On the Convergence of FedAvg on Non-IID Data

Abstract: Federated learning enables a large amount of edge computing devices to jointly learn a model without data sharing. As a leading algorithm in this setting, Federated Averaging (\texttt{FedAvg}) runs Stochastic Gradient Descent (SGD) in parallel on a small subset of the total devices and averages the sequences only once in a while. Despite its simplicity, it lacks theoretical guarantees under realistic settings. In this paper, we analyze the convergence of \texttt{FedAvg} on non-iid data and establish a convergence rate of $\mathcal{O}(\frac{1}{T})$ for strongly convex and smooth problems, where $T$ is the number of SGDs. Importantly, our bound demonstrates a trade-off between communication-efficiency and convergence rate. As user devices may be disconnected from the server, we relax the assumption of full device participation to partial device participation and study different averaging schemes; low device participation rate can be achieved without severely slowing down the learning. Our results indicate that heterogeneity of data slows down the convergence, which matches empirical observations. Furthermore, we provide a necessary condition for \texttt{FedAvg} on non-iid data: the learning rate $\eta$ must decay, even if full-gradient is used; otherwise, the solution will be $\Omega (\eta)$ away from the optimal.

• The paper establishes that FedAvg achieves an O(1/T) convergence rate for strongly convex and smooth functions under non-IID conditions.
• It demonstrates that optimal local SGD steps and controlled device participation reduce communication rounds and enhance learning efficiency.
• The analysis reveals that a decaying learning rate is vital for achieving optimal convergence in environments with heterogeneous data distributions.

On the Convergence of FedAvg on Non-IID Data

The paper "On the Convergence of FedAvg on Non-IID Data" explores the theoretical analysis of the Federated Averaging (FedAvg) algorithm applied to Federated Learning (FL) environments with non-IID data distributions. The study provides insights into the convergence behavior and performance guarantees of FedAvg in scenarios where data heterogeneity is prevalent, addressing a key challenge in decentralized learning systems.

Federated Learning and FedAvg Algorithm

Federated Learning allows decentralized devices to collaboratively train models without sharing raw data, preserving privacy and reducing communication overhead. The FedAvg algorithm, a pivotal method within FL, operates by periodically averaging model updates computed across multiple devices using local Stochastic Gradient Descent (SGD).

FedAvg performs multiple local SGD steps on devices and exchanges updated model parameters intermittently. This reduces communication frequency compared to standard SGD approaches. However, theoretical guarantees on its convergence, especially with non-IID data distribution and partial device participation, have been limited.

Convergence Analysis Under Non-IID Settings

This study establishes convergence properties of FedAvg when confronted with non-IID data, contributing significantly to extending theoretical frameworks applicable to real-world FL scenarios. The primary contribution includes demonstrating that FedAvg achieves an $\mathcal{O}(1/T)$ convergence rate for strongly convex and smooth functions without relying on impractical assumptions such as IID data or full device participation.

Key Results

• Convergence Rate: The paper shows $\mathcal{O}(1/T)$ convergence for FedAvg in non-IID settings, validating the algorithm's efficacy in practical applications with heterogeneous data distributions.
• Communication Efficiency: The research reveals a balance between communication efficiency and convergence rate, stressing the importance of optimizing local SGD steps and device selection to minimize communication rounds.
• Learning Rate Decay: A fundamental observation is that for FedAvg to converge to an optimal solution in non-IID scenarios, it is necessary for the learning rate to diminish over time. A constant learning rate leads to suboptimal convergence.

Analytical Insights

The convergence analysis relies on defining mathematical bounds that relate the degree of data heterogeneity to the convergence behavior. The variance in communication steps between devices and the necessary decay of learning rates form a critical part of the theoretical proofs. The bounds illustrate how specific parameters, like the number of local steps $E$ and the number of participating devices $K$, influence convergence. The study suggests neither extremely small nor large values for $E$, advocating for an optimal range that balances local computations against communication costs.

Implementation Strategies

For practical implementations, the paper recommends considering sampling and averaging strategies to optimize FedAvg's performance. It contrasts several schemes, showing how they affect convergence rates and stability under different data distribution scenarios:

• Scheme I (With Replacement Sampling): Guarantees average convergence particularly well suited for scenarios where sampling probabilities can be controlled.
• Scheme II (Without Replacement Sampling): Requires balanced data distributions for effective convergence, illustrating its limitations in heterogeneous data environments.
• Adaptive Sampling Techniques: Highlighted as critical for improving performance in systems with uneven data distribution, ensuring fairness and minimizing the straggler effect.

Conclusion

The paper provides invaluable theoretical support for deploying FedAvg in federated learning environments with non-IID data distributions. It addresses practical concerns related to fine-tuning learning rates, optimizing local versus global updates, and strategizing device participation to maintain efficiency and convergence. These insights pave the way for more robust federated learning models capable of handling real-world data challenges.

By thoroughly analyzing FedAvg within non-IID settings, the study lays a solid foundation for future developments in federated learning, encouraging adaptations and innovations to enhance decentralized model training strategies, ensuring they are both efficient and effective across diverse and unbalanced datasets.