On the Convergence of FedProx with Extrapolation and Inexact Prox (2410.01410v1)

Abstract: Enhancing the FedProx federated learning algorithm (Li et al., 2020) with server-side extrapolation, Li et al. (2024a) recently introduced the FedExProx method. Their theoretical analysis, however, relies on the assumption that each client computes a certain proximal operator exactly, which is impractical since this is virtually never possible to do in real settings. In this paper, we investigate the behavior of FedExProx without this exactness assumption in the smooth and globally strongly convex setting. We establish a general convergence result, showing that inexactness leads to convergence to a neighborhood of the solution. Additionally, we demonstrate that, with careful control, the adverse effects of this inexactness can be mitigated. By linking inexactness to biased compression (Beznosikov et al., 2023), we refine our analysis, highlighting robustness of extrapolation to inexact proximal updates. We also examine the local iteration complexity required by each client to achieved the required level of inexactness using various local optimizers. Our theoretical insights are validated through comprehensive numerical experiments.

• The paper establishes that FedExProx converges within a calculable error neighborhood even with inexact proximal evaluations.
• The authors introduce a novel error control method that directly links approximation accuracy to convergence, mitigating bias in computations.
• Numerical experiments confirm that combining server-side extrapolation with inexact prox control accelerates local iteration efficiency.

On the Convergence of FedExProx with Extrapolation and Inexact Prox

Abstract

The paper "On the Convergence of FedExProx with Extrapolation and Inexact Prox" by Hanmin Li and Peter Richtarik investigates the practical applicability of the FedProx federated learning algorithm in the presence of inexact computations. The authors build on the previously established benefits of server-side extrapolation introduced to FedProx and provide a comprehensive theoretical analysis under the more realistic assumption that proximal operators at each client are computed inexactly. They demonstrate that the inexactness leads to convergence within a neighborhood of the optimal solution and propose methods to control and mitigate the adverse effects of this inexact evaluation, making the algorithm robust against such imperfections.

Introduction

Federated Learning (FL) involves distributed optimization where multiple clients collaboratively train a shared global model while keeping their data localized to preserve privacy. One of the key algorithms in FL is the federated averaging algorithm (FedAvg). However, FedAvg suffers from client drift due to data heterogeneity. Addressing this, FedProx introduces a proximal term to each client's local objective, effectively stabilizing the variations but relying on the exact proximal evaluations which are impractical in real-world scenarios.

The paper by Li and Richtarik introduces the FedExProx algorithm, enhancing FedProx with server-side extrapolation to counteract the issues arising from inexact prox computations. Their theoretical analysis explores two types of inexact evaluations: absolute and relative approximations.

Contributions

1. General Convergence with Inexact Prox The authors establish a theoretical framework showing that the inexact evaluations lead to convergence to a neighborhood of the optimal solution under globally strongly convex functions. This foundational result elucidates that the inexactness does not diverge the learning process but bounds it within a calculable error range.
2. Refining Inexactness Control By defining a new method of managing inexactness, they eliminate the aforementioned neighborhood effect in convergence. The improved approach ties the error bounds more tightly to the approximation’s accuracy, demonstrating that a sufficiently small extrapolation parameter retains the algorithm’s efficacy despite the inexact prox.
3. Biased Compression Link The paper draws a parallel between inexactness in proximal evaluations and biased gradient compression, leveraging the theoretical underpinnings of the latter to improve the analysis of FedExProx. This novel insight results in a faster convergence rate while maintaining robustness to inexact evaluations.
4. Local Iteration Complexity Analyzing the local computation requirements, the paper provides specific iteration complexities for achieving desired levels of inexactness using standard optimization algorithms such as Gradient Descent (GD) and accelerated methods like Nesterov’s Accelerated Gradient Descent (NAGD).
5. Numerical Validation Comprehensive numerical experiments validate the theoretical findings, showing that relative approximation can indeed remove bias and, in some cases, even outperform exact solutions enhanced with server-side extrapolation.

Theoretical and Practical Implications

The implications of this research are multifaceted:

• Robust Federated Learning: The paper makes federated learning more robust to real-world conditions where exact proximal solutions are infeasible.
• Versatile Application: By linking the analysis to biased gradient compression, the results hold broader applicability, potentially benefiting other distributed learning and optimization scenarios.
• Efficient Implementation: The established local iteration complexities guide the practical implementation of federated algorithms, optimizing client-side computation without compromising global convergence.
• Adaptive Techniques: The exploration into adaptive extrapolation techniques, although requiring further research, sets a foundation for developing dynamic and context-aware federated learning frameworks.

Future Research

The paper opens several avenues for future developments:

1. Adaptive Convergence Techniques: Developing and further refining adaptive rules like error feedback mechanisms to handle biased updates in federated settings.
2. Broader Applicability: Extending the framework beyond strongly convex functions to more general convex or even non-convex settings common in modern deep learning applications.
3. Algorithmic Innovations: Designing algorithms that leverage the insights from biased compression to achieve high-fidelity aggregation in federated learning, especially under heterogeneous and stochastic conditions.

In conclusion, the paper contributes significantly to both the theoretical and practical facets of federated learning, advancing our understanding of convergence under realistic constraints and setting the stage for more robust and efficient federated systems.