--Federated learning, as a distributed architecture, shows great promise for applications in Cyber-Physical-Social Systems (CPSS). In order to mitigate the privacy risks inherent in CPSS, the integration of differential privacy with federated learning has attracted considerable attention. Existing research mainly focuses on dynamically adjusting the noise added or discarding certain gradients to mitigate the noise introduced by differential privacy. However, these approaches fail to remove the noise that hinders convergence and correct the gradients affected by the noise, which significantly reduces the accuracy of model classification. T o overcome these challenges, this paper proposes a novel framework for differentially private federated learning that balances rigorous privacy guarantees with accuracy by introducing a server-side gradient correction mechanism. Specifically, after clients perform gradient clipping and noise perturbation, our framework detects deviations in the noisy local gradients and employs a projection mechanism to correct them, mitigating the negative impact of noise. Simultaneously, gradient projection promotes the alignment of gradients from different clients and guides the model towards convergence to a global optimum. We evaluate our framework on several benchmark datasets, and the experimental results demonstrate that it achieves state-of-the-art performance under the same privacy budget. HE proliferation of Cyber-Physical Systems (CPSS) has revolutionized a wide range of sectors, including smart manufacturing, autonomous driving [1], and healthcare [2], [3], driving substantial advancements in both efficiency and intelligence [4]. By seamlessly integrating computation, communication, and physical processes, CPSS enables sophisticated real-time monitoring and control within complex systems, offering significant improvements in operational performance and decision-making processes [5]. Jiayi Wan is with the School of Software Engineering, Nanjing University of Information Science and Technology, Nanjing, China (email: wanjiayi117@gmail.com). Xiang Zhu is with the College of Meteorology and Oceanography, National University of Defense Technology, Changsha, China (email: zhuxi-ang@nudt.edu.cn).

Recent theoretical results demonstrate that the convergence rates of permutation-based SGD (e.g., random reshuffling SGD) are faster than uniform-sampling SGD; however, these studies focus mainly on the large epoch regime, where the number of epochs $K$ exceeds the condition number $κ$. In contrast, little is known when $K$ is smaller than $κ$, and it is still a challenging open question whether permutation-based SGD can converge faster in this small epoch regime (Safran and Shamir, 2021). As a step toward understanding this gap, we study the naive deterministic variant, Incremental Gradient Descent (IGD), on smooth and strongly convex functions. Our lower bounds reveal that for the small epoch regime, IGD can exhibit surprisingly slow convergence even when all component functions are strongly convex. Furthermore, when some component functions are allowed to be nonconvex, we prove that the optimality gap of IGD can be significantly worse throughout the small epoch regime. Our analyses reveal that the convergence properties of permutation-based SGD in the small epoch regime may vary drastically depending on the assumptions on component functions. Lastly, we supplement the paper with tight upper and lower bounds for IGD in the large epoch regime.

We study gradient descent (GD) with a constant stepsize for $\ell_2$-regularized logistic regression with linearly separable data. Classical theory suggests small stepsizes to ensure monotonic reduction of the optimization objective, achieving exponential convergence in $\widetilde{\mathcal{O}}(κ)$ steps with $κ$ being the condition number. Surprisingly, we show that this can be accelerated to $\widetilde{\mathcal{O}}(\sqrtκ)$ by simply using a large stepsize -- for which the objective evolves nonmonotonically. The acceleration brought by large stepsizes extends to minimizing the population risk for separable distributions, improving on the best-known upper bounds on the number of steps to reach a near-optimum. Finally, we characterize the largest stepsize for the local convergence of GD, which also determines the global convergence in special scenarios. Our results extend the analysis of Wu et al. (2024) from convex settings with minimizers at infinity to strongly convex cases with finite minimizers.

We study the advantages of accelerated gradient methods, specifically based on the Frank-Wolfe method and projected gradient descent, for privacy and heavy-tailed robustness. Our approaches are as follows: For the Frank-Wolfe method, our technique is based on a tailored learning rate and a uniform lower bound on the gradient of the $\ell_2$-norm over the constraint set. For accelerating projected gradient descent, we use the popular variant based on Nesterov's momentum, and we optimize our objective over $\mathbb{R}^p$. These accelerations reduce iteration complexity, translating into stronger statistical guarantees for empirical and population risk minimization. Our analysis covers three settings: non-random data, random model-free data, and parametric models (linear regression and generalized linear models). Methodologically, we approach both privacy and robustness based on noisy gradients. We ensure differential privacy via the Gaussian mechanism and advanced composition, and we achieve heavy-tailed robustness using a geometric median-of-means estimator, which also sharpens the dependency on the dimension of the covariates. Finally, we compare our rates to existing bounds and identify scenarios where our methods attain optimal convergence.

Knowledge Distillation (KD) transfers knowledge from a large teacher model to a smaller student model by minimizing the divergence between their output distributions, typically using forward Kullback-Leibler divergence (FKLD) or reverse KLD (RKLD). It has become an effective training paradigm due to the broader supervision information provided by the teacher distribution compared to one-hot labels. We identify that the core challenge in KD lies in balancing two mode-concentration effects: the \textbf{\textit{Hardness-Concentration}} effect, which refers to focusing on modes with large errors, and the \textbf{\textit{Confidence-Concentration}} effect, which refers to focusing on modes with high student confidence. Through an analysis of how probabilities are reassigned during gradient updates, we observe that these two effects are entangled in FKLD and RKLD, but in extreme forms. Specifically, both are too weak in FKLD, causing the student to fail to concentrate on the target class. In contrast, both are too strong in RKLD, causing the student to overly emphasize the target class while ignoring the broader distributional information from the teacher. To address this imbalance, we propose ABKD, a generic framework with $α$-$β$-divergence. Our theoretical results show that ABKD offers a smooth interpolation between FKLD and RKLD, achieving an effective trade-off between these effects. Extensive experiments on 17 language/vision datasets with 12 teacher-student settings confirm its efficacy. The code is available at https://github.com/ghwang-s/abkd.

Recent dimension-free communication frameworks in Federated Learning (FL), such as DeComFL, significantly reduce per-round communication by transmitting only scalars via zeroth-order stochastic gradient descent (ZO-SGD). This method is particularly advantageous for federated fine-tuning of Large Language Models (LLMs). Yet, the high variance in ZO gradient estimation typically leads to slow convergence. Although leveraging Hessian information is known to enhance optimization speed, integrating this into FL presents significant challenges. These include clients' restrictions on local data and the critical need to maintain the dimension-free communication property. To overcome this limitation, we first introduce a generalized scalar-only communication FL framework that decouples dimension-free communication from standard ZO-SGD, enabling the integration of more advanced optimization strategies. Building on this framework, we propose HiSo, a fast federated fine-tuning method via Hessian-informed zeroth-order optimization and Scalar-only communication. Specifically, it leverages global curvature information to accelerate convergence while preserving the same minimal communication cost per round. Theoretically, we establish convergence guarantees that are independent of the global Lipschitz constant, and further show that HiSo achieves faster rates when the global Hessian exhibits a low effective rank -- a common phenomenon in LLMs. Extensive experiments on benchmark datasets and LLM fine-tuning tasks confirm that HiSo significantly outperforms existing ZO-based FL methods in both convergence speed and communication efficiency.

We introduce the attention-indexed model (AIM), a theoretical framework for analyzing learning in deep attention layers. Inspired by multi-index models, AIM captures how token-level outputs emerge from layered bilinear interactions over high-dimensional embeddings. Unlike prior tractable attention models, AIM allows full-width key and query matrices, aligning more closely with practical transformers. Using tools from statistical mechanics and random matrix theory, we derive closed-form predictions for Bayes-optimal generalization error and identify sharp phase transitions as a function of sample complexity, model width, and sequence length. We propose a matching approximate message passing algorithm and show that gradient descent can reach optimal performance. AIM offers a solvable playground for understanding learning in modern attention architectures.

Using stochastic differential equation (SDE) approximations, we study the dynamics of Distributed SGD, Distributed Compressed SGD, and Distributed SignSGD under $(L_0,L_1)$-smoothness and flexible noise assumptions. Our analysis provides insights -- which we validate through simulation -- into the intricate interactions between batch noise, stochastic gradient compression, and adaptivity in this modern theoretical setup. For instance, we show that \textit{adaptive} methods such as Distributed SignSGD can successfully converge under standard assumptions on the learning rate scheduler, even under heavy-tailed noise. On the contrary, Distributed (Compressed) SGD with pre-scheduled decaying learning rate fails to achieve convergence, unless such a schedule also accounts for an inverse dependency on the gradient norm -- de facto falling back into an adaptive method.

Modern machine learning models are often trained in a setting where the number of parameters exceeds the number of training samples. To understand the implicit bias of gradient descent in such overparameterized models, prior work has studied diagonal linear neural networks in the regression setting. These studies have shown that, when initialized with small weights, gradient descent tends to favor solutions with minimal $\ell^1$-norm - an effect known as implicit regularization. In this paper, we investigate implicit regularization in diagonal linear neural networks of depth $D\ge 2$ for overparameterized linear regression problems. We focus on analyzing the approximation error between the limit point of gradient flow trajectories and the solution to the $\ell^1$-minimization problem. By deriving tight upper and lower bounds on the approximation error, we precisely characterize how the approximation error depends on the scale of initialization $α$. Our results reveal a qualitative difference between depths: for $D \ge 3$, the error decreases linearly with $α$, whereas for $D=2$, it decreases at rate $α^{1-\varrho}$, where the parameter $\varrho \in [0,1)$ can be explicitly characterized. Interestingly, this parameter is closely linked to so-called null space property constants studied in the sparse recovery literature. We demonstrate the asymptotic tightness of our bounds through explicit examples. Numerical experiments corroborate our theoretical findings and suggest that deeper networks, i.e., $D \ge 3$, may lead to better generalization, particularly for realistic initialization scales.

Mixture of Experts (MoE), an ensemble of specialized models equipped with a router that dynamically distributes each input to appropriate experts, has achieved successful results in the field of machine learning. However, theoretical understanding of this architecture is falling behind due to its inherent complexity. In this paper, we theoretically study the sample and runtime complexity of MoE following the stochastic gradient descent (SGD) when learning a regression task with an underlying cluster structure of single index models. On the one hand, we prove that a vanilla neural network fails in detecting such a latent organization as it can only process the problem as a whole. This is intrinsically related to the concept of information exponent which is low for each cluster, but increases when we consider the entire task. On the other hand, we show that a MoE succeeds in dividing this problem into easier subproblems by leveraging the ability of each expert to weakly recover the simpler function corresponding to an individual cluster. To the best of our knowledge, this work is among the first to explore the benefits of the MoE framework by examining its SGD dynamics in the context of nonlinear regression.