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Straggler-Resilient Federated Learning: Leveraging the Interplay Between Statistical Accuracy and System Heterogeneity

Amirhossein Reisizadeh, Isidoros Tziotis, Hamed Hassani, Aryan Mokhtari, Ramtin Pedarsani

arXiv:2012.14453v1cs.LGcs.DCstat.ML

TL;DR

Federated learning can be slowed by heterogeneous client speeds and stragglers. The paper proposes FLANP, which progressively activates faster-to-slower clients when each stage reaches statistical accuracy, using prior models as warm-starts. It reports up to O(log(Ns)) analytical speedup and significant wall-clock improvements over federated benchmarks.

  • Problem

    System heterogeneity makes federated training slow because standard methods wait for straggling clients with lower computational capability.

  • Method

    FLANP adaptively starts with the fastest clients and gradually adds slower clients after each participating set reaches statistical accuracy, using warm-start models.

  • Results

    Up to O(log(Ns)) analytical speedup is reported for strongly convex risks, while experiments significantly improve wall-clock time over benchmarks for convex and non-convex risks.

  • Takeaways & Limitations

    Adaptive participation can reduce straggler-related wall-clock time while ultimately involving all clients in the training procedure.

  • Takeaways & Limitations

    FLANP requires constants such as µ, c, and V_ns, although heuristic threshold tuning achieved performance close to FLANP.

Abstract

from arXiv · show

Federated Learning is a novel paradigm that involves learning from data samples distributed across a large network of clients while the data remains local. It is, however, known that federated learning is prone to multiple system challenges including system heterogeneity where clients have different computation and communication capabilities. Such heterogeneity in clients' computation speeds has a negative effect on the scalability of federated learning algorithms and causes significant slow-down in their runtime due to the existence of stragglers. In this paper, we propose a novel straggler-resilient federated learning method that incorporates statistical characteristics of the clients' data to adaptively select the clients in order to speed up the learning procedure. The key idea of our algorithm is to start the training procedure with faster nodes and gradually involve the slower nodes in the model training once the statistical accuracy of the data corresponding to the current participating nodes is reached. The proposed approach reduces the overall runtime required to achieve the statistical accuracy of data of all nodes, as the solution for each stage is close to the solution of the subsequent stage with more samples and can be used as a warm-start. Our theoretical results characterize the speedup gain in comparison to standard federated benchmarks for strongly convex objectives, and our numerical experiments also demonstrate significant speedups in wall-clock time of our straggler-resilient method compared to federated learning benchmarks.

1 Introduction

Federated learning faces substantial delays from heterogeneous client speeds and stragglers. FLANP adaptively adds clients from fastest to slowest as each stage reaches statistical accuracy, yielding theoretical and empirical wall-clock speedups.

  • Motivation: System heterogeneity causes slow clients to delay federated training across computationally diverse networks.Traditional methods can wait for slow clients throughout training.
  • Approach: FLANP starts with the fastest clients, then progressively adds slower clients after each participating set reaches statistical accuracy.The previous stage’s model supplies a warm-start for the next stage.
  • Contribution: The method leverages statistical accuracy and device heterogeneity through adaptive client activation.
  • Results: Up to O(log(Ns)) speedup gain is analytically established against standard federated benchmarks for strongly convex risks.
  • Results: Numerical experiments significantly improve wall-clock time over benchmarks with full or partial node participation for convex and non-convex risks.
  • Related Work: Prior work addresses heterogeneity through asynchronous methods, active sampling, normalized averaging, or deadlines, while FLANP uses adaptive participation tied to statistical accuracy.

2 Federated Learning Setup

The setup models federated learning with fixed local samples, empirical-risk objectives, statistical-accuracy thresholds, and heterogeneous client update times. It distinguishes this setting from both data-homogeneous resampling and more general data-heterogeneous models.

  • Federated Learning Setup: Each node stores fixed local samples, and the global objective is empirical risk minimization over the samples held by all available nodes.The samples are selected at the beginning and are not redrawn during training.
  • Empirical Risk: L_n(w) denotes the average empirical loss over the n × s samples stored at the first n nodes.
  • Assumptions: Unlike standard homogeneous settings with newly sampled data each round, this setup uses fixed samples and therefore does not ensure unbiased local gradients.
  • Statistical Accuracy: Statistical accuracy is reached when optimization error is no larger than the estimation-error bound V_ns, beyond which further optimization does not improve expected-risk accuracy.
  • System Heterogeneity: Client update time T_i represents computational speed, with larger T_i identifying slower clients or stragglers.

3 Adaptive Node Participation Approach

FLANP is a meta-algorithm that solves progressively larger empirical-risk problems, beginning with fast clients and doubling participation until all nodes reach statistical accuracy. Warm-starting makes later stages feasible while reducing exposure to slow clients early in training.

  • Problem: Existing federated methods often involve all clients throughout training, forcing each round to wait for the slowest node.
  • Stagewise Participation: FLANP begins with the n0 fastest nodes, optimizes their empirical risk to statistical accuracy, and then activates larger client sets.
  • Stagewise Participation: Participation grows geometrically until all N nodes are included and the final model reaches statistical accuracy for the global loss.
  • Accuracy Check: For strongly convex losses, the gradient condition ∥∇L_n(w_n)∥2 ≤ 2µV_ns provides a feasible statistical-accuracy check.
  • Warm-Starts: A prior stage’s statistically accurate model provides a warm-start for a larger node set containing the previous participants.
  • FLANP via FedGATE: FedGATE is one instantiation of FLANP, using gradient tracking to handle local gradients that are not unbiased estimators of the global gradient.

4 Theoretical Results

Under strong-convexity and statistical-accuracy assumptions, the analysis establishes FLANP’s optimization guarantees and compares its expected wall-clock time with non-adaptive federated learning. The results show that adaptive participation can reduce runtime and achieve a logarithmic speedup under exponential client computation times.

  • Optimization Guarantees: The guarantees assume µ-strongly convex losses with L-Lipschitz gradients and empirical-risk approximation error bounded by V_ns.The analysis also assumes bounded stochastic-gradient variance and uses κ := L/µ as the condition number.
  • Optimization Guarantees: 12κ ln(6) communication rounds suffice to reach statistical accuracy when each node performs τ_n = 1.5sσ^2/c local updates.Here, c defines the statistical-accuracy term V_ns = c/ns.
  • Optimization Guarantees: FLANP reaches the full-data statistical accuracy after stages that double participating clients, using O(s) local iterations and O(κ) communication rounds per stage.The final model satisfies E[L_N(w_N) − L_N(w*)] ≤ V_Ns.
  • Wall-Clock Time Analysis: FLANP’s expected runtime is order-wise O(κs(T_n0 + T_2n0 + ··· + T_N)), because each stage waits only for its slowest participating node.The stages begin with the n0 fastest nodes and progressively add slower nodes.
  • Wall-Clock Time Analysis: FedGATE’s runtime depends on the slowest client because all N nodes participate from the beginning and the server waits for every local update.The slowest node has computation time T_N, the largest among the client computation times.
  • Wall-Clock Time Analysis: Under i.i.d. exponential client computation times, FLANP achieves an O(log(Ns)) speedup over the non-adaptive FedGATE benchmark.For homogeneous clients, the expected wall-clock time is reduced by a factor log(Ns)/log(N).

5 Numerical Experiments

The experiments evaluate FLANP against federated benchmarks across convex and non-convex risks, datasets, network sizes, computation-speed models, and participation settings. FLANP consistently improves wall-clock training time, while heuristic tuning remains close to the parameterized method.

  • Experimental setup: Experiments compare FLANP with FedAvg, FedGATE, and FedNova using full and partial node participation across convex and non-convex risks.The evaluation covers communication rounds and wall-clock time.
  • Uniform computation speeds: 2.1× speedup over FedGATE is achieved for logistic regression on MNIST with 50 heterogeneous clients.FLANP is initially slightly less accurate but benefits from starting with the fastest nodes.
  • Uniform computation speeds: 10× faster training than non-adaptive FedGATE is observed for linear regression on synthetic data distributed across 100 heterogeneous nodes.The comparison concerns the optimality gap versus communication round and wall-clock time.
  • Uniform computation speeds: Up to 3× and 4× acceleration over FedNova is obtained for fully connected neural networks on MNIST and CIFAR10, respectively.The experiments use 20 clients and fixed stepsizes for each dataset.
  • Partial node participation: With partial participation, FLANP is significantly faster than FedGATE, while using only the fastest fixed clients yields higher final training-error saturation.The fastest-client method eventually outperforms FLANP in wall-clock time but produces an inaccurate final model because only k clients contribute data.
  • Effect of N and s and practicality: Increasing either the number of clients or samples per client results in faster training time for FLANP, and heuristic phase-transition tuning performs close to FLANP.The parameter-sensitivity experiments use random exponential computation speeds; the heuristic tunes thresholds by monitoring the global-gradient norm.

A Proof of Proposition 1

The proof of Proposition 1 bounds how well a model optimized for a smaller participating-node subset performs on the enlarged subset. It decomposes the suboptimality into empirical-risk, optimality-gap, and gradient-approximation terms under strong convexity and sampling assumptions.

  • Lemma and proof structure: Lemma 1 bounds the enlarged-set risk suboptimality when the model is statistically accurate for a smaller node subset.The loss is assumed to be µ-strongly convex, and the smaller-set model satisfies a gradient-norm accuracy condition.
  • Proof structure: The proof decomposes the enlarged-risk suboptimality into four difference terms before bounding each component.The terms compare risks and optima associated with the smaller and larger participating-node sets.
  • Approximation bounds: Empirical-risk and gradient differences are controlled with high-probability approximation bounds under Assumption 2.The bounds apply to samples associated with both individual nodes and participating-node subsets.
  • Optimality gap: Strong convexity converts the smaller-set statistical-accuracy condition into a bound on its optimality gap.The proof uses the gradient condition at the smaller-set model together with strong convexity.
  • Conclusion: Combining the four bounds proves inequality (5), while the gradient-difference argument proves inequality (6).The proof then bounds local gradients to establish the remaining claim in Proposition 1.

B Proof of Theorem 1

The proof of Theorem 1 analyzes one FLANP stage after doubling the participating clients and initializing from the previous stage's model. It selects stepsizes and round counts so the stage reaches the statistical accuracy required for its enlarged empirical risk.

  • Stage initialization: A stage with n = 2m participating nodes runs FedGATE from the previous model w_m after the smaller stage.The stage begins with a warm start and performs local and global updates for R_n communication rounds and τ_n local updates.
  • Convergence analysis: The previous-stage accuracy bound is substituted into the FedGATE convergence guarantee for the enlarged stage.This transfers the statistical-accuracy relationship between successive participant sets into the stage analysis.
  • Accuracy target: The proof requires each of six terms in the convergence bound to be at most V_ns/6 to attain the target statistical accuracy.The target is E[L_n(w_n) − L_n(w*_n)] ≤ V_ns.
  • Parameter selection: Stepsizes η_n and γ_n are chosen to satisfy the convergence conditions, and the resulting local-update and communication-round counts determine the stage complexity.The proof derives these counts from the conditions imposed on the stepsizes and algorithm parameters.

C Proof of Proposition 3

The proof of Proposition 3 derives the runtime of full-participation FedGATE needed to reach final statistical accuracy. Because every communication round waits for the slowest client, the total wall-clock time scales with that client’s computation time.

  • Full-participation baseline: FedGATE is analyzed with all N clients, starting from an arbitrary model and targeting the statistical accuracy of the global loss.The proof determines the local-update and communication-round parameters needed to reach that target.
  • Convergence conditions: The parameter conditions simplify the convergence expression and yield sufficient inequalities for reaching final statistical accuracy.The proof selects stepsizes satisfying the required relations before deriving the update and round counts.
  • Initial conditions: The initial suboptimality and gradient terms are treated as constants independent of N under the stated assumptions.The proof assumes Δ_0 = O(1) and an analogous constant bound for the initial global gradient.
  • Runtime calculation: Each FedGATE communication round takes τT_N because the server waits for the slowest node, whose processing time is T_N.The total expected wall-clock time follows by multiplying the per-round runtime by the required number of rounds.
  • Runtime conclusion: Substituting the round count and per-round runtime gives the expected total wall-clock time stated in Proposition 3.The proposition characterizes the full-participation baseline used for the speedup comparison.

D Proof of Theorem 2

The proof derives expected runtimes for FLANP with FedGATE and FedGATE alone, then bounds their ratio under exponentially distributed client computation times. These bounds establish the resulting speedup characterization under the stated sorting, normalization, and power-of-two assumptions.

  • The analysis compares FLANP with FedGATE against the expected runtime of FedGATE using a runtime ratio.The proof first characterizes each method's expected runtime before deriving their ratio.
  • Client computation times are modeled as i.i.d. exponential variables, sorted from fastest to slowest, with n0 = 1 and λ = 1 for simplification.The order statistics are denoted T(1) ≤ T(2) ≤ · · · ≤ T(N).
  • The expected order-statistic computation times are decomposed into sums of successive spacings and the first order statistic.This decomposition supports the subsequent runtime bounds.
  • Assuming N = 2^K, harmonic-number bounds are used to simplify the runtime-ratio expressions.The proof applies ln(n) + γ ≤ H_n ≤ ln(n + 1) + γ for n ≥ 2.
  • The resulting expressions precisely characterize FLANP's speedup gain relative to FedGATE.The proof concludes after combining the numerator and denominator bounds and evaluating the ratio of expected runtimes.
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