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The rapid increase in data generation, combined with the impracticality of centralizing large-scale datasets and the growing complexity of machine learning tasks, has driven the development of distributed learning techniques. Among these, Federated Learning (FL) has gained significant attention due to its privacy-preserving approach, where multiple clients collaboratively train a global model without sharing their local data. However, FL faces several key challenges, including data heterogeneity, high computational costs, and communication inefficiencies. These issues become more pronounced in real-world scenarios where client data distributions are non-IID, computational resources are limited, and communication is constrained.

This thesis addresses these challenges through the development of efficient algorithms for Personalized Federated Learning (pFL) and Constrained Federated Learning. The proposed approaches are designed to handle heterogeneous data, minimize computational overhead, and reduce communication costs while maintaining strong theoretical guarantees.

Specifically, the thesis introduces three key contributions: (1) pFLMF, a novel pFL formulation based on low-rank matrix optimization, leveraging Burer-Monteiro factorization to enable personalization without relying on predefined distance metrics. (2) PerMFL, an algorithm for multi-tier pFL that introduces personalized decision variables for both teams and individual devices, enabling efficient optimization in scenarios with hierarchical client structures. (3) FedFW, a projection-free algorithm for constrained FL, which emphasizes low computational cost, privacy preservation, and communication efficiency through sparse signal exchanges.

By addressing critical issues in FL, such as data heterogeneity, computation costs, and communication bottlenecks, the proposed algorithms advance the field of Federated Learning, providing robust and scalable solutions for real-world applications. 

This paper introduces a novel ridgelet transform-based method for Poisson image denoising. Our work focuses on harnessing the Poisson noise's unique non-additive and signal-dependent properties, distinguishing it from Gaussian noise. The core of our approach is a new thresholding scheme informed by theoretical insights into the ridgelet coefficients of Poisson-distributed images and adaptive thresholding guided by Stein's method. We verify our theoretical model through numerical experiments and demonstrate the potential of ridgelet thresholding across assorted scenarios. Our findings represent a significant step in enhancing the understanding of Poisson noise and offer an effective denoising method for images corrupted with it.

Federated learning (FL) has gained a lot of attention in recent years for building privacy-preserving collaborative learning systems. However, FL algorithms for constrained machine learning problems are still limited, particularly when the projection step is costly. To this end, we propose a Federated Frank-Wolfe Algorithm (FedFW). FedFW features data privacy, low per-iteration cost, and communication of sparse signals. In the deterministic setting, FedFW achieves an ε-suboptimal solution within O(ε^-2) iterations for smooth and convex objectives, and O(ε^-3) iterations for smooth but non-convex objectives. Furthermore, we present a stochastic variant of FedFW and show that it finds a solution within O(ε^-3) iterations in the convex setting. We demonstrate the empirical performance of FedFW on several machine learning tasks.

Personalized Federated Learning (pFL) has gained attention for building a suite of models tailored to different clients. In pFL, the challenge lies in balancing the reliance on local datasets, which may lack representativeness, against the diversity of other clients’ models, whose quality and relevance are uncertain. Focusing on the clustered FL scenario, where devices are grouped based on similarities intheir data distributions without prior knowledge of cluster memberships, we develop a mathematical model for pFL using low-rank matrix optimization. Building on this formulation, we propose a pFL approach leveraging the Burer-Monteiro factorization technique. We examine the convergence guarantees of the proposed method, and present numerical experiments on training deep neural networks, demonstrating the empirical performance of the proposed method in scenarios where personalization is crucial. 

The key challenge of personalized federated learning (PerFL) is to capture the statistical heterogeneity properties of data with inexpensive communications and gain customized performance for participating devices. To address these, we introduced personalized federated learning in multi-tier architecture (PerMFL) to obtain optimized and personalized local models when there are known team structures across devices. We provide theoretical guarantees of PerMFL, which offers linear convergence rates for smooth strongly convex problems and sub-linear convergence rates for smooth non-convex problems. We conduct numerical experiments demonstrating the robust empirical performance of PerMFL, outperforming the state-of-the-art in multiple personalized federated learning tasks.

Federated learning (FL) has gained much attention in recent years for building privacy-preserving collaborative learning systems. However, FL algorithms for constrained machine learning problems are still very limited, particularly when the projection step is costly. To this end, we propose a Federated Frank-Wolfe Algorithm (FedFW). FedFW provably finds an ε-suboptimal solution of the constrained empirical risk-minimization problem after O(ε−2) iterations if the objective function is convex. The rate becomes O(ε−3) if the objective is non-convex. The method enjoys data privacy, low per-iteration cost and communication of sparse signals. We demonstrate empirical performance of the FedFW algorithm on several machine learning tasks.