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Decentralized Federated Learning with Unreliable Communications

Hao Ye, Le Liang, Geoffrey Li

arXiv:2108.02397v1cs.IT

TL;DR

Existing decentralized training assumes reliable communication, although wireless links lose packets and reliable protocols impose overhead and restrict connectivity. Soft-DSGD uses UDP-compatible partial messages and reliability-aware mixing, and the paper proves convergence at the same asymptotic rate as vanilla decentralized SGD while experiments show faster convergence by using unreliable links.

  • Problem

    Existing decentralized optimization methods generally require reliable networks, whereas wireless links experience packet loss and errors, and TCP-style recovery adds overhead while limiting reliable neighbors.

  • Method

    Soft-DSGD uses UDP-based unreliable communication, updates with partially received messages, and chooses consensus mixing weights according to link reliability.

  • Results

    Soft-DSGD achieves the same asymptotic convergence rate as vanilla decentralized SGD with perfect communications and leverages unreliable links to accelerate convergence.

  • Takeaways & Limitations

    Unreliable communication links can be incorporated into decentralized training without sacrificing the stated asymptotic convergence guarantee.

Abstract

from arXiv · show

Decentralized federated learning, inherited from decentralized learning, enables the edge devices to collaborate on model training in a peer-to-peer manner without the assistance of a server. However, existing decentralized learning frameworks usually assume perfect communication among devices, where they can reliably exchange messages, e.g., gradients or parameters. But the real-world communication networks are prone to packet loss and transmission errors. Transmission reliability comes with a price. The commonly-used solution is to adopt a reliable transportation layer protocol, e.g., transmission control protocol (TCP), which however leads to significant communication overhead and reduces connectivity among devices that can be supported. For a communication network with a lightweight and unreliable communication protocol, user datagram protocol (UDP), we propose a robust decentralized stochastic gradient descent (SGD) approach, called Soft-DSGD, to address the unreliability issue. Soft-DSGD updates the model parameters with partially received messages and optimizes the mixing weights according to the link reliability matrix of communication links. We prove that the proposed decentralized training system, even with unreliable communications, can still achieve the same asymptotic convergence rate as vanilla decentralized SGD with perfect communications. Moreover, numerical results confirm the proposed approach can leverage all available unreliable communication links to speed up convergence.

I. INTRODUCTION

Decentralized federated learning replaces the centralized server with peer-to-peer collaboration, addressing scalability and communication bottlenecks but exposing training to unreliable links. Soft-DSGD uses UDP-compatible partial messages and reliability-aware mixing to preserve convergence while exploiting available links.

  • Motivation: Federated learning lets edge devices train with local data while exchanging parameters or gradients, avoiding centralized collection of private data.The approach is motivated by limited bandwidth and privacy requirements at the edge.
  • Centralized and decentralized training: Centralized federated learning requires every participating device to send updates to one server, creating a communication bottleneck that limits scalability.Decentralized federated learning instead uses peer-to-peer exchanges among neighboring devices to reach consensus.
  • Communication challenge: Decentralized training methods often reduce communication through compression, local updates, or asynchronous communication, but existing methods generally assume reliable device-to-device networks.Real wireless links instead suffer packet loss, noise, fading, and interference.
  • Communication challenge: TCP-style reliability adds retransmissions and control overhead while limiting the number of reliable neighbors, slowing decentralized training.These costs motivate lightweight, unreliable communication protocols for edge collaboration.
  • Proposed approach: Soft-DSGD updates parameters with partially received messages and optimizes consensus mixing weights using the reliability matrix of communication links.The method is designed for UDP-based, connectionless communication in which packet loss and transmission errors occur randomly.
  • Decentralized optimization: The paper analyzes decentralized SGD over connected device graphs, where local objectives and stochastic losses are optimized through peer-to-peer gradient and consensus updates.Convergence requires a symmetric, doubly stochastic mixing matrix with a strictly positive spectrum gap.

3) Decentralized training of machine learning models:

The paper formulates decentralized training over unreliable UDP links and develops Soft-DSGD to perform consensus with partially received messages. It replaces lost packet values locally and selects mixing weights using link reliability when available.

  • Decentralized optimization distributes training losses across devices, which retain local parameters and gradients while supporting scalable, privacy-preserving training.
  • A. Unreliable communications with UDP: UDP provides connectionless packet delivery without retransmission, so Soft-DSGD must handle random packet loss and transmission errors.
  • A. Unreliable communications with UDP: Soft-DSGD divides model parameters into independently transmitted packets, broadcasts them, and receives packets stochastically across devices.
  • B. Algorithm: Each training iteration combines a local SGD update with a consensus update using messages received from other devices.
  • B. Algorithm: Missing message values are filled with the receiving device’s local parameters, avoiding additional memory for historically received data.
  • B. Algorithm: In expectation, unreliable consensus updates are equivalent to reliable consensus updates with W, which is doubly stochastic.
  • B. Algorithm: When link reliability P is unavailable, Soft-DSGD uses uniform mixing; when P is available, it optimizes W by minimizing the largest eigenvalue of W2 − J.

IV. CONVERGENCE ANALYSIS

The convergence analysis establishes that Soft-DSGD retains the asymptotic convergence rate of vanilla decentralized SGD despite unreliable communications.

  • Soft-DSGD achieves the same asymptotic convergence rate as vanilla decentralized SGD with perfect communications, even in unreliable communication networks.

A. Assumptions

The analysis assumes standard smoothness, variance, and unbiased-gradient conditions for local objectives, together with symmetric, independent, and stable communication links.

  • Assumptions on functions: Each local objective is assumed to be smooth with L-Lipschitzian gradients.
  • Assumptions on functions: Stochastic-gradient variance and cross-device data-distribution discrepancy are bounded by σ2 and ζ2, respectively.
  • Assumptions on functions: Stochastic gradients at each device are assumed to be unbiased estimates of the local objectives’ real gradients.
  • Assumptions on communication networks: Communication links are assumed to be symmetric, independent across links, and stable throughout training.

B. Soft-Consensus Algorithm

Soft-Consensus analyzes how decentralized averaging behaves when communication links are unreliable, then establishes convergence guarantees for Soft-DSGD. The analysis shows that suitable mixing preserves the asymptotic behavior of vanilla decentralized SGD and can retain linear device speedup.

  • Consensus properties: The consensus update preserves the average model in expectation at each step, while expected deviation is bounded by the variance of the device models.The communication-network resistance is captured by κ; deterministic links make κ zero.
  • Consensus properties: Unreliable communications still yield exponential convergence of consensus under the proposed update policy.This matches the exponential rate stated for reliable communication in the paper’s comparison remark.
  • Soft-DSGD convergence: The Soft-DSGD convergence theorem incorporates communication resistance through κ and the consensus factor ρ.The bound’s dependence on ρ motivates optimizing the mixing weights.
  • Soft-DSGD convergence: With a properly configured learning rate, the method achieves linear speedup with the number of devices while matching vanilla decentralized SGD’s rate.The result is stated in the corollary under the theorem’s conditions.
  • Soft-DSGD convergence: Soft-DSGD with unreliable communications achieves the same asymptotic convergence rate as vanilla decentralized SGD with reliable communications.The paper concludes that communication unreliability does not negatively affect asymptotic convergence.

V. EXPERIMENTS

Experiments evaluate Soft-DSGD on unreliable communication networks using image classification and compare it with TCP-based vanilla decentralized SGD. Soft-DSGD uses unreliable links and partially received messages, achieving competitive convergence while reducing communication-round requirements.

  • Experimental setup: Experiments train ResNet-20 models on CIFAR-10 across simulated geometric random networks of 16 devices.The setup uses 50,000 training images, 10,000 test images, and randomly located devices.
  • Effectiveness of Soft-DSGD: Soft-DSGD converges similarly to vanilla decentralized SGD with pδ = 0.5 and faster than vanilla decentralized SGD with pδ = 0.7.It leverages all unreliable links and partially received messages, whereas vanilla decentralized SGD uses only selected neighbours.
  • Communication efficiency: TCP requires more communication rounds than Soft-DSGD because packets are retransmitted after loss or transmission errors.Lower pδ increases the average number of neighbours but decreases average link quality, worsening communication requirements.
  • Effects of link reliability: Soft-DSGD training slows as communication links degrade, while it continues to work when TCP's communication graph is disconnected at k = 0.3.The link-reliability distributions are varied by changing k while device positions remain fixed.
  • Effects of link reliability: The performance gap between uniform and optimal mixing weights increases as link quality degrades.Uniform weights give equal weight to links with low transmission-success probabilities, impeding convergence.

APPENDIX

The appendix represents each parameter dimension with a transmission-dependent mixing matrix and combines these dimension-specific updates across the model.

  • Dimension-wise mixing: Soft-DSGD constructs a separate mixing matrix for each parameter dimension based on successful transmissions.The transmission-dependent matrix reflects whether each parameter is successfully received over each link.
  • Dimension-wise mixing: The expected contribution from link i to j is weighted by its reliability p_i,j and mixing weight w_i,j.The expectation of the transmission-dependent mixing term is p_i,j w_i,j.
  • Dimension-wise mixing: The full model update is obtained by combining the mixing behavior across all parameter dimensions.

B. Proof of Lemma 2

The proof of Lemma 2 establishes the expected model-update property under the doubly stochastic mixing matrix and characterizes transmission-induced randomness.

  • Proof conclusion: The lemma concludes after combining the expected-update and transmission-randomness arguments.
  • Expected update: The proof uses the doubly stochastic property of W to preserve the expected average model state.It applies E{x_t+1} = 1/N 1^T E{X_t+1} = 1.
  • Transmission randomness: Transmission-related terms are modeled as zero-mean variables with variance 2p_i,j(1 − p_i,j).

C. Proof of Lemma 3

The proof of Lemma 3 analyzes consensus error by projecting the model onto the disagreement subspace and recursively bounding its expected norm.

  • Disagreement analysis: The proof defines β_l(t) = (I − J)X_l(t) to isolate disagreement in the l-th parameter dimension.
  • Disagreement analysis: It bounds the expected norm of β_l(t) conditioned on the previous disagreement state.
  • Recursive bound: The dimension-wise bounds are combined into β(t) = (I−J)X_t and iterated over time.
  • Proof conclusion: The proof concludes the lemma after establishing the recursive disagreement bound.

D. Proof of Theorem

The proof derives bounds for the theorem by applying smoothness and lemmas to component terms, then rearranging and averaging the resulting inequality.

  • The proof begins from the Lipschitz smoothness of f and repeatedly applies Lemma 2 and Lemma 3 to bound intermediate terms.
  • The analysis separately bounds the first and second terms, including T1, T2, F, and PT, before combining them.
  • The final algebraic steps apply minor rearrangement to obtain the theorem’s bound.
  • The proof uses the common initialization x0 with β_l(0) = 0 when analyzing the l-th dimension.
  • After substituting intermediate bounds and defining D, the proof rearranges the inequality, sums over t, and takes the average.

E. Proof of Convexity of Weight Optimization Problem in (13)

The proof establishes convexity of the weight optimization objective by showing that the feasible set is convex and the expected objective preserves convexity under mixing.

  • The feasible set S is convex because convex combinations of symmetric, stochastic matrices with entries in [0,1] remain in S.
  • For a fixed transmission-success matrix A, the effective matrix f_W is affine in W under convex combinations.
  • Taking expectations after comparing the objective for convexly combined weights yields the convexity of the weight optimization objective.
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