arXiv subjects

Zhongjie Shi

Publications and source records attributed to Zhongjie Shi.

11 recordsLinked to original sources

arXiv2026

Transformers as Cross-Task Learners: Shared Structure Drives Sample Efficiency in In-Context Learning

Transformers achieve remarkable performance by jointly learning broad families of tasks during pretraining and adapting to unseen tasks from only a short prompt. Yet a rigorous mathematical and statistical understanding of this phenomenon remains limited. This paper aims to study how Transformers exploit shared cross-task structure and how this structure affects the sample complexity of in-context learning (ICL). Specifically, we characterize task-space complexity through covering numbers under a prescribed metric, thereby quantifying the low-dimensional cross-task structure without requiring an explicit parametric representation. The resulting cover provides a set of anchor functions, which we use to introduce a task-identification-and-evaluation procedure: context observations localize an unseen task among the anchor functions, and the response at a query is predicted by aggregating the corresponding anchor function query evaluations. For approximation, we explicitly construct a Transformer with Softmax attention to approximate this procedure. For generalization, we derive an error bound that separates the effects of the number of pretraining tasks and the prompt length. The scaling with respect to the number of pretraining tasks is governed by the intrinsic dimensions of the task space and input domain; once sufficiently many tasks are available, the dependence on the prompt context length becomes dimension-free. To the best of our knowledge, this is the first work to quantify cross-task complexity for general nonlinear task families and explicitly construct a Transformer that exploits their low-dimensional structure to perform ICL. Our theory provides a quantitative explanation of how joint pretraining across related tasks improves in-context generalization.

stat.ML↗

arXiv2026

Learning Theory of Transformers: Local-to-Global Approximation via Softmax Partition of Unity

This paper investigates the learning theory of Transformer networks for regression tasks on the compact Euclidean domain $[0,1]^d$ and $d$-dimensional compact Riemannian manifolds. We propose a novel constructive approximation framework for Transformers that builds local approximations of the target function and aggregates them into a global approximation via softmax partition of unity. This approach leverages the attention mechanism to achieve spatial localization through affine transformations of the input. The softmax activation plays a crucial role in aggregating local approximations to a global output. From an approximation perspective, we prove that a dense Transformer equipped with only two encoder blocks and standard single-hidden-layer point-wise feed-forward networks can achieve a uniform $\varepsilon$-approximation error for $α$-Hölder continuous functions with $α\in (0,1]$ using $\mathcal{O}(\varepsilon^{-d/α})$ total parameters. Building upon this approximation guarantee, we establish a near minimax-optimal generalization error bound of order $\mathcal{O}\big(n^{-\frac{2α}{2α+d}} \log n\big)$ for the empirical risk minimizer, where $n$ is the training data size. The Transformer architecture studied in this paper is dense, shallow and wide, and employs softmax activation and sinusoidal positional encodings, closely reflecting practical implementations.

stat.ML↗

arXiv2026

High-Probability Convergence Theory for Distributed Composite Optimization with Sub-Weibull Noises

With the rapid development of distributed optimization (DO) theory, the distributed stochastic gradient methods (DSGMs) occupy an important position. Although the theory of different DSGMs has been widely established, the main-stream results of existing work are still derived under the condition of light-tailed stochastic gradient noises. Increasing examples from various fields, indicate that, the light-tailed noise model is overly idealized in many practical instances, failing to capture the complexity and variability of noises in real-world scenarios, such as the presence of outliers or extreme values from data science and statistical learning. To address this issue, we propose a new DO framework that incorporates stochastic gradients under sub-Weibull randomness. We study a distributed composite stochastic mirror descent scheme with sub-Weibull gradient noise (DCSMD-SW) for solving a convex distributed composite optimization (DCO) problem over the time-varying multi-agent network. By investigating sub-Weibull randomness in DCSMD for the first time, we show that the algorithm is applicable in some common heavier-tailed noise environments while also guaranteeing good convergence properties. We comprehensively study the convergence performance of DCSMD-SW. Satisfactory high-probability convergence rates are derived for DCSMD-SW without any smoothness requirement. The work also offers a unified analytical framework for several critical cases of both algorithms and noise environments.

math.OC↗

arXiv2026

Towards Understanding Generalization in DP-GD: A Case Study in Training Two-Layer CNNs

Modern deep learning techniques focus on extracting intricate information from data to achieve accurate predictions. However, the training datasets may be crowdsourced and include sensitive information, such as personal contact details, financial data, and medical records. As a result, there is a growing emphasis on developing privacy-preserving training algorithms for neural networks that maintain good performance while preserving privacy. In this paper, we investigate the generalization and privacy performances of the differentially private gradient descent (DP-GD) algorithm, which is a private variant of the gradient descent (GD) by incorporating additional noise into the gradients during each iteration. Moreover, we identify a concrete learning task where DP-GD can achieve superior generalization performance compared to GD in training two-layer Huberized ReLU convolutional neural networks (CNNs). Specifically, we demonstrate that, under mild conditions, a small signal-to-noise ratio can result in GD producing training models with poor test accuracy, whereas DP-GD can yield training models with good test accuracy and privacy guarantees if the signal-to-noise ratio is not too small. This indicates that DP-GD has the potential to enhance model performance while ensuring privacy protection in certain learning tasks. Numerical simulations are further conducted to support our theoretical results.

stat.ML↗

arXiv2025

Generic Frameworks for Distributed Functional Optimization and Learning over Time-Varying Networks

In this paper, we establish a distributed functional optimization (DFO) theory over time-varying networks. The vast majority of existing distributed optimization theories are developed based on Euclidean decision variables. However, for many scenarios in machine learning and statistical learning, such as reproducing kernel spaces or probability measure spaces that use functions or probability measures as fundamental variables, the development of existing distributed optimization theories exhibit obvious theoretical and technical deficiencies. This paper addresses these issues by developing a novel general DFO theory on Banach spaces, allowing functional learning problems in the aforementioned scenarios to be incorporated into our framework for resolution. We study both convex and nonconvex DFO problems and rigorously establish a comprehensive convergence theory of distributed functional mirror descent and distributed functional gradient descent algorithm to solve them. Satisfactory convergence rates are fully derived. The work has provided generic analyzing frameworks for DFO. The established theory is shown to have crucial application value in the kernel-based distributed learning theory over networks.

math.OC↗

arXiv2025

Theory of Decentralized Robust Kernel-Based Learning

We propose a new decentralized robust kernel-based learning algorithm within the framework of reproducing kernel Hilbert spaces (RKHSs) by utilizing a networked system that can be represented as a connected graph. The robust loss function $\huaL_σ$ induced by a windowing function $W$ and a robustness scaling parameter $σ>0$ can encompass a broad spectrum of robust losses. Consequently, the proposed algorithm effectively provides a unified decentralized learning framework for robust regression, which fundamentally differs from the existing distributed robust kernel-based learning schemes, all of which are divide-and-conquer based. We rigorously establish a learning theory and offer comprehensive convergence analysis for the algorithm. We show each local robust estimator generated from the decentralized algorithm can be utilized to approximate the regression function. Based on kernel-based integral operator techniques, we derive general high confidence convergence bounds for the local approximating sequence in terms of the mean square distance, RKHS norm, and generalization error, respectively. Moreover, we provide rigorous selection rules for local sample size and show that, under properly selected step size and scaling parameter $σ$, the decentralized robust algorithm can achieve optimal learning rates (up to logarithmic factors) in both norms. The parameter $σ$ is shown to be essential for enhancing robustness and ensuring favorable convergence behavior. The intrinsic connection among decentralization, sample selection, robustness of the algorithm, and its convergence is clearly reflected.

cs.LG↗

arXiv2025

Nonlinear functional regression by functional deep neural network with kernel embedding

Recently, deep learning has been widely applied in functional data analysis (FDA) with notable empirical success. However, the infinite dimensionality of functional data necessitates an effective dimension reduction approach for functional learning tasks, particularly in nonlinear functional regression. In this paper, we introduce a functional deep neural network with an adaptive and discretization-invariant dimension reduction method. Our functional network architecture consists of three parts: first, a kernel embedding step that features an integral transformation with an adaptive smooth kernel; next, a projection step that utilizes eigenfunction bases based on a projection Mercer kernel for the dimension reduction; and finally, a deep ReLU neural network is employed for the prediction. Explicit rates of approximating nonlinear smooth functionals across various input function spaces by our proposed functional network are derived. Additionally, we conduct a generalization analysis for the empirical risk minimization (ERM) algorithm applied to our functional net, by employing a novel two-stage oracle inequality and the established functional approximation results. Ultimately, we conduct numerical experiments on both simulated and real datasets to demonstrate the effectiveness and benefits of our functional net.

stat.ML↗

arXiv2024

Distributed Gradient Descent for Functional Learning

In recent years, different types of distributed and parallel learning schemes have received increasing attention for their strong advantages in handling large-scale data information. In the information era, to face the big data challenges {that} stem from functional data analysis very recently, we propose a novel distributed gradient descent functional learning (DGDFL) algorithm to tackle functional data across numerous local machines (processors) in the framework of reproducing kernel Hilbert space. Based on integral operator approaches, we provide the first theoretical understanding of the DGDFL algorithm in many different aspects of the literature. On the way of understanding DGDFL, firstly, a data-based gradient descent functional learning (GDFL) algorithm associated with a single-machine model is proposed and comprehensively studied. Under mild conditions, confidence-based optimal learning rates of DGDFL are obtained without the saturation boundary on the regularity index suffered in previous works in functional regression. We further provide a semi-supervised DGDFL approach to weaken the restriction on the maximal number of local machines to ensure optimal rates. To our best knowledge, the DGDFL provides the first divide-and-conquer iterative training approach to functional learning based on data samples of intrinsically infinite-dimensional random functions (functional covariates) and enriches the methodologies for functional data analysis.

stat.ML↗

arXiv2024

Can overfitted deep neural networks in adversarial training generalize? -- An approximation viewpoint

Adversarial training is a widely used method to improve the robustness of deep neural networks (DNNs) over adversarial perturbations. However, it is empirically observed that adversarial training on over-parameterized networks often suffers from the \textit{robust overfitting}: it can achieve almost zero adversarial training error while the robust generalization performance is not promising. In this paper, we provide a theoretical understanding of the question of whether overfitted DNNs in adversarial training can generalize from an approximation viewpoint. Specifically, our main results are summarized into three folds: i) For classification, we prove by construction the existence of infinitely many adversarial training classifiers on over-parameterized DNNs that obtain arbitrarily small adversarial training error (overfitting), whereas achieving good robust generalization error under certain conditions concerning the data quality, well separated, and perturbation level. ii) Linear over-parameterization (meaning that the number of parameters is only slightly larger than the sample size) is enough to ensure such existence if the target function is smooth enough. iii) For regression, our results demonstrate that there also exist infinitely many overfitted DNNs with linear over-parameterization in adversarial training that can achieve almost optimal rates of convergence for the standard generalization error. Overall, our analysis points out that robust overfitting can be avoided but the required model capacity will depend on the smoothness of the target function, while a robust generalization gap is inevitable. We hope our analysis will give a better understanding of the mathematical foundations of robustness in DNNs from an approximation view.

stat.ML↗

arXiv2023

Learning Theory of Distribution Regression with Neural Networks

In this paper, we aim at establishing an approximation theory and a learning theory of distribution regression via a fully connected neural network (FNN). In contrast to the classical regression methods, the input variables of distribution regression are probability measures. Then we often need to perform a second-stage sampling process to approximate the actual information of the distribution. On the other hand, the classical neural network structure requires the input variable to be a vector. When the input samples are probability distributions, the traditional deep neural network method cannot be directly used and the difficulty arises for distribution regression. A well-defined neural network structure for distribution inputs is intensively desirable. There is no mathematical model and theoretical analysis on neural network realization of distribution regression. To overcome technical difficulties and address this issue, we establish a novel fully connected neural network framework to realize an approximation theory of functionals defined on the space of Borel probability measures. Furthermore, based on the established functional approximation results, in the hypothesis space induced by the novel FNN structure with distribution inputs, almost optimal learning rates for the proposed distribution regression model up to logarithmic terms are derived via a novel two-stage error decomposition technique.

stat.ML↗

arXiv2021

Theory of Deep Convolutional Neural Networks III: Approximating Radial Functions

We consider a family of deep neural networks consisting of two groups of convolutional layers, a downsampling operator, and a fully connected layer. The network structure depends on two structural parameters which determine the numbers of convolutional layers and the width of the fully connected layer. We establish an approximation theory with explicit approximation rates when the approximated function takes a composite form $f\circ Q$ with a feature polynomial $Q$ and a univariate function $f$. In particular, we prove that such a network can outperform fully connected shallow networks in approximating radial functions with $Q(x) =|x|^2$, when the dimension $d$ of data from $\mathbb{R}^d$ is large. This gives the first rigorous proof for the superiority of deep convolutional neural networks in approximating functions with special structures. Then we carry out generalization analysis for empirical risk minimization with such a deep network in a regression framework with the regression function of the form $f\circ Q$. Our network structure which does not use any composite information or the functions $Q$ and $f$ can automatically extract features and make use of the composite nature of the regression function via tuning the structural parameters. Our analysis provides an error bound which decreases with the network depth to a minimum and then increases, verifying theoretically a trade-off phenomenon observed for network depths in many practical applications.

cs.LG↗