Robust Federated Personalised Mean Estimation for the Gaussian Mixture Model (2504.19955v1)

Abstract: Federated learning with heterogeneous data and personalization has received significant recent attention. Separately, robustness to corrupted data in the context of federated learning has also been studied. In this paper we explore combining personalization for heterogeneous data with robustness, where a constant fraction of the clients are corrupted. Motivated by this broad problem, we formulate a simple instantiation which captures some of its difficulty. We focus on the specific problem of personalized mean estimation where the data is drawn from a Gaussian mixture model. We give an algorithm whose error depends almost linearly on the ratio of corrupted to uncorrupted samples, and show a lower bound with the same behavior, albeit with a gap of a constant factor.

• The paper introduces a theoretical model and methods for robust personalized mean estimation in federated learning environments featuring Gaussian mixture models and adversarial data corruption.
• It presents a robust clustering algorithm for the server to assess and cluster client data, reducing the influence of corrupted samples to improve component mean estimation.
• The research provides theoretical error bounds demonstrating that the mean squared error in personalized estimation increases linearly with the fraction of corrupt clients.

Robust Federated Personalized Mean Estimation for the Gaussian Mixture Model: A Critical Overview

The research paper "Robust Federated Personalized Mean Estimation for the Gaussian Mixture Model" introduces a specialized domain of federated learning, wherein the focus is personalized mean estimation within the context of federated data that are drawn from a Gaussian mixture model. This paper particularly emphasizes the need for robustness in the presence of corrupted or adversarial data within heterogeneous client environments, a prominent concern in federated learning systems.

Summary of Contributions

The paper presents a theoretical model that delineates a method for robust personalized mean estimation across client systems characterized by data heterogeneity. It employs a Gaussian mixture model for data generation and targets data corruption scenarios where a certain fraction of clients are adversarial.

The structured insights presented in the paper include:

1. Robust Clustering Algorithm: An innovative algorithm designed for a server in federated learning to assess and cluster data points from clients in a manner that reduces the influence of corrupted data. The clustering relies on filtrations and distances in data, ensuring component means are effectively estimated despite adversarial noise.
2. Component-Specific Mean Estimation: By leveraging the Gaussian mixture model's structure, the authors developed a mechanism whereby clients integrate server-derived cluster estimates with their local verified samples to fine-tune personalized mean estimates. This integration is analytically supported with bounds demonstrating improved estimation accuracy relative to isolated local sample use.
3. Theoretical Upper and Lower Bounds: The paper provides comprehensive theoretical analysis, yielding both lower and asymptotic upper error bounds for personalized mean estimation under the studied adversarial corruption model. These bounds highlight that the mean squared error increases linearly with the fraction of corrupt clients, a finding consonant with observed results in robust federated learning literature.

Implications and Future Directions

This paper's insights have profound implications for federated systems where data validity may be compromised due to client-side adversarial attacks. The methodological advancement offers potential to improve performance reliability and accuracy in real-world federated applications across sectors such as healthcare and finance, where client data heterogeneity and security concerns are prevalent.

From a theoretical standpoint, the findings enrich the robust statistics domain by providing tangible error bounds and highlight the challenges of personalized learning in distributed environments. The framework for component-specific analysis could inform future developments in multi-dimensional data spaces or contexts with differing component variances.

Further exploration is warranted along several lines:

• Extension to High Dimensions: The methodology adapts well to scalar data but scaling to higher dimensions poses computational and analytical complexity challenges.
• Comprehensive Finite Sample Analysis: Practical federated learnings often operate under finite sample settings, making such analysis critical in bridging theory with real-world applicability.
• Diversification in Mixture Model Parameters: Future research might encompass non-uniform component distributions with unknown mixing parameters to enhance the versatility of the model.

The paper substantiates the notion that collaborative filtering mechanisms in federated learning can be robust against corruption, even under varying data conditions and adversarial pressures, fostering advancements in secure and personalized distributed computing.