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Robust Federated Learning in a Heterogeneous Environment

Avishek Ghosh, Justin Hong, Dong Yin, Kannan Ramchandran

arXiv:1906.06629v2cs.LGstat.ML

TL;DR

The paper addresses Federated Learning with heterogeneous data distributions and Byzantine machines. It proposes a statistical model and modular algorithm, obtaining optimal estimation error and exponential decay of misclustered points under robust Lloyd clustering.

  • Problem

    Federated Learning must handle heterogeneous data distributions across devices and Byzantine machines while also addressing adversarial clustering and communication constraints.

  • Method

    The paper proposes a statistical model and a 3-stage algorithm, including clustering with Lloyd’s algorithm augmented by robust estimation.

  • Results

    The algorithm achieves estimation error optimal in dimension and number of data points, while robust Lloyd clustering’s misclustered-point count decays exponentially under proper initialization.

  • Takeaways & Limitations

    Robust estimation enables a modular solution for heterogeneous Federated Learning and provides statistical guarantees for clustering with adversarial data points.

  • Takeaways & Limitations

    The analysis assumes sub-Gaussian noise and bounded covariance eigenvalues, while weakening the sub-Gaussian assumption remains future work.

Abstract

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We study a recently proposed large-scale distributed learning paradigm, namely Federated Learning, where the worker machines are end users' own devices. Statistical and computational challenges arise in Federated Learning particularly in the presence of heterogeneous data distribution (i.e., data points on different devices belong to different distributions signifying different clusters) and Byzantine machines (i.e., machines that may behave abnormally, or even exhibit arbitrary and potentially adversarial behavior). To address the aforementioned challenges, first we propose a general statistical model for this problem which takes both the cluster structure of the users and the Byzantine machines into account. Then, leveraging the statistical model, we solve the robust heterogeneous Federated Learning problem \emph{optimally}; in particular our algorithm matches the lower bound on the estimation error in dimension and the number of data points. Furthermore, as a by-product, we prove statistical guarantees for an outlier-robust clustering algorithm, which can be considered as the Lloyd algorithm with robust estimation. Finally, we show via synthetic as well as real data experiments that the estimation error obtained by our proposed algorithm is significantly better than the non-Byzantine-robust algorithms; in particular, we gain at least by 53\% and 33\% for synthetic and real data experiments, respectively, in typical settings.

1 Introduction

The paper addresses Federated Learning with heterogeneous user data and Byzantine machines, proposing a modular statistical and algorithmic framework with optimal estimation guarantees. It also analyzes robust Lloyd clustering and validates the approach on synthetic and real data.

  • Motivation: Federated Learning must handle both heterogeneous data distributions across devices and Byzantine machines that may behave arbitrarily.These challenges arise because devices hold decentralized data and individual machines may be faulty, compromised, or malicious.
  • Contributions: The algorithm achieves estimation error optimal in dimension and number of data points.The paper presents this as a solution to robust heterogeneous Federated Learning under its statistical model.
  • Contributions: Robust Lloyd clustering is analyzed statistically, with the number of misclustered points decaying exponentially under proper initialization.This extends Lloyd’s algorithm with robust estimation to adversarial data points.
  • Contributions: The proposed framework uses local solutions, centralized robust clustering, and cluster-level robust distributed optimization.The framework is designed to exploit user cluster structure while tolerating Byzantine participants.
  • Experiments: Experiments on synthetic mixture-of-regressions data and Yahoo! Learning to Rank show that the proposed method outperforms standard non-robust algorithms.The experiments assess both theoretical validation and practical implementation.

2 Problem setup

The problem setup models non-Byzantine devices as belonging to multiple distributional clusters while allowing an α-fraction of machines to be arbitrarily corrupted. The goal is to estimate the population-risk minimizer for every cluster despite adversarial nodes.

  • Statistical model: The system contains m compute nodes, with αm Byzantine nodes for α < 0.5 and K clusters among the remaining machines.Each non-Byzantine cluster corresponds to a distinct data distribution, while corrupt-node data are uncontrolled.
  • Statistical model: Each non-Byzantine node in cluster C_k holds n i.i.d. samples from distribution D_k.The cluster structure represents heterogeneous data across devices.
  • Objective: The objective is to find the minimizers of all K population risk functions rather than one parameter for the entire system.This reflects the paper’s heterogeneous multi-cluster formulation.
  • Challenges: The problem requires adversary-tolerant clustering and robust distributed optimization because corrupted nodes cannot be perfectly clustered.The setup also seeks to reduce uplink communication cost.

3 A modular algorithm for robust Federated Learning in heterogeneous environment

The proposed solution is a three-stage modular pipeline: compute local empirical solutions, robustly cluster them, and optimize jointly within each cluster. Robust estimation is used to handle Byzantine nodes, especially when local sample sizes are small or dimension is high.

  • Stage I: Stage I computes each worker’s local empirical risk minimizer and sends it to the center machine.A streaming online-to-batch routine can provide a similar predictor without storing all samples in advance.
  • Stage II: Stage II clusters the local ERMs centrally using an outlier-robust clustering algorithm.The center seeks K clusters despite potentially Byzantine ERMs.
  • Stage II: When each worker has sufficiently many samples, threshold clustering exploits closeness within clusters and separation across clusters.The passage contrasts this setting with small-n Federated Learning, where thresholding may fail.
  • Stage II: For small local sample sizes, the method uses a robust version of Lloyd’s algorithm with robust mean estimation.Candidate estimators include geometric median, coordinate-wise median, trimmed mean, and high-dimensional robust mean methods.
  • Stage III: Stage III runs communication-efficient Byzantine-tolerant distributed optimization separately within each estimated cluster.Trimmed mean is used for low to moderate dimension, while iterative filtering targets high dimension.

4 Main results

The paper presents a modular approach to robust heterogeneous Federated Learning, combining local estimation, robust clustering, and Byzantine-tolerant optimization. Its guarantees cover oracle comparisons, dimensional and sample-size dependence, and assumptions for the learning setting.

  • Oracle comparison: The modular algorithm matches the oracle bound with high probability when non-Byzantine machines are clustered correctly.The oracle is assumed to know all non-Byzantine cluster identities.
  • Robust clustering: Robust clustering replaces Lloyd's sample-mean update with geometric-median-based trimming while retaining nearest-center assignments.Trimmed K-means estimates centers from points inside a ball around each bucket's geometric median.
  • Oracle comparison: Unknown clustering can reduce precision by a factor involving (˜αi − ˆαi)d/√n relative to the oracle setting.Here ˆαi is the oracle's worst-case Byzantine fraction and ˜αi is the corresponding fraction after clustering.
  • Statistical rates: For K = 2 with centers θ1 = −θ2, the dependence on α is near optimal, while dependence on n and Mi is optimal under the compared lower bounds.The special case uses ˜αi ≈ αm/Mi when Mi ≫ αm.
  • Statistical rates: The dependence on dimension d is optimal against the cited lower bound when iterative filtering is used, although the dependence on ˜αi then becomes sub-optimal.Trimmed mean is used for low to moderate dimension, while iterative filtering improves dimensional dependence in high dimension.

5 Robust clustering

The paper studies robust clustering of local ERMs when non-Byzantine machines form heterogeneous clusters and Byzantine points are arbitrary. Robust Lloyd-style methods achieve exponential misclustering decay under suitable initialization, while iterative filtering removes the dimension dependence in the symmetric two-cluster setting.

  • Robust clustering setup: Local ERMs from non-Byzantine machines satisfy y_i = θ_{z_i} + τ_i, whereas Byzantine observations are arbitrary.The robust clustering problem must recover unknown cluster labels despite adversarial observations.
  • Robust clustering setup: Classical Lloyd can fail arbitrarily because adversaries may place points far from the centers and corrupt sample-mean updates.The proposed methods replace vulnerable mean-based estimation with robust estimation.
  • Symmetric 2 clusters with Gaussian mixture: For two symmetric Gaussian clusters, the misclustering fraction decays exponentially with iterations under constant SNR and initialization slightly better than random guessing.Theorem 3 provides the corresponding high-probability guarantee for trimmed robust Lloyd iterations.
  • Symmetric 2 clusters with Gaussian mixture: If ∥θ*∥/σ ≳ √log m, then after 3 log m steps no non-Byzantine points are misclustered, matching the oracle bound.The same analysis tolerates α ∼ 1/d in this setting, while the general K-cluster result improves that tolerance.
  • K clusters with sub-Gaussian mixture: For general K clusters with sub-Gaussian noise, the analysis defines cluster-wise and trimmed cluster-wise misclustering fractions and establishes high-probability guarantees after constant iterations.The separation condition is O(√K), matching the standard separation condition stated in the passage.
  • Robust clustering in high dimension: In high dimension, Algorithm 3 uses iterative filtering under a Huber contamination model to estimate θ* and recover inlier labels.Iterative filtering uses higher-order statistics such as sample covariance to remove outliers.
  • Robust clustering in high dimension: Theorem 5 permits α ≤ 1/16 and gives convergence after T = Θ(log(1/β)) iterations with probability at least 1 − η.Its stated tolerance level has no dependence on dimension, improving over Theorem 3.

6 Experiments

Experiments evaluate robust clustering and optimization against non-robust alternatives on synthetic mixture-of-regressions and Yahoo! Learning to Rank data. Robust methods achieve lower estimation or misclustering error, with the strongest reported gains under adversarial contamination.

  • Synthetic data: The synthetic study uses a mixture of linear regressions with K = 5 clusters and m = 100 machines, including adversarial machines.It compares Lloyd, Trimmed K-means, K-geomedians, trimmed-mean optimization, sample-mean optimization, and Federated Averaging.
  • Synthetic data: ≥53% higher error occurs with K-means clustering than with Trimmed K-means or K-geomedians in synthetic experiments.Trimmed-mean optimization also outperforms sample-mean optimization by ≥29% under robust clustering.
  • Yahoo! Learning to Rank dataset: The real-data experiment constructs four clusters from Yahoo! Learning to Rank queries and introduces adversarial splits by synthetic perturbation.Queries are grouped using an ℓ2-distance threshold, then divided into batches of 50 points.
  • Yahoo! Learning to Rank dataset: 0.125 estimation error is achieved by Trimmed K-means with trimmed-mean optimization, compared with 0.256 for K-means with sample-mean optimization.The paper describes the latter result as relatively two times worse and reports similar final errors for Trimmed K-means and K-geomedians.

7 Conclusion and future work

The paper presents a modular three-stage solution for heterogeneous Federated Learning with Byzantine machines and analyzes its robust clustering component. Its framework is modular and achieves optimal error in several parameters, while future work includes weaker assumptions and improved initialization.

  • Conclusion and future work: The proposed framework finds local solutions, performs centralized robust clustering, and conducts joint robust distributed optimization.The stages admit flexible algorithm choices and are designed for modular implementation.
  • Conclusion and future work: The algorithm incurs estimation error that is optimal in several problem parameters for heterogeneous Federated Learning with Byzantine machines.The paper frames this as a three-step modular solution.
  • Conclusion and future work: The robust clustering analysis treats Lloyd’s algorithm with robust estimation and establishes exponential decay of the misclustering rate when initialized properly.The analysis concerns adversarial data points and robust estimation within the Lloyd procedure.
  • Conclusion and future work: Future work includes weakening the sub-Gaussian assumption and developing a better initialization scheme.These directions are explicitly retained as future endeavors.

A.4.3 Proof of final results

The final-results proof iterates recursive error bounds for clustering and center estimation. Under stated assumptions and sufficiently large constants, the bounds yield a controlled error floor after a constant number of iterations.

  • Proof of final results: With specified parameter choices, the proof guarantees G_U_s ≤ 0.35 under the stated assumptions.The passage sets ϵ = 12√r, C_1 ≥ 500, and r_1 ≥ 160.
  • Proof of final results: The proof derives an error-floor recursion for G_U_s+1 and iterates it for S constant iterations.The recursion combines bounds from the preceding lemmas.
  • Proof of final results: For S ≥ 2, the first term in the recursive bound is order-wise negligible, yielding G_S ≤ ϱ.This is obtained after substituting the specified δ′ values.

A.5 High dimension: proof of Theorem 5

The high-dimensional proof controls iterative label errors and robust mean-estimation errors under sub-Gaussian assumptions. It uses induction, concentration inequalities, and iterative filtering to establish probabilistic guarantees over successive iterations.

  • High dimension: proof of Theorem 5: The proof partitions each iteration’s data into correctly labeled inliers, mislabeled inliers, and outliers.This decomposition supports separate concentration bounds for label errors and adversarial contamination.
  • High dimension: proof of Theorem 5: Iterative filtering robustly estimates θ∗ from a corrupted sample, with error bounded by c_0σ̂√(A_t + 3α).The bound applies to the output θ̂(t) relative to the empirical mean.
  • High dimension: proof of Theorem 5: The proof uses virtual inliers to represent outlier replacements for analysis, while the implementation operates on the observed transformed data.The virtual points are drawn from the inlier distribution with mean θ∗.
  • High dimension: proof of Theorem 5: ERM closeness guarantees are established only for non-Byzantine machines.The paper compares direct ERM computation with online optimization over n episodes as order-wise identical under the stated conditions.

B.2 Stage III-robust distributed optimization

Stage III applies Byzantine-robust distributed optimization separately to each estimated cluster. Its convergence combines contraction toward the optimum with an error term determined by robust gradient estimation and clustering contamination.

  • B.2 Stage III-robust distributed optimization: The third stage learns one model per estimated cluster using Byzantine-robust distributed optimization.The framework jointly uses machines assigned to each cluster, even though adversarial clustering can be imperfect.
  • B.2 Stage III-robust distributed optimization: Each iteration sends the current model to workers, collects gradients, robustly estimates the gradients, and performs a gradient-descent update.
  • B.2 Stage III-robust distributed optimization: The optimization guarantee has a geometrically contracting initialization term plus a residual proportional to the robust-estimation error ∆′.The stated bound is ∥wT − w∗∥ ≤ (1 − λF/(L1+λF))^T∥w0 − w∗∥ + 2∆′/λF.
  • B.2 Stage III-robust distributed optimization: The error term satisfies ∆′ = e O(ˆαk d√n + d√nM) for iterative filtering.
  • B.2 Stage III-robust distributed optimization: Iterative trimmed means achieve an error rate of order optimal in dimension.The passage attributes this conclusion to the unavoidable d√nM term.

C Proof of Theorem 2

The proof of Theorem 2 is obtained by combining a prior theorem with the robust distributed optimization guarantee, under a bound on the effective contamination fraction.

  • C Proof of Theorem 2: The proof of Theorem 2 directly combines Theorem 4 and Theorem 7.
  • C Proof of Theorem 2: The effective contamination fraction ˜αi represents the worst-case Byzantine fraction for cluster i.It is defined using the fraction of non-Byzantine machines that are mis-clustered.
  • C Proof of Theorem 2: The result follows by assuming maxi∈[K] ˜αi < 1/2 and invoking Theorem 7.

D Technical lemmas

The technical section collects auxiliary concentration and matrix bounds used in the main proofs, under Gaussian or sub-Gaussian noise assumptions.

  • D Technical lemmas: The section lists technical lemmas whose proofs appear in the Appendix.
  • D Technical lemmas: The remaining results continue the notation established in Sections 5.2 and related technical developments.
  • D Technical lemmas: For independent Gaussian vectors τi ∼ N(0, σ^2Id), the maximum eigenvalue is at most 1.62(n + 4d)σ^2 with probability greater than 1 − exp(−0.1t).
  • D Technical lemmas: The analysis also considers independent zero-mean sub-Gaussian vectors with parameter σ^2.
  • D Technical lemmas: Several displayed probability bounds are stated for all clusters or all distinct cluster pairs with probability at least 1 − 1/m^3.
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