordination between a master and workers. The master initializes and broadcasts variables, accepts qualifying worker messages, updates its state, and repeats until a stopping criterion is met.

  • Termination: After processing an iteration, the master increments k and continues until a predefined stopping criterion is satisfied.The same termination structure appears in the asynchronous distributed ADMM procedures.
  • Algorithm variant: Algorithm 4 is presented as an alternative implementation of asynchronous distributed ADMM.Its master and worker procedures are specified separately.
  • Master initialization: The master initializes x^0, broadcasts it to the workers, and sets k = 0 together with message-tracking counters.The counters are initialized as d_1 = · · · = d_N = 0.
  • Master coordination: At each iteration, the master waits for messages from a qualifying worker subset A_k satisfying |A_k| ≥ A and d_i < τ − 1.This acceptance rule incorporates both a minimum number of workers and a delay constraint.
  • Worker updates: The workers receive master variables, perform their asynchronous update steps, and send the resulting worker quantities back to the master.The worker routines explicitly wait for master messages before continuing.
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