Lower Bounds and Optimal Algorithms for Personalized Federated Learning (2010.02372v1)

Abstract: In this work, we consider the optimization formulation of personalized federated learning recently introduced by Hanzely and Richt\'arik (2020) which was shown to give an alternative explanation to the workings of local {\tt SGD} methods. Our first contribution is establishing the first lower bounds for this formulation, for both the communication complexity and the local oracle complexity. Our second contribution is the design of several optimal methods matching these lower bounds in almost all regimes. These are the first provably optimal methods for personalized federated learning. Our optimal methods include an accelerated variant of {\tt FedProx}, and an accelerated variance-reduced version of {\tt FedAvg}/Local {\tt SGD}. We demonstrate the practical superiority of our methods through extensive numerical experiments.

• The paper establishes the first known lower bounds for communication and local oracle complexity in personalized federated learning, and proposes optimal algorithms.
• The authors propose optimal algorithms like Accelerated Proximal Gradient Descent variants and demonstrate their practical efficiency through experiments.
• The findings justify the use of local algorithms for federated learning problems with heterogeneous data, enhancing real-world deployments like mobile keyboards and recommendations.
• The findings justify the use of local algorithms for federated learning problems with heterogeneous data, enhancing real-world deployments like mobile keyboards and recommendations.

Lower Bounds and Optimal Algorithms for Personalized Federated Learning

In the paper titled "Lower Bounds and Optimal Algorithms for Personalized Federated Learning," the authors tackle the challenges inherent in personalized federated learning (FL), particularly focusing on communication complexity and local oracle complexity. The paper is thorough in establishing lower bounds for these complexities and introduces optimal algorithms that match these bounds. The authors design several advanced methods, including an accelerated variant of FedProx and an enhanced version of FedAvg/Local SGD, demonstrating their effectiveness through robust numerical experiments.

Key Contributions

1. Optimization Formulation: The paper provides a new perspective on personalized FL, distinguishing from traditional FL objectives, which prioritize minimizing overall population loss. The authors argue that given the distinct data distributions on clients, a personalized model might yield superior outcomes for individual nodes. They employ a modified objective that enables local models to differ while penalizing dissimilarities, thus fostering personalization within federated settings.
2. Lower Complexity Bounds: The authors establish the first known lower bounds on both communication and local oracle complexities within this personalized paradigm. They demonstrate that attaining an ε-neighborhood of the optimal requires at least sqrt(min{L, λ}/μ) log(1/ε) communication rounds, where L and λ denote smoothness and penalty parameters, respectively. Similarly, they elucidate the necessity for local proximal oracle calls under smooth, strongly convex local objectives.
3. Optimal Algorithm Design: Several optimal algorithms are proposed, aligning with the lower bounds derived. They present two variants of Accelerated Proximal Gradient Descent (APGD): APGD1 and APGD2, each tailored to different smoothness regimes, ensuring optimal communication complexity and local gradient complexity. Additionally, inexact APGD approaches using local solvers (AGD and Katyusha) address practical scenarios where local proximal computation might be infeasible.
4. Algorithm Evaluation: Through empirical evaluation using LIBSVM datasets, the authors evidence the strong practical performance of their proposed algorithms against baseline methods. Notably, AL2SGD+ outperformed others in communication efficiency due to its ability to balance the local model accuracy and the global model convergence. This practically validates the theoretical insights regarding the adoption of local algorithms in federated learning settings.

Implications and Future Directions

The development of these lower bounds and optimal algorithms substantively contributes to the theoretical framework governing personalized federated learning. By illustrating that local algorithms are optimal for FL problems with heterogeneous data, the authors justify common FL practices and potentially guide future algorithmic innovations. Practically, these insights can enhance the deployment of FL in real-world applications, such as mobile keyboards and personalized recommendations, where data heterogeneity is pronounced.

The paper opens avenues for further exploration into optimizing communication in FL systems, accommodating more varied client conditions or data structures. Additionally, investigating the implications of data heterogeneity on model personalization and generalization, as well as exploring privacy-preserving mechanisms alongside these optimal procedures, remains an essential area for ongoing research. Integrating these algorithms with privacy constraints could yield algorithms robust to both data variability and privacy norms.

This paper makes substantial contributions to the discourse on optimization in personalized federated learning by establishing foundational complexity bounds and demonstrating methods that efficiently navigate these limits, paving pathways for advanced methods tailored to distributed settings.