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On the Convergence of Federated Learning Algorithms without Data Similarity

Ali Beikmohammadi, Sarit Khirirat, Sindri Magnússon

TL;DR

This paper presents a novel and unified framework for analyzing the convergence of federated learning algorithms without the need for data similarity conditions, and derives precise expressions for three widely used step size schedules: fixed, diminishing, and step-decay step sizes, which are independent of data similarity conditions.

Abstract

Data similarity assumptions have traditionally been relied upon to understand the convergence behaviors of federated learning methods. Unfortunately, this approach often demands fine-tuning step sizes based on the level of data similarity. When data similarity is low, these small step sizes result in an unacceptably slow convergence speed for federated methods. In this paper, we present a novel and unified framework for analyzing the convergence of federated learning algorithms without the need for data similarity conditions. Our analysis centers on an inequality that captures the influence of step sizes on algorithmic convergence performance. By applying our theorems to well-known federated algorithms, we derive precise expressions for three widely used step size schedules: fixed, diminishing, and step-decay step sizes, which are independent of data similarity conditions. Finally, we conduct comprehensive evaluations of the performance of these federated learning algorithms, employing the proposed step size strategies to train deep neural network models on benchmark datasets under varying data similarity conditions. Our findings demonstrate significant improvements in convergence speed and overall performance, marking a substantial advancement in federated learning research.

On the Convergence of Federated Learning Algorithms without Data Similarity

TL;DR

This paper presents a novel and unified framework for analyzing the convergence of federated learning algorithms without the need for data similarity conditions, and derives precise expressions for three widely used step size schedules: fixed, diminishing, and step-decay step sizes, which are independent of data similarity conditions.

Abstract

Data similarity assumptions have traditionally been relied upon to understand the convergence behaviors of federated learning methods. Unfortunately, this approach often demands fine-tuning step sizes based on the level of data similarity. When data similarity is low, these small step sizes result in an unacceptably slow convergence speed for federated methods. In this paper, we present a novel and unified framework for analyzing the convergence of federated learning algorithms without the need for data similarity conditions. Our analysis centers on an inequality that captures the influence of step sizes on algorithmic convergence performance. By applying our theorems to well-known federated algorithms, we derive precise expressions for three widely used step size schedules: fixed, diminishing, and step-decay step sizes, which are independent of data similarity conditions. Finally, we conduct comprehensive evaluations of the performance of these federated learning algorithms, employing the proposed step size strategies to train deep neural network models on benchmark datasets under varying data similarity conditions. Our findings demonstrate significant improvements in convergence speed and overall performance, marking a substantial advancement in federated learning research.
Paper Structure (32 sections, 11 theorems, 142 equations, 9 figures, 1 table, 4 algorithms)

This paper contains 32 sections, 11 theorems, 142 equations, 9 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notations
  4. Prior Works
  5. Data Similarity Assumptions
  6. Communication-efficient Federated Optimization
  7. Step Size Schedules for Stochastic Optimization
  8. Main Convergence Theorems
  9. Applications in Federated Learning
  10. Full-precision Federated Learning Algorithms
  11. FedAvg
  12. FedProx
  13. Error-feedback Federated Learning Algorithms
  14. Error-feedback FedAvg
  15. Error-feedback FedProx
  16. ...and 17 more sections

Key Result

Theorem 1

Consider the system eqn:mainIneq. If $\gamma_k = \gamma = c/\sqrt{K}$ for $c>0$ and $K\in\mathbb{N}$, then

Figures (9)

  • Figure 1: Visual workflow of our analysis.
  • Figure 2: Visual workflow of the full-precision federated learning algorithms.
  • Figure 3: Visual workflow of the error-feedback federated learning algorithms.
  • Figure 4: Performance of FedAvg, error-feedback FedAvg, FedProx, and error-feedback FedProx with the fixed step size in (left plots -) training loss and (right plots -) test accuracy on MNIST dataset considering three different partitioned data among the workers.
  • Figure 5: Performance of FedAvg, error-feedback FedAvg, FedProx, and error-feedback FedProx with the fixed step size in (left plots -) training loss and (right plots -) test accuracy on FashionMNIST dataset considering three different partitioned data among the workers.
  • ...and 4 more figures

Theorems & Definitions (16)

  • Theorem 1: Fixed step sizes
  • Theorem 2: Diminishing step sizes
  • Theorem 3: Step-decay step sizes
  • proof
  • Proposition 1: FedAvg
  • Proposition 2: FedProx
  • Proposition 3: Error-feedback FedAvg
  • Proposition 4: Error-feedback FedProx
  • Lemma 1
  • proof
  • ...and 6 more