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ByRDiE: Byzantine-resilient distributed coordinate descent for decentralized learning

Zhixiong Yang, Waheed U. Bajwa

arXiv:1708.08155v4cs.LGcs.DCmath.OCstat.ML

TL;DR

The paper addresses Byzantine failures in high-dimensional fully distributed learning, where existing methods do not provide efficient and practical protection. It develops ByRDiE, a Byzantine-resilient coordinate-descent algorithm, and analyzes its behavior. ByRDiE remains resilient in convex and nonconvex experiments, including performance close to centralized coordinate descent in one neural-network task.

  • Problem

    Existing distributed learning methods assume faultless nodes, while Byzantine-resilient approaches have limitations for high-dimensional fully distributed learning.

  • Method

    ByRDiE splits distributed empirical risk minimization into coordinate-wise scalar subproblems and solves them using Byzantine-resilient procedures.

  • Results

    ByRDiE achieves theoretical guarantees under mild assumptions and numerical resilience in convex and nonconvex distributed learning; on Iris, it comes close to centralized CD while DGD breaks down with one Byzantine node.

  • Takeaways & Limitations

    ByRDiE provides a coordinate-descent approach for distributed learning that tolerates Byzantine failures across convex and nonconvex tasks.

Abstract

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Distributed machine learning algorithms enable learning of models from datasets that are distributed over a network without gathering the data at a centralized location. While efficient distributed algorithms have been developed under the assumption of faultless networks, failures that can render these algorithms nonfunctional occur frequently in the real world. This paper focuses on the problem of Byzantine failures, which are the hardest to safeguard against in distributed algorithms. While Byzantine fault tolerance has a rich history, existing work does not translate into efficient and practical algorithms for high-dimensional learning in fully distributed (also known as decentralized) settings. In this paper, an algorithm termed Byzantine-resilient distributed coordinate descent (ByRDiE) is developed and analyzed that enables distributed learning in the presence of Byzantine failures. Theoretical analysis (convex settings) and numerical experiments (convex and nonconvex settings) highlight its usefulness for high-dimensional distributed learning in the presence of Byzantine failures.

I. INTRODUCTION

Distributed learning addresses data spread across networked nodes, but Byzantine failures can undermine existing algorithms. The paper motivates resilient, scalable learning through coordinate descent in fully distributed settings.

  • I. INTRODUCTION: Distributed learning minimizes risk using data held across different locations rather than gathered centrally.The paper frames this as distributed optimization over local empirical risks.
  • I. INTRODUCTION: Byzantine failures occur when nodes arbitrarily deviate from intended behavior and can jeopardize the whole network.Examples include cyber attacks, equipment malfunctions, and undetected failures.
  • I. INTRODUCTION: A single Byzantine node can cause most state-of-the-art distributed learning algorithms to fail.This motivates algorithms that explicitly tolerate Byzantine behavior.
  • I. INTRODUCTION: Existing Byzantine-resilient methods face scalability and correctness limitations for high-dimensional, fully distributed learning.Some vector-valued methods require neighborhood sizes that grow linearly with dimensionality, while others target minimizers of convex combinations rather than exact average losses.
  • I. INTRODUCTION: ByRDiE addresses these challenges by decomposing vector-valued learning into scalar coordinate subproblems solved with Byzantine-resilient procedures.The algorithm uses an outer coordinate-descent loop and an inner resilient scalar-optimization loop.
  • I. INTRODUCTION: The paper studies consensus, statistical convergence, and algorithmic convergence for distributed empirical risk minimization under Byzantine failures.The formulation assumes connected networks, local training sets, and i.i.d. samples, with theoretical analysis under stated regularity conditions.

A. Byzantine Failure Model

The paper models decentralized learning with Byzantine nodes that may deviate from prescribed updates or broadcasts, under a graph-redundancy assumption designed to support resilience. Its goal is consensus among nonfaulty nodes and convergence toward the statistical optimum despite such failures.

  • Failure model: Byzantine nodes may use arbitrary update functions or broadcast values other than their intended updates.
  • Failure model: The network is assumed to contain at most b Byzantine nodes, with nonfaulty nodes forming the set J′.The exact value of b need not be known; an upper bound may be used.
  • Graph assumption: Reduced graphs remove Byzantine nodes and up to b additional incoming edges per nonfaulty node, and each must contain a source component of at least b + 1 nodes.
  • Graph assumption: Assumption 3 provides redundancy for Byzantine tolerance, but efficiently certifying it remains an open problem.For the Erdős–Rényi graphs used in experiments, the assumption was typically satisfied when average incoming degree was sufficiently large relative to the number of Byzantine nodes.
  • Learning objectives: The learning objectives are consensus among nonfaulty nodes as iterations grow and convergence to the statistical optimum as their training samples grow.These objectives require both statistical and algorithmic convergence analysis for distributed empirical risk minimization.
  • Learning objectives: ByRDiE addresses these objectives by applying Byzantine-resilient coordinate descent to distributed learning, with the algorithm detailed in Algorithm 1.The approach breaks vector-valued optimization into scalar-valued coordinate problems and uses Byzantine-resilient processing for them.

A. Algorithmic Details

ByRDiE cycles through scalar coordinate subproblems and uses broadcast, screening, and update operations to limit Byzantine influence. Its analysis establishes consensus and convergence properties under stated assumptions, while parameter T trades off consensus and convergence rate.

  • Algorithmic Details: ByRDiE splits the ERM problem into P one-dimensional coordinate subproblems and solves each with a Byzantine-resilient scalar method.The outer loop performs coordinate descent, while the inner loop handles each scalar optimization problem.
  • Algorithmic Details: During each inner-loop iteration, nonfaulty nodes broadcast coordinate values, screen received values, and update their local coordinate.Screening rejects values that are too large or too small relative to values generated by nonfaulty nodes.
  • Algorithmic Details: The algorithm uses diminishing stepsizes that are square-summable but not summable, and completes each coordinate-descent iteration after all P subproblems are solved.
  • Algorithmic Details: The parameter T trades off consensus among nonfaulty nodes against convergence rate, with theory focused on T →∞ and T = 1.
  • Theoretical Guarantees: Consensus: ByRDiE achieves consensus among nonfaulty nodes as r̄ →∞ and/or T →∞, with a sublinear consensus rate under ρ(r) = O(1/r) when T = 1.
  • Theoretical Guarantees: Consensus: For T →∞ and T = 1, the ByRDiE output converges to the statistical optimum as r̄ becomes large.
  • Algorithmic Details: The screening rule lets nonfaulty nodes replace Byzantine data with convex combinations of data received from nonfaulty neighbors.
  • Theoretical Guarantees: Consensus: The row-stochastic matrix products governing the updates converge to a steady-state matrix with identical stochastic rows.

C. Theoretical Guarantees: Convergence for T →∞

For T →∞, ByRDiE’s coordinate-wise subproblems approach minimizers of convex combinations of nonfaulty nodes’ empirical risks, while nonfaulty nodes reach consensus. Its statistical and algorithmic convergence are established under the stated assumptions.

  • C. Theoretical Guarantees: Convergence for T →∞: As T →∞, each ByRDiE coordinate subproblem converges to a minimizer of a convex combination of nonfaulty nodes’ local empirical risks.The resulting consensus point is characterized coordinate by coordinate.
  • C. Theoretical Guarantees: Convergence for T →∞: With high probability, ByRDiE’s coordinate-wise outputs achieve nearly the same statistical risk as the corresponding statistical risk minimizers.The result applies uniformly across coordinate updates under Case I.
  • C. Theoretical Guarantees: Convergence for T →∞: Under the stated assumptions, ByRDiE’s iterates converge to the statistical risk minimizer as the number of training samples grows.Strict convexity gives uniqueness of the minimum in the corresponding convergence statement.
  • C. Theoretical Guarantees: Convergence for T →∞: The effective sample size satisfies N ≤ N_eff ≤ |J′|N, placing ByRDiE’s learning rate between local and idealized centralized learning rates.N_eff is defined as N/ā^2, and the bounds hold under the stated conditions.
  • C. Theoretical Guarantees: Convergence for T →∞: ByRDiE requires r = O(1/ϵ) iterations to reduce excess statistical risk to ϵ, yielding sublinear algorithmic convergence with high probability.This guarantee holds even in the presence of Byzantine failures.

D. Theoretical Guarantees: Convergence for T = 1

For T = 1, ByRDiE updates one coordinate at a time and analyzes the resulting sequence of iterates. Under the stated assumptions, the nonfaulty nodes asymptotically reach consensus and converge in probability to the statistical risk minimizer.

  • D. Theoretical Guarantees: Convergence for T = 1: T = 1 corresponds to taking one descent step in each coordinate before switching to another coordinate.This is the second convergence case analyzed for ByRDiE.
  • D. Theoretical Guarantees: Convergence for T = 1: The sequence Q(q) captures ByRDiE’s optimization variable after each coordinate-wise update.Each update replaces one coordinate while keeping the others fixed.
  • D. Theoretical Guarantees: Convergence for T = 1: For T = 1, nonfaulty-node iterates asymptotically reach consensus and converge in probability to the statistical risk minimizer.The convergence result is stated coordinate by coordinate.

E. How to Choose the Parameter T in ByRDiE?

The parameter T trades off consensus quality against convergence speed measured in communication iterations. Large T reduces early disagreement but slows convergence, whereas T = 1 reaches consensus only asymptotically.

  • E. How to Choose the Parameter T in ByRDiE?: Each ByRDiE iteration uses TP scalar-valued communication exchanges, so T directly affects communication cost.The tradeoff is evaluated using the communication-iteration measure t_c.
  • E. How to Choose the Parameter T in ByRDiE?: Larger T reduces early disagreement among nonfaulty nodes but slows convergence to the minimizer as a function of communication iterations.This follows from the two-time-scale behavior of the inner loop.
  • E. How to Choose the Parameter T in ByRDiE?: T = 1 guarantees consensus only asymptotically, so early iterations may have substantial disagreement among nonfaulty nodes.The choice still has asymptotic consensus and convergence guarantees.
  • E. How to Choose the Parameter T in ByRDiE?: An adaptive T_r that starts large and gradually decreases could improve the consensus–convergence tradeoff, but analyzing it is beyond the paper’s scope.The paper presents this as a plausible direction rather than an established result.

IV. NUMERICAL RESULTS

The numerical evaluation covers convex learning on infinite MNIST and nonconvex learning on Iris. Figure 1 compares ByRDiE with local coordinate descent and distributed gradient descent for linear-SVM training.

  • IV. NUMERICAL RESULTS: The experiments evaluate ByRDiE on binary classification using the infinite MNIST dataset distributed across a network.This experiment satisfies the assumptions used in the theoretical analysis.
  • IV. NUMERICAL RESULTS: Figure 1 plots average test classification accuracy against algorithmic iterations for ByRDiE, local CD, and DGD at different values of N.The plotted task is distributed linear-SVM training on infinite MNIST.
  • IV. NUMERICAL RESULTS: A second experiment trains a small neural network on distributed Iris data, representing a nonconvex learning problem outside the main theorem assumptions.The experiments examine whether ByRDiE continues to perform well in this setting.

A. Distributed SVM Using Infinite MNIST Dataset

The experiments evaluate ByRDiE for distributed SVM and neural-network training under Byzantine failures, varying data, communication, and fault levels. Results report resilience in convex and nonconvex settings, while identifying a practical limit under sufficiently many Byzantine nodes.

  • Distributed SVM Using Infinite MNIST Dataset: ByRDiE is evaluated on distributed linear SVM training for MNIST digits ‘5’ and ‘8’, which are described as difficult to separate.The experiments use a linear SVM and focus on two MNIST classes.
  • Distributed SVM Using Infinite MNIST Dataset: The SVM experiments use Erdős–Rényi networks, Byzantine nodes broadcasting random data, and coordinate updates repeated T times per dimension.Nonfaulty nodes receive equally divided class samples, while Byzantine nodes transmit random values in each iteration.
  • Distributed SVM Using Infinite MNIST Dataset: With M = 50 and b = 10 Byzantine nodes, ByRDiE is compared against local CD and DGD while varying each nonfaulty node’s sample size N from 10 to 30.Results are averaged over 10 independent Monte Carlo trials.
  • Distributed SVM Using Infinite MNIST Dataset: ByRDiE’s parameter T is studied through accuracy and consensus behavior as communication iterations tc increase, using M = 50, b = 5, and N = 60.Figure 2 compares four T values over 10 independent trials.
  • Distributed SVM Using Infinite MNIST Dataset: For M = 20, N = 10, and T = 3, b ≥5 Byzantine nodes—at least 25% of the network—often renders ByRDiE unusable under the worst-case resilience condition.The stated condition requires each node’s neighborhood to contain at least (2b+1) nodes.
  • Distributed Neural Network Using Iris Dataset: In nonconvex Iris neural-network training with one Byzantine node, ByRDiE remains resilient and approaches centralized CD, whereas DGD completely breaks down.The comparison measures average outer iterations needed to reach 95% classification accuracy.

APPENDIX A PROOF OF THEOREM 1

The proof establishes convergence of the coordinate-wise ByRDiE iterates by relating finite communication rounds to an asymptotic consensus limit. As the number of rounds or inner updates grows, each coordinate approaches its corresponding limiting value.

  • APPENDIX A PROOF OF THEOREM 1: The proof uses bounded coordinate gradients and the diminishing behavior of ρ̄(m) to control the relevant error terms.It invokes a coordinate-wise bound L∇≤L′ and the diminishing nature of ρ̄(m).
  • APPENDIX A PROOF OF THEOREM 1: The limiting relationship follows because the effective iteration index m̄ grows without bound when either the communication-round index r̄ or T tends to infinity.The proof explicitly states m̄ →∞ as r̄ →∞ and/or T →∞.
  • APPENDIX A PROOF OF THEOREM 1: For every coordinate k, the ByRDiE iterate [w̄r,T_j]k converges to vk(m̄) as r̄ →∞ and/or T →∞.The proof concludes this coordinate-wise convergence for all k = 1, 2, …, P.

APPENDIX B PROOF OF THEOREM 3

The proof derives a high-probability uniform bound for the empirical coordinate functions over the relevant parameter sets. It combines bounded independent samples, finite coverings, and concentration arguments to extend pointwise control uniformly.

  • APPENDIX B PROOF OF THEOREM 3: The proof starts from independent, bounded training-sample contributions and applies Hoeffding’s inequality to obtain a probability bound.The boundedness follows from Assumption 2, while independence follows from independent training samples.
  • APPENDIX B PROOF OF THEOREM 3: The resulting probability bound is extended from a fixed simplex element to every v ∈∆.The proof explicitly states that the bound in (44) holds for any v in the standard simplex.
  • APPENDIX B PROOF OF THEOREM 3: Finite ξ- and ζ-coverings of the simplex-related set Sα and parameter set W remove dependence on individual points in the uniform bound.The proof defines covering sets and uses their approximation radii to control deviations over the full sets.
  • APPENDIX B PROOF OF THEOREM 3: The proof concludes by selecting approximation parameters and combining the covering and concentration bounds to obtain the desired uniform result.The choices of ξ and ζ are tied to the target error ε and constants C, |J′|, and L′.
  • APPENDIX B PROOF OF THEOREM 3: The covering-number bounds depend on the ambient dimensions of Sα and W, enabling the union-bound argument to control the complete parameter domains.The proof bounds Sα through a sphere in R^{|J′|−1} and W through a sphere in R^P.

APPENDIX C PROOF OF THEOREM 4

The proof shows that the coordinate-descent objective decreases under the stated gradient condition and that the resulting iterates approach the minimum statistical risk. The conclusion relies on the high-probability bound established earlier and on boundedness assumptions.

  • APPENDIX C PROOF OF THEOREM 4: With the high-probability result from Theorem 3, the statistical risk of the limiting iterate approaches the minimum statistical risk.The proof states this convergence in the large-iteration limit.
  • APPENDIX C PROOF OF THEOREM 4: The coordinate-descent objective h̄r_k is strictly monotonically decreasing when the relevant gradient magnitude satisfies the stated threshold condition.The proof states this decrease for every coordinate k.
  • APPENDIX C PROOF OF THEOREM 4: Convexity of E[f(·, (x, y))] and the Cauchy–Schwarz inequality are used to relate coordinate updates to objective improvement.These steps connect the gradient condition with the descent argument.
  • APPENDIX C PROOF OF THEOREM 4: The proof’s argument depends on bounded iterates and related conditions that are revisited when discussing the validity of the coordinate bounds.The discussion notes a condition involving Γ and the boundedness of the relevant coordinates.

APPENDIX D PROOF OF THEOREM 5

The proof tracks ByRDiE’s iterates through an expected-risk notation and derives a recursion for their progress. A telescoping argument then shows the risk decreases while the optimality gap exceeds ϵ.

  • The proof defines expected risk ¯f(w) and uses bwr for the output produced by ByRDiE at each iteration.This notation applies across iterations and nonfaulty nodes.
  • Conditioned on the relevant probability event, the recursion holds while ¯f(bwr) − ¯f* > ϵ.The guarantee holds with probability at least 1 − δ(ϵ, N, ¯a).
  • The telescoping sum yields ¯f(bwr) < ¯f0 − rϵ′, equivalently ¯f(bwr) − ¯f* < ¯f0 − ¯f* − rϵ′.The initial iterate is specified as bw0 = 0.

APPENDIX E PROOF OF LEMMA 4

The proof analyzes the statistical risk of the sequence Q(q), establishing eventual monotonic decrease and convergence in probability under the stated assumptions. It concludes that convergence occurs only when every coordinate of the expected gradient is sufficiently small.

  • The proof defines ¯f(Q(q)) as the statistical risk incurred by Q(q) and seeks convergence toward a strong minimizer w* in probability.The convergence statement is established as N tends to infinity.
  • For sufficiently large q, ¯f(Q(q)) is strictly decreasing, lower bounded by ¯f(w*), and therefore convergent with probability 1 as N → ∞.The argument uses monotone convergence and the sublevel-set definition of W.
  • The update changes only coordinate k, and the proof rewrites the update relation to derive a condition ensuring strict monotonicity of ¯f(Q(q)).The relevant algebra follows from the coordinate-sparse difference Q(q + 1) − Q(q).
  • The weighted local-gradient expression equals the expected gradient coordinate [∇¯f(Q(q))]k under the stated sampling relation.This identity connects distributed gradient estimates to the population gradient.
  • The proof treats positive and negative values of [∇¯f(Q(q))]k separately and obtains sufficient conditions for the desired update inequality in probability.Gradient and local-gradient magnitudes are bounded using Assumption 1 and the definition of W.
  • The error terms can be made arbitrarily small as q grows, while convergence cannot occur if any gradient coordinate remains larger than ϵ∇ in absolute value.The contradiction uses summability of η(q)^2 and the lower boundedness of ¯f.
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