聯邦學習（FL）是一種新的範式，它使許多客戶能夠在中央參數伺服器（PS）的協調下共同訓練機器學習模型，同時保持本地數據不直接暴露給任何第三方。有效的聯邦學習算法的發展面臨多個實際挑戰，包括高通信成本、數據異構性和客戶隱私保護。本論文嘗試通過在聯邦監督和非監督學習任務領域進行詳細的理論分析來應對這些挑戰。本論文的主要貢獻分為兩個部分。

本論文的第一部分探討了一類以凸但非平滑損失函數為特徵的聯邦問題。這些函數在聯邦學習應用中普遍存在，由於其複雜的非平滑性質和通信效率與隱私保護之間相沖突的要求，我們提出了一種新穎的帶有差分隱私（DP）的聯邦隨機 原-對偶演算法，稱為 FedSPD-DP，專為解決非平滑聯的邦學習問題。我們在理論上分析了DP噪聲、本地多步隨機梯度下降（local SGD）和部分客戶參與（PCP）對模型收斂性能的影響。具體而言，我們的理論分析表明，數據採樣策略和PCP可以增強數據隱私，而本地多步SGD可能增加隱私泄漏，揭示了算法通信效率和隱私保護之間的存在折中。實驗結果展示了我們提出的演算法在聯邦分類任務上的實際性能，並證明其相對於現有最先進演算法的優越性，同時驗證了所有的分析結果。

本論文的第二部分深入研究了聯邦聚類（FedC）問題， 其目標是在中央參數伺服器的協調下，準確地將分佈在眾多客戶端上的未標記數據樣本進行有效劃分，同時考慮數據隱私。儘管這是一個涉及表示簇中心和表示每個數據樣本的簇成員歸屬的實變量的 NP-hard 優化問題，但我們巧妙地將FedC問題重新建模為只有一個凸約束的非凸優化問題，從而得到一個擁有軟聚類解的問題。 然後，提出了一種採用DP技術的新型FedC演算法，稱為DP-FedC，該演算法考慮了部分客戶參與和本地模型多步更新策略。此外，通過對隱私保護和收斂速度的理論分析，特別是對於非獨立分佈的數據（non-i.i.d.）情況，獲得了所提出DP-FedC的各種屬性，這理論上可作為設計聯邦學習系統的指導方針。然後，我們在兩個真實數據集上展示了一些實驗結果，以驗證所提出DP-FedC演算法在聯邦聚類任務上的有效性， 以及相對於一些現有最先進的聯邦聚類算法的優越性，並與所有呈現的分析結果保持一致。

Federated learning (FL) is a new paradigm that enables many clients to jointly train a machine learning model under the orchestration of a central parameter server (PS) while keeping the local data not directly exposed to any third party. The development of effective FL algorithms faces multiple practical challenges, including high communication costs, data heterogeneity, and clients' privacy protection. This dissertation attempts to deal with these challenges with detailed theoretical analyses in the realm of both federated supervised and unsupervised learning tasks. The main contributions of this dissertation include two parts.

The first part of this dissertation explores a class of FL problems characterized by convex but non-smooth loss functions. These functions are prevalent in FL applications, posing a challenge due to their intricate non-smooth nature and the conflicting requirements of communication efficiency and privacy protection.
We propose a novel federated stochastic primal-dual algorithm with differential privacy (DP), referred to as FedSPD-DP, tailored for non-smooth FL problems.
We theoretically analyze the impact of DP noise, multiple steps of local stochastic gradient descent (local SGD) and partial client participation (PCP) on convergence performance. Specifically, our analysis reveals that the data sampling strategy and PCP can enhance data privacy, whereas a larger number of local SGD steps could increase privacy leakage, revealing a non-trivial tradeoff between algorithm communication efficiency and privacy protection. Experimental results are presented to evaluate the practical performance of the proposed algorithm on classification tasks and demonstrate its superior performance compared to state-of-the-art methods, together with the validation of all the analytical results and properties.

The second part of this dissertation delves into the federated clustering (FedC) problem, which aims to accurately partition unlabeled data samples distributed over numerous clients into finite clusters under the orchestration of the PS, while taking data privacy into consideration.
Though it is an NP-hard optimization problem involving real variables denoting cluster centroids and binary variables denoting the cluster membership of each data sample, we judiciously reformulate the FedC problem into a non-convex optimization problem with only one convex constraint, accordingly yielding a soft clustering solution. Then a novel FedC algorithm using DP technique, termed DP-FedC, is proposed in which PCP and local SGD are also considered. Furthermore, various attributes of the proposed DP-FedC are obtained through theoretical analyses of privacy protection and convergence rate, especially for the case of non-identically and independently distributed (non-i.i.d.) data, that ideally serve as the guidelines for the design of the proposed DP-FedC. Then some experimental results on two real datasets are provided to demonstrate the efficacy of the proposed DP-FedC on clustering tasks together with its much superior performance over some state-of-the-art FedC algorithms, and the consistency with all the presented analytical results.

Chinese Abstract ii
Abstract iv
Acknowledgments vi
List of Figures xi
List of Tables xiv
List of Notations xv
1 Introduction 1
2 The DP-FL System: A Review 6
2.1 FL System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Preliminaries of DP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 The DP-FL System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Noise Reduction in DP-FL System . . . . . . . . . . . . . . . . . . 11
2.3.2 Noise Reduction by Privacy Amplification . . . . . . . . . . . . . . 12
2.3.3 Noise Reduction by Model Sparsification . . . . . . . . . . . . . . 14
2.3.4 Noise Reduction by Reducing Sensitivity . . . . . . . . . . . . . . 16
2.4 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Differentially Private Federated Supervised Learning via PDM 20
3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Proposed FedSPD-DP Algorithm . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Privacy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.3 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.2 Impact of DP Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.3 Impact of Local SGD . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.4 Performance Comparison with State-of-the-Art Works . . . . . . . 35
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Differentially Private Federated Unsupervised Learning 37
4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.1 Centralized k-means Clustering . . . . . . . . . . . . . . . . . . . 37
4.1.2 Federated Clustering . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.3 Federated Clustering Model . . . . . . . . . . . . . . . . . . . . . 41
4.2 Proposed DP-FedC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Update of W and H i . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 Privacy Concern . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.2 Privacy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.3 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.2 Impact of DP Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.3 Impact of the Number of Participated Clients (K) . . . . . . . . . . 54
4.4.4 Comparison with Existing Distributed Clustering Methods . . . . . 55
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Conclusions and Future Works 59
A Proofs of Theorems in Chapter 2 61
A.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
B Proofs of Theorems and Lemmas in Chapter 3 63
B.1 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
B.2 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
B.3 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
B.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
B.3.2 Sketch of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 69
B.4 Proof of Key Lemmas in Theorem 4 . . . . . . . . . . . . . . . . . . . . . 74
B.4.1 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B.4.2 Proof of Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 75
B.4.3 Proof of Lemma 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B.4.4 Proof of Lemma 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.4.5 Proof of Lemma 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 85
C Proofs of Theorems and Lemmas in Chapter 4 87
C.1 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
C.2 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
C.3 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
C.4 Proofs of Key Lemmas for Theorem 7 . . . . . . . . . . . . . . . . . . . . 96
C.4.1 Proof of Lemma 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 96
C.4.2 Proof of Lemma 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 97
C.4.3 Proof of Lemma 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 98
C.5 Proof of Remark 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Bibliography 104
Publication List of The Author 112

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