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Distributed Statistical Machine Learning in Adversarial Settings: Byzantine Gradient Descent

Yudong Chen, Lili Su, Jiaming Xu

arXiv:1705.05491v2cs.DCcs.CRcs.LGstat.ML

TL;DR

The paper studies distributed statistical learning when Federated Learning workers may be adversarial and their failures create unspecified dependencies across iterations. It proposes Byzantine Gradient Descent using geometric median-of-means aggregation, proving exponential convergence under a bounded bad-batch fraction and addressing practical scope limits.

  • Problem

    Distributed learning must estimate a model when workers may be Byzantine and their arbitrary behavior creates unspecified dependencies across communication rounds.

  • Method

    The method partitions received gradients into batches, averages each batch, takes the geometric median of those means, and performs a gradient descent step.

  • Results

    The method converges exponentially fast with high probability when the total fraction of statistically bad and Byzantine-containing batches is below 1/2.

  • Takeaways & Limitations

    The approach provides a communication-efficient robust gradient method for adversarial distributed learning and extends to the linear regression application studied in the paper.

  • Takeaways & Limitations

    The paper leaves asynchronous communication and characterization of the privacy sacrificed in Federated Learning for future work.

Abstract

from arXiv · show

We consider the problem of distributed statistical machine learning in adversarial settings, where some unknown and time-varying subset of working machines may be compromised and behave arbitrarily to prevent an accurate model from being learned. This setting captures the potential adversarial attacks faced by Federated Learning -- a modern machine learning paradigm that is proposed by Google researchers and has been intensively studied for ensuring user privacy. Formally, we focus on a distributed system consisting of a parameter server and $m$ working machines. Each working machine keeps $N/m$ data samples, where $N$ is the total number of samples. The goal is to collectively learn the underlying true model parameter of dimension $d$. In classical batch gradient descent methods, the gradients reported to the server by the working machines are aggregated via simple averaging, which is vulnerable to a single Byzantine failure. In this paper, we propose a Byzantine gradient descent method based on the geometric median of means of the gradients. We show that our method can tolerate $q \le (m-1)/2$ Byzantine failures, and the parameter estimate converges in $O(\log N)$ rounds with an estimation error of $\sqrt{d(2q+1)/N}$, hence approaching the optimal error rate $\sqrt{d/N}$ in the centralized and failure-free setting. The total computational complexity of our algorithm is of $O((Nd/m) \log N)$ at each working machine and $O(md + kd \log^3 N)$ at the central server, and the total communication cost is of $O(m d \log N)$. We further provide an application of our general results to the linear regression problem. A key challenge arises in the above problem is that Byzantine failures create arbitrary and unspecified dependency among the iterations and the aggregated gradients. We prove that the aggregated gradient converges uniformly to the true gradient function.

1 Introduction

The paper develops a Byzantine-robust distributed gradient method for Federated Learning settings with adversarial workers, limited local data, and scarce communication. It provides statistical convergence guarantees, computational and communication costs, and an analysis overcoming unspecified dependencies across iterations.

  • Motivation: Federated Learning keeps training data on users’ devices, but those devices may be externally controlled and behave adversarially.The setting also involves small local datasets and communication constraints.
  • Contribution: The proposed gradient descent variant tolerates arbitrary adversarial failures, learns complex models with low local data volume, and converges in logarithmic communication rounds.Because gradient descent is already widely adopted, the method requires only small code modifications.
  • System model: The system distributes N samples evenly across m working machines and uses synchronous communication between those machines and a parameter server.The paper notes that heterogeneous data sizes of the same order can be accommodated, while asynchronous communication remains future work.
  • Adversarial setting: Up to q working machines may be Byzantine, with the attacker allowed to change the controlled set between communication rounds.This arbitrary behavior creates unspecified dependencies across rounds, complicating convergence analysis.
  • Analysis: Uniform convergence of the geometric median of means of gradients to the true gradient function addresses the arbitrary probabilistic dependencies created by Byzantine workers.The analysis targets dependencies among iterates and aggregated gradients that cannot be specified in advance.

2 Algorithms and Summary of Convergence Results

The Byzantine Gradient Descent Method replaces vulnerable gradient averaging with geometric median aggregation and establishes exponentially fast convergence under adversarial failures. Its guarantees balance Byzantine tolerance, statistical error, and computational efficiency through the batch count k.

  • 2.1 Byzantine Gradient Descent Method: A single Byzantine machine can arbitrarily skew simple gradient averaging, whereas geometric median aggregation is designed to remain robust when a majority of points are reliable.The geometric median inherits a majority-based robustness property in multiple dimensions.
  • 2.1 Byzantine Gradient Descent Method: The method partitions local gradients into k batches, averages each batch, computes their geometric median, and uses it for the gradient descent update.This robust aggregation rule replaces simple averaging of received gradients.
  • 2.2 Summary of Convergence Results: 2(1 + ǫ)q ≤ k ≤ m is sufficient for the informal convergence theorem, with δ ≤ α − q/k and N/k required to exceed a dimension-dependent threshold.The theorem also assumes mild technical conditions and fixed constants α and ǫ.
  • 2.2 Summary of Convergence Results: The method converges exponentially fast when the total expected fraction of statistically bad and Byzantine-contaminated batches is below 1/2.The bad-batch fraction is represented by δ + q/k.
  • 2.2 Summary of Convergence Results: c2√(dq/N) is the asymptotic estimation error rate under the theorem’s conditions, with probability at least 1 − e^−Ω(qd).The result follows from the stated concentration condition on N/k and the theorem’s convergence guarantee.
  • 2.2 Summary of Convergence Results: Larger k increases Byzantine-failure tolerance but also increases the statistical estimation error, although the authors note this trade-off may be proof-dependent.The algorithm’s summarized costs are also computation- and communication-efficient under the selected k.

3 Convergence Results and Analysis

The analysis separates deterministic convergence on a uniformly good event from the stochastic proof that this event occurs with high probability. Uniform control of batch-gradient deviations enables geometric-median aggregation to support exponential convergence despite Byzantine behavior.

  • Probability analysis: The functions Zℓ are independently and identically distributed because they depend on disjoint batches of i.i.d. local data.This independence supports the stochastic analysis of the good event.
  • Proof strategy: The proof has deterministic and stochastic stages: exponential convergence on Eα,ξ1,ξ2, followed by a high-probability bound for that event.The event requires uniform control over batch sample-gradient deviations.
  • Robust aggregation: The geometric median is translation-invariant, allowing the aggregation analysis to transfer uniform batch-gradient bounds to the received gradients.This connects the batch-level deviation event to the approximate gradient used by the server.
  • Good event: On Eα,ξ1,ξ2, at least k(1−α)+q batches satisfy the uniform deviation bound, leaving at least k(1−α) Byzantine-free received batches controlled.For Byzantine-free batches, the perturbed deviation function equals the original batch deviation function.
  • Deterministic convergence: With η = L/(2M^2), approximate gradient descent converges under a uniform deviation bound, and Algorithm 2 consequently converges exponentially fast on Eα,ξ1,ξ2.The contraction result is established through Lemmas 3 and 4 and applied in Theorem 2.
  • Uniform convergence: An ϵ-net argument, concentration inequalities, and union bounds establish uniform convergence of averaged random gradients to the population gradient under Assumptions 2–4.The resulting proposition and theorem provide the stochastic foundation for the convergence guarantees.

4 Application to Linear Regression

The paper applies its Byzantine gradient-descent guarantees to Gaussian linear regression. Under the stated least-squares model and assumptions, the method solves the problem exponentially fast with high probability.

  • Model: The linear regression model uses Gaussian covariates with identity covariance and independent Gaussian additive noise around the signal ⟨w_i,θ∗⟩.The squared-loss formulation defines the application and its data-generating assumptions.
  • Population objective: In this model, the population gradient is θ−θ∗, so the population risk is 1-Lipschitz and 1-strongly convex with stepsize η = 1/2.These properties verify Assumption 1 for the application.
  • Gradient approximation: Because the input distribution is unknown, the population gradient cannot be computed exactly; the algorithm instead uses random sample gradients of the form w⟨w,θ−θ∗⟩−wζ.This is the statistical estimation setting addressed by the distributed method.
  • Guarantee: Under the least-squares model and the corollary’s conditions, Byzantine Gradient Descent solves linear regression exponentially fast with high probability.The guarantee applies when the search space, batch count, failure fraction, and sample-size conditions in Corollary 1 hold.
  • Scope: The corollary assumes a bounded or controlled search space Θ, whose range may grow with d and N/k.The paper describes this as a mild assumption supported by practical prior knowledge about θ∗.

5 Related Work

The paper differs from prior adversarial distributed optimization by studying statistical estimation with machine-local random data and explicit estimation accuracy. It also develops geometric-median gradient-function convergence using concentration and empirical-process techniques.

  • Adversarial distributed optimization: Unlike prior adversarial optimization work, this paper gives each machine its own samples from an unknown distribution and estimates the optimal model parameter θ∗.The comparison concerns statistical learning rather than deterministic shared-data optimization.
  • Adversarial distributed optimization: The paper focuses on statistical estimation accuracy, whereas the cited prior work proves almost-sure convergence without explicit convergence speed or estimation-error characterizations.This distinguishes the paper’s stated analytical goals from those of the related optimization study.
  • Robust estimation: Geometric median has a breakdown point of 0.5, motivating its use for robust aggregation under sufficiently many corrupted vectors.The paper places its aggregation method within robust parameter-estimation literature.
  • Robust estimation: High-dimensional robust mean and covariance estimators do not directly solve this setting because the target is a d-dimensional gradient function within gradient descent.The paper identifies a mismatch between those results and its distributed gradient-estimation problem.
  • Technical relationship: The convergence proof establishes uniform convergence for empirical gradients and their geometric-median version, not only for empirical risk functions.Pointwise concentration is strengthened to uniform convergence through an ϵ-net argument.

6 Discussion

The paper presents a robust, communication-efficient gradient aggregation method for unsecured training environments, while identifying privacy and synchronization as unresolved directions.

  • Method: Geometric median-of-means aggregation groups gradients into batches and takes their median to reduce Byzantine interruptions.The parameter server first forms nonoverlapping batches, then aggregates their batch gradients using the median.
  • Open Directions: Federated Learning privacy remains incompletely characterized because high-quality training requires extracting information about users’ data.The paper leaves precise characterization of the minimum privacy sacrifice to future work.
  • Open Directions: Asynchronous operation remains an open practical challenge because slow machines can significantly delay otherwise logarithmic-round synchronous execution.The authors propose adapting the algorithms to asynchronous settings in future work.
  • Open Directions: The Byzantine fault model assumes adversaries know the parameter server’s random bits, an assumption that may be relaxable in some applications.The discussion suggests gradient subset selection and averaging as a possible defense under relaxed faults.

A Proof of Lemma 2.1

The proof isolates vectors within radius r, derives bounds for that subset, and combines the resulting inequalities using its lower-bounded size.

  • Proof: The proof defines S as the set of indices whose vectors have norm at most r.This identifies the subset on which the subsequent bound is applied.
  • Proof: For every index in S, the proof establishes the relevant norm bound before combining it with the remaining inequality.The displayed equations provide the intermediate bounds used in the argument.
  • Proof: The final combination relies on |S| ≥ (1 − α)n, which follows from the lemma’s assumption.This lower bound on the subset size completes the stated inequality step.

B Proof of Lemma 3.2

The proof combines inequalities derived from the gradient condition and Assumption 1, then selects η = L/2M^2 to obtain the conclusion.

  • Proof: The proof begins from equation (13) and the optimality condition ∇F(θ*) = 0.These facts anchor the first inequality in the argument.
  • Proof: Assumption 1 supplies the second bound needed by the proof.The argument explicitly invokes the assumption before combining the displayed inequalities.
  • Proof: Summing the two preceding inequalities yields the penultimate step of the proof.The conclusion follows after this combination.
  • Proof: The proof concludes by choosing η = L/2M^2.This parameter choice is stated as the final step establishing the result.

C Concentration Inequality for Sub-exponential Random Variables

This appendix defines sub-exponential random variables and states a closure theorem for sums of independent sub-exponential variables.

  • Definition: A random variable is sub-exponential when positive scaling parameters ν and α satisfy the stated moment-generating condition.The definition assumes the variable has mean μ.
  • Theorem: Theorem 6 considers independent variables with individual sub-exponential scaling parameters and means.The theorem applies to X1, …, Xn.
  • Theorem: Their sum is also sub-exponential, with aggregate scaling parameters (ν*, α*).The theorem specifies the resulting parameters through the subsequent displayed expressions.
  • Theorem: The stated tail expression gives the resulting bound in the relevant case.The displayed formula includes an exponential term involving t and α*.
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