Ding-Xuan Zhou

Tong Mao, Ding-Xuan Zhou

Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from Hölder spaces by these networks is crucial for understanding the efficiency of the induced learning algorithms. Although the topic has been well investigated in the setting of deep neural networks with many layers of hidden neurons, it is still open for shallow networks having only one hidden layer. In this paper, we provide rates of uniform approximation by these networks. We show that ReLU shallow neural networks with $m$ hidden neurons can uniformly approximate functions from the Hölder space $W_\infty^r([-1, 1]^d)$ with rates $O((\log m)^{\frac{1}{2} +d}m^{-\frac{r}{d}\frac{d+2}{d+4}})$ when $r<d/2 +2$. Such rates are very close to the optimal one $O(m^{-\frac{r}{d}})$ in the sense that $\frac{d+2}{d+4}$ is close to $1$, when the dimension $d$ is large.

Yunfei Yang, Ding-Xuan Zhou

We study the approximation capacity of some variation spaces corresponding to shallow ReLU$^k$ neural networks. It is shown that sufficiently smooth functions are contained in these spaces with finite variation norms. For functions with less smoothness, the approximation rates in terms of the variation norm are established. Using these results, we are able to prove the optimal approximation rates in terms of the number of neurons for shallow ReLU$^k$ neural networks. It is also shown how these results can be used to derive approximation bounds for deep neural networks and convolutional neural networks (CNNs). As applications, we study convergence rates for nonparametric regression using three ReLU neural network models: shallow neural network, over-parameterized neural network, and CNN. In particular, we show that shallow neural networks can achieve the minimax optimal rates for learning Hölder functions, which complements recent results for deep neural networks. It is also proven that over-parameterized (deep or shallow) neural networks can achieve nearly optimal rates for nonparametric regression.

Yunwen Lei, Tianbao Yang, Yiming Ying et al.

Recently, contrastive learning has found impressive success in advancing the state of the art in solving various machine learning tasks. However, the existing generalization analysis is very limited or even not meaningful. In particular, the existing generalization error bounds depend linearly on the number $k$ of negative examples while it was widely shown in practice that choosing a large $k$ is necessary to guarantee good generalization of contrastive learning in downstream tasks. In this paper, we establish novel generalization bounds for contrastive learning which do not depend on $k$, up to logarithmic terms. Our analysis uses structural results on empirical covering numbers and Rademacher complexities to exploit the Lipschitz continuity of loss functions. For self-bounding Lipschitz loss functions, we further improve our results by developing optimistic bounds which imply fast rates in a low noise condition. We apply our results to learning with both linear representation and nonlinear representation by deep neural networks, for both of which we derive Rademacher complexity bounds to get improved generalization bounds.

Zihan Zhang, Lei Shi, Ding-Xuan Zhou

Deep neural networks (DNNs) trained with the logistic loss (i.e., the cross entropy loss) have made impressive advancements in various binary classification tasks. However, generalization analysis for binary classification with DNNs and logistic loss remains scarce. The unboundedness of the target function for the logistic loss is the main obstacle to deriving satisfactory generalization bounds. In this paper, we aim to fill this gap by establishing a novel and elegant oracle-type inequality, which enables us to deal with the boundedness restriction of the target function, and using it to derive sharp convergence rates for fully connected ReLU DNN classifiers trained with logistic loss. In particular, we obtain optimal convergence rates (up to log factors) only requiring the Hölder smoothness of the conditional class probability $η$ of data. Moreover, we consider a compositional assumption that requires $η$ to be the composition of several vector-valued functions of which each component function is either a maximum value function or a Hölder smooth function only depending on a small number of its input variables. Under this assumption, we derive optimal convergence rates (up to log factors) which are independent of the input dimension of data. This result explains why DNN classifiers can perform well in practical high-dimensional classification problems. Besides the novel oracle-type inequality, the sharp convergence rates given in our paper also owe to a tight error bound for approximating the natural logarithm function near zero (where it is unbounded) by ReLU DNNs. In addition, we justify our claims for the optimality of rates by proving corresponding minimax lower bounds. All these results are new in the literature and will deepen our theoretical understanding of classification with DNNs.

Puyu Wang, Yunwen Lei, Yiming Ying et al.

Recently there is a large amount of work devoted to the study of Markov chain stochastic gradient methods (MC-SGMs) which mainly focus on their convergence analysis for solving minimization problems. In this paper, we provide a comprehensive generalization analysis of MC-SGMs for both minimization and minimax problems through the lens of algorithmic stability in the framework of statistical learning theory. For empirical risk minimization (ERM) problems, we establish the optimal excess population risk bounds for both smooth and non-smooth cases by introducing on-average argument stability. For minimax problems, we develop a quantitative connection between on-average argument stability and generalization error which extends the existing results for uniform stability \cite{lei2021stability}. We further develop the first nearly optimal convergence rates for convex-concave problems both in expectation and with high probability, which, combined with our stability results, show that the optimal generalization bounds can be attained for both smooth and non-smooth cases. To the best of our knowledge, this is the first generalization analysis of SGMs when the gradients are sampled from a Markov process.

Yunfei Yang, Ding-Xuan Zhou

It is shown that over-parameterized neural networks can achieve minimax optimal rates of convergence (up to logarithmic factors) for learning functions from certain smooth function classes, if the weights are suitably constrained or regularized. Specifically, we consider the nonparametric regression of estimating an unknown $d$-variate function by using shallow ReLU neural networks. It is assumed that the regression function is from the Hölder space with smoothness $α<(d+3)/2$ or a variation space corresponding to shallow neural networks, which can be viewed as an infinitely wide neural network. In this setting, we prove that least squares estimators based on shallow neural networks with certain norm constraints on the weights are minimax optimal, if the network width is sufficiently large. As a byproduct, we derive a new size-independent bound for the local Rademacher complexity of shallow ReLU neural networks, which may be of independent interest.

Linhao Song, Jun Fan, Di-Rong Chen et al.

In recent years, functional neural networks have been proposed and studied in order to approximate nonlinear continuous functionals defined on $L^p([-1, 1]^s)$ for integers $s\ge1$ and $1\le p<\infty$. However, their theoretical properties are largely unknown beyond universality of approximation or the existing analysis does not apply to the rectified linear unit (ReLU) activation function. To fill in this void, we investigate here the approximation power of functional deep neural networks associated with the ReLU activation function by constructing a continuous piecewise linear interpolation under a simple triangulation. In addition, we establish rates of approximation of the proposed functional deep ReLU networks under mild regularity conditions. Finally, our study may also shed some light on the understanding of functional data learning algorithms.

Jianfei Li, Han Feng, Ding-Xuan Zhou

Deep neural networks (DNNs) have garnered significant attention in various fields of science and technology in recent years. Activation functions define how neurons in DNNs process incoming signals for them. They are essential for learning non-linear transformations and for performing diverse computations among successive neuron layers. In the last few years, researchers have investigated the approximation ability of DNNs to explain their power and success. In this paper, we explore the approximation ability of DNNs using a different activation function, called SignReLU. Our theoretical results demonstrate that SignReLU networks outperform rational and ReLU networks in terms of approximation performance. Numerical experiments are conducted comparing SignReLU with the existing activations such as ReLU, Leaky ReLU, and ELU, which illustrate the competitive practical performance of SignReLU.

Jianfei Li, Han Feng, Ding-Xuan Zhou

Deep learning based on deep neural networks has been very successful in many practical applications, but it lacks enough theoretical understanding due to the network architectures and structures. In this paper we establish some analysis for linear feature extraction by a deep multi-channel convolutional neural networks (CNNs), which demonstrates the power of deep learning over traditional linear transformations, like Fourier, wavelets, redundant dictionary coding methods. Moreover, we give an exact construction presenting how linear features extraction can be conducted efficiently with multi-channel CNNs. It can be applied to lower the essential dimension for approximating a high dimensional function. Rates of function approximation by such deep networks implemented with channels and followed by fully-connected layers are investigated as well. Harmonic analysis for factorizing linear features into multi-resolution convolutions plays an essential role in our work. Nevertheless, a dedicate vectorization of matrices is constructed, which bridges 1D CNN and 2D CNN and allows us to have corresponding 2D analysis.

Puyu Wang, Yunwen Lei, Yiming Ying et al.

Modern machine learning algorithms aim to extract fine-grained information from data to provide accurate predictions, which often conflicts with the goal of privacy protection. This paper addresses the practical and theoretical importance of developing privacy-preserving machine learning algorithms that ensure good performance while preserving privacy. In this paper, we focus on the privacy and utility (measured by excess risk bounds) performances of differentially private stochastic gradient descent (SGD) algorithms in the setting of stochastic convex optimization. Specifically, we examine the pointwise problem in the low-noise setting for which we derive sharper excess risk bounds for the differentially private SGD algorithm. In the pairwise learning setting, we propose a simple differentially private SGD algorithm based on gradient perturbation. Furthermore, we develop novel utility bounds for the proposed algorithm, proving that it achieves optimal excess risk rates even for non-smooth losses. Notably, we establish fast learning rates for privacy-preserving pairwise learning under the low-noise condition, which is the first of its kind.

Guanhang Lei, Zhen Lei, Lei Shi et al.

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on surfaces, including spheres. However, theoretical understanding of the numerical performance of PINNs, especially PINNs on surfaces or manifolds, is still lacking. In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere. By using and improving the latest approximation results of deep convolutional neural networks and spherical harmonic analysis, we prove an upper bound for the approximation error with respect to the Sobolev norm. Subsequently, we integrate this with innovative localization complexity analysis to establish fast convergence rates for PICNN. Our theoretical results are also confirmed and supplemented by our experiments. In light of these findings, we explore potential strategies for circumventing the curse of dimensionality that arises when solving high-dimensional PDEs.

Zhongjie Shi, Zhan Yu, Ding-Xuan Zhou

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.

Shao-Bo Lin, Di Wang, Ding-Xuan Zhou

This paper proposes a sketching strategy based on spherical designs, which is applied to the classical spherical basis function approach for massive spherical data fitting. We conduct theoretical analysis and numerical verifications to demonstrate the feasibility of the proposed { sketching} strategy. From the theoretical side, we prove that sketching based on spherical designs can reduce the computational burden of the spherical basis function approach without sacrificing its approximation capability. In particular, we provide upper and lower bounds for the proposed { sketching} strategy to fit noisy data on spheres. From the experimental side, we numerically illustrate the feasibility of the sketching strategy by showing its comparable fitting performance with the spherical basis function approach. These interesting findings show that the proposed sketching strategy is capable of fitting massive and noisy data on spheres.

Jianfei Li, Shuo Huang, Han Feng et al.

Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited interpretability. This work investigates how sparsity can help address these challenges in functional learning, a central ingredient in operator learning. We propose a framework that employs convolutional architectures to extract sparse features from a finite number of samples, together with deep fully connected networks to effectively approximate nonlinear functionals. Using universal discretization methods, we show that sparse approximators enable stable recovery from discrete samples. In addition, both the deterministic and the random sampling schemes are sufficient for our analysis. These findings lead to improved approximation rates and reduced sample sizes in various function spaces, including those with fast frequency decay and mixed smoothness. They also provide new theoretical insights into how sparsity can alleviate the curse of dimensionality in functional learning.

Ziyan Chen, Ding-Xuan Zhou

Scaling laws provide compact descriptions of how prediction error varies with compute, model size, and data, but existing theory mainly treats single-sample SGD or full data reuse, leaving the role of mini-batching unclear. We study batch scaling laws for sketched linear regression under a power-law covariance spectrum and a source condition on the target parameter. We analyze one-pass batch SGD, multi-pass batch SGD with replacement, and multi-pass batch SGD without replacement. Our first result is a risk decomposition: all three procedures share the same irreducible and approximation terms, while their stochastic terms depend on the sampling protocol. One-pass batch SGD splits into bias and variance, whereas the two multi-pass methods split into GD bias, GD variance, and a fluctuation term around a common GD reference trajectory. We then prove source-condition scaling laws for one-pass and multi-pass mini-batch methods. For one-pass batch SGD, mini-batching preserves the approximation and optimization-bias exponents, while the variance scales as $O(\min(M,(T_{\mathrm{eff}}γ)^{1/a})/(B T_{\mathrm{eff}}))$. Thus the usual $1/B$ covariance reduction holds at fixed update count $T$, but in the one-pass regime $T=N/B$ it is partly offset by the shorter optimization horizon. For multi-pass batch SGD, with- and without-replacement sampling have identical approximation and GD bias/variance terms; they differ only in the fluctuation covariance prefactor, which is $1/B$ with replacement and $ρ_{N,B}=(N-B)/(B(N-1))$ without replacement. Hence without-replacement sampling is less noisy for $B>1$, and when $B=N$ the fluctuation vanishes, recovering deterministic gradient descent. These results place batch size on the same theoretical footing as compute, data, and model dimension in sketched linear regression.

Zhi Han, Baichen Liu, Shao-Bo Lin et al.

This paper studies the performance of deep convolutional neural networks (DCNNs) with zero-padding in feature extraction and learning. After verifying the roles of zero-padding in enabling translation-equivalence, and pooling in its translation-invariance driven nature, we show that with similar number of free parameters, any deep fully connected networks (DFCNs) can be represented by DCNNs with zero-padding. This demonstrates that DCNNs with zero-padding is essentially better than DFCNs in feature extraction. Consequently, we derive universal consistency of DCNNs with zero-padding and show its translation-invariance in the learning process. All our theoretical results are verified by numerical experiments including both toy simulations and real-data running.

Di Wang, Xiaotong Liu, Shao-Bo Lin et al.

Data silos, mainly caused by privacy and interoperability, significantly constrain collaborations among different organizations with similar data for the same purpose. Distributed learning based on divide-and-conquer provides a promising way to settle the data silos, but it suffers from several challenges, including autonomy, privacy guarantees, and the necessity of collaborations. This paper focuses on developing an adaptive distributed kernel ridge regression (AdaDKRR) by taking autonomy in parameter selection, privacy in communicating non-sensitive information, and the necessity of collaborations in performance improvement into account. We provide both solid theoretical verification and comprehensive experiments for AdaDKRR to demonstrate its feasibility and effectiveness. Theoretically, we prove that under some mild conditions, AdaDKRR performs similarly to running the optimal learning algorithms on the whole data, verifying the necessity of collaborations and showing that no other distributed learning scheme can essentially beat AdaDKRR under the same conditions. Numerically, we test AdaDKRR on both toy simulations and two real-world applications to show that AdaDKRR is superior to other existing distributed learning schemes. All these results show that AdaDKRR is a feasible scheme to defend against data silos, which are highly desired in numerous application regions such as intelligent decision-making, pricing forecasting, and performance prediction for products.

Jianfei Li, Han Feng, Ding-Xuan Zhou

In this work, we explore the intersection of sparse coding theory and deep learning to enhance our understanding of feature extraction capabilities in advanced neural network architectures. We begin by introducing a novel class of Deep Sparse Coding (DSC) models and establish a thorough theoretical analysis of their uniqueness and stability properties. By applying iterative algorithms to these DSC models, we derive convergence rates for convolutional neural networks (CNNs) in their ability to extract sparse features. This provides a strong theoretical foundation for the use of CNNs in sparse feature-learning tasks. We additionally extend this convergence analysis to more general neural network architectures, including those with diverse activation functions, as well as self-attention and transformer-based models. This broadens the applicability of our findings to a wide range of deep learning methods for the extraction of deep-sparse features. Inspired by the strong connection between sparse coding and CNNs, we also explore training strategies to encourage neural networks to learn sparser features. Through numerical experiments, we demonstrate the effectiveness of these approaches, providing valuable insight for the design of efficient and interpretable deep learning models.

Shao-Bo Lin, Tao Li, Shaojie Tang et al.

With the help of massive data and rich computational resources, deep Q-learning has been widely used in operations research and management science and has contributed to great success in numerous applications, including recommender systems, supply chains, games, and robotic manipulation. However, the success of deep Q-learning lacks solid theoretical verification and interpretability. The aim of this paper is to theoretically verify the power of depth in deep Q-learning. Within the framework of statistical learning theory, we rigorously prove that deep Q-learning outperforms its traditional version by demonstrating its good generalization error bound. Our results reveal that the main reason for the success of deep Q-learning is the excellent performance of deep neural networks (deep nets) in capturing the special properties of rewards namely, spatial sparseness and piecewise constancy, rather than their large capacities. In this paper, we make fundamental contributions to the field of reinforcement learning by answering to the following three questions: Why does deep Q-learning perform so well? When does deep Q-learning perform better than traditional Q-learning? How many samples are required to achieve a specific prediction accuracy for deep Q-learning? Our theoretical assertions are verified by applying deep Q-learning in the well-known beer game in supply chain management and a simulated recommender system.

Guangrui Yang, Jianfei Li, Ming Li et al.

In this paper, we explore the approximation theory of functions defined on graphs. Our study builds upon the approximation results derived from the $K$-functional. We establish a theoretical framework to assess the lower bounds of approximation for target functions using Graph Convolutional Networks (GCNs) and examine the over-smoothing phenomenon commonly observed in these networks. Initially, we introduce the concept of a $K$-functional on graphs, establishing its equivalence to the modulus of smoothness. We then analyze a typical type of GCN to demonstrate how the high-frequency energy of the output decays, an indicator of over-smoothing. This analysis provides theoretical insights into the nature of over-smoothing within GCNs. Furthermore, we establish a lower bound for the approximation of target functions by GCNs, which is governed by the modulus of smoothness of these functions. This finding offers a new perspective on the approximation capabilities of GCNs. In our numerical experiments, we analyze several widely applied GCNs and observe the phenomenon of energy decay. These observations corroborate our theoretical results on exponential decay order.

Xiaotong Liu, Shao-Bo Lin, Jun Fan et al.

Synthetic data have gained increasing attention across various domains, with a growing emphasis on their performance in downstream prediction tasks. However, most existing synthesis strategies focus on maintaining statistical information. Although some studies address prediction performance guarantees, their single-stage synthesis designs make it challenging to balance the privacy requirements that necessitate significant perturbations and the prediction performance that is sensitive to such perturbations. We propose a two-stage synthesis strategy. In the first stage, we introduce a synthesis-then-hybrid strategy, which involves a synthesis operation to generate pure synthetic data, followed by a hybrid operation that fuses the synthetic data with the original data. In the second stage, we present a kernel ridge regression (KRR)-based synthesis strategy, where a KRR model is first trained on the original data and then used to generate synthetic outputs based on the synthetic inputs produced in the first stage. By leveraging the theoretical strengths of KRR and the covariant distribution retention achieved in the first stage, our proposed two-stage synthesis strategy enables a statistics-driven restricted privacy--prediction trade-off and guarantee optimal prediction performance. We validate our approach and demonstrate its characteristics of being statistics-driven and restricted in achieving the privacy--prediction trade-off both theoretically and numerically. Additionally, we showcase its generalizability through applications to a marketing problem and five real-world datasets.

Zhongjie Shi, Jun Fan, Linhao Song et al.

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.

Junyu Zhou, Puyu Wang, Ding-Xuan Zhou

While considerable theoretical progress has been devoted to the study of metric and similarity learning, the generalization mystery is still missing. In this paper, we study the generalization performance of metric and similarity learning by leveraging the specific structure of the true metric (the target function). Specifically, by deriving the explicit form of the true metric for metric and similarity learning with the hinge loss, we construct a structured deep ReLU neural network as an approximation of the true metric, whose approximation ability relies on the network complexity. Here, the network complexity corresponds to the depth, the number of nonzero weights and the computation units of the network. Consider the hypothesis space which consists of the structured deep ReLU networks, we develop the excess generalization error bounds for a metric and similarity learning problem by estimating the approximation error and the estimation error carefully. An optimal excess risk rate is derived by choosing the proper capacity of the constructed hypothesis space. To the best of our knowledge, this is the first-ever-known generalization analysis providing the excess generalization error for metric and similarity learning. In addition, we investigate the properties of the true metric of metric and similarity learning with general losses.

Yunfei Yang, Han Feng, Ding-Xuan Zhou

We study approximation and learning capacities of convolutional neural networks (CNNs) with one-side zero-padding and multiple channels. Our first result proves a new approximation bound for CNNs with certain constraint on the weights. Our second result gives new analysis on the covering number of feed-forward neural networks with CNNs as special cases. The analysis carefully takes into account the size of the weights and hence gives better bounds than the existing literature in some situations. Using these two results, we are able to derive rates of convergence for estimators based on CNNs in many learning problems. In particular, we establish minimax optimal convergence rates of the least squares based on CNNs for learning smooth functions in the nonparametric regression setting. For binary classification, we derive convergence rates for CNN classifiers with hinge loss and logistic loss. It is also shown that the obtained rates for classification are minimax optimal in some common settings.

Qin Fang, Lei Shi, Min Xu et al.

This paper investigates approximation capabilities of two-dimensional (2D) deep convolutional neural networks (CNNs), with Korobov functions serving as a benchmark. We focus on 2D CNNs, comprising multi-channel convolutional layers with zero-padding and ReLU activations, followed by a fully connected layer. We propose a fully constructive approach for building 2D CNNs to approximate Korobov functions and provide rigorous analysis of the complexity of the constructed networks. Our results demonstrate that 2D CNNs achieve near-optimal approximation rates under the continuous weight selection model, significantly alleviating the curse of dimensionality. This work provides a solid theoretical foundation for 2D CNNs and illustrates their potential for broader applications in function approximation.

Zihan Zhang, Lei Shi, Ding-Xuan Zhou

In this paper, we study the binary classification problem on $[0,1]^d$ under the Tsybakov noise condition (with exponent $s \in [0,\infty]$) and the compositional assumption. This assumption requires the conditional class probability function of the data distribution to be the composition of $q+1$ vector-valued multivariate functions, where each component function is either a maximum value function or a Hölder-$β$ smooth function that depends only on $d_*$ of its input variables. Notably, $d_*$ can be significantly smaller than the input dimension $d$. We prove that, under these conditions, the optimal convergence rate for the excess 0-1 risk of classifiers is $$ \left( \frac{1}{n} \right)^{\frac{β\cdot(1\wedgeβ)^q}{{\frac{d_*}{s+1}+(1+\frac{1}{s+1})\cdotβ\cdot(1\wedgeβ)^q}}}\;\;\;, $$ which is independent of the input dimension $d$. Additionally, we demonstrate that ReLU deep neural networks (DNNs) trained with hinge loss can achieve this optimal convergence rate up to a logarithmic factor. This result provides theoretical justification for the excellent performance of ReLU DNNs in practical classification tasks, particularly in high-dimensional settings. The technique used to establish these results extends the oracle inequality presented in our previous work. The generalized approach is of independent interest.

Zhan Yu, Zhongjie Shi, Ding-Xuan Zhou

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.

Nathanael Tepakbong, Ding-Xuan Zhou, Xiang Zhou

We study the classical binary classification problem for hypothesis spaces of Deep Neural Networks (DNNs) with ReLU activation under Tsybakov's low-noise condition with exponent $q>0$, and its limit-case $q\to\infty$ which we refer to as the "hard-margin condition". We show that DNNs which minimize the empirical risk with square loss surrogate and $\ell_p$ penalty can achieve finite-sample excess risk bounds of order $\mathcal{O}\left(n^{-α}\right)$ for arbitrarily large $α>0$ under the hard-margin condition, provided that the regression function $η$ is sufficiently smooth. The proof relies on a novel decomposition of the excess risk which might be of independent interest.

Junyu Zhou, Shuo Huang, Han Feng et al.

In this paper, we are concerned with the generalization performance of non-parametric estimation for pairwise learning. Most of the existing work requires the hypothesis space to be convex or a VC-class, and the loss to be convex. However, these restrictive assumptions limit the applicability of the results in studying many popular methods, especially kernel methods and neural networks. We significantly relax these restrictive assumptions and establish a sharp oracle inequality of the empirical minimizer with a general hypothesis space for the Lipschitz continuous pairwise losses. As an example, we apply our general results to study pairwise least squares regression and derive an excess population risk bound that matches the minimax lower bound for the pointwise least squares regression. The key novelty lies in constructing a structured deep ReLU neural network to approximate the true predictor, and in designing a targeted hypothesis space composed of networks with this structure and controllable complexity. Experiments validate the effectiveness of the proposed method. This example demonstrates that the obtained general results indeed help us to explore the generalization performance on a variety of problems that cannot be handled by existing approaches.

Puyu Wang, Yunwen Lei, Di Wang et al.

Recently, significant progress has been made in understanding the generalization of neural networks (NNs) trained by gradient descent (GD) using the algorithmic stability approach. However, most of the existing research has focused on one-hidden-layer NNs and has not addressed the impact of different network scaling parameters. In this paper, we greatly extend the previous work \cite{lei2022stability,richards2021stability} by conducting a comprehensive stability and generalization analysis of GD for multi-layer NNs. For two-layer NNs, our results are established under general network scaling parameters, relaxing previous conditions. In the case of three-layer NNs, our technical contribution lies in demonstrating its nearly co-coercive property by utilizing a novel induction strategy that thoroughly explores the effects of over-parameterization. As a direct application of our general findings, we derive the excess risk rate of $O(1/\sqrt{n})$ for GD algorithms in both two-layer and three-layer NNs. This sheds light on sufficient or necessary conditions for under-parameterized and over-parameterized NNs trained by GD to attain the desired risk rate of $O(1/\sqrt{n})$. Moreover, we demonstrate that as the scaling parameter increases or the network complexity decreases, less over-parameterization is required for GD to achieve the desired error rates. Additionally, under a low-noise condition, we obtain a fast risk rate of $O(1/n)$ for GD in both two-layer and three-layer NNs.

Zhan Yu, Jun Fan, Zhongjie Shi et al.

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.

Zhiying Fang, Yidong Ouyang, Ding-Xuan Zhou et al.

Deep learning models have been widely applied in various aspects of daily life. Many variant models based on deep learning structures have achieved even better performances. Attention-based architectures have become almost ubiquitous in deep learning structures. Especially, the transformer model has now defeated the convolutional neural network in image classification tasks to become the most widely used tool. However, the theoretical properties of attention-based models are seldom considered. In this work, we show that with suitable adaptations, the single-head self-attention transformer with a fixed number of transformer encoder blocks and free parameters is able to generate any desired polynomial of the input with no error. The number of transformer encoder blocks is the same as the degree of the target polynomial. Even more exciting, we find that these transformer encoder blocks in this model do not need to be trained. As a direct consequence, we show that the single-head self-attention transformer with increasing numbers of free parameters is universal. These surprising theoretical results clearly explain the outstanding performances of the transformer model and may shed light on future modifications in real applications. We also provide some experiments to verify our theoretical result.

Han Feng, Shao-Bo Lin, Ding-Xuan Zhou

This paper proposes a distributed weighted regularized least squares algorithm (DWRLS) based on spherical radial basis functions and spherical quadrature rules to tackle spherical data that are stored across numerous local servers and cannot be shared with each other. Via developing a novel integral operator approach, we succeed in deriving optimal approximation rates for DWRLS and theoretically demonstrate that DWRLS performs similarly as running a weighted regularized least squares algorithm with the whole data on a large enough machine. This interesting finding implies that distributed learning is capable of sufficiently exploiting potential values of distributively stored spherical data, even though every local server cannot access all the data.

Shao-Bo Lin, Yao Wang, Ding-Xuan Zhou

In this paper, we study the generalization performance of global minima for implementing empirical risk minimization (ERM) on over-parameterized deep ReLU nets. Using a novel deepening scheme for deep ReLU nets, we rigorously prove that there exist perfect global minima achieving almost optimal generalization error bounds for numerous types of data under mild conditions. Since over-parameterization is crucial to guarantee that the global minima of ERM on deep ReLU nets can be realized by the widely used stochastic gradient descent (SGD) algorithm, our results indeed fill a gap between optimization and generalization.

Tong Mao, Zhongjie Shi, Ding-Xuan Zhou

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.

Shao-Bo Lin, Kaidong Wang, Yao Wang et al.

Compared with avid research activities of deep convolutional neural networks (DCNNs) in practice, the study of theoretical behaviors of DCNNs lags heavily behind. In particular, the universal consistency of DCNNs remains open. In this paper, we prove that implementing empirical risk minimization on DCNNs with expansive convolution (with zero-padding) is strongly universally consistent. Motivated by the universal consistency, we conduct a series of experiments to show that without any fully connected layers, DCNNs with expansive convolution perform not worse than the widely used deep neural networks with hybrid structure containing contracting (without zero-padding) convolution layers and several fully connected layers.

Zhan Yu, Daniel W. C. Ho, Ding-Xuan Zhou

Regularization schemes for regression have been widely studied in learning theory and inverse problems. In this paper, we study distribution regression (DR) which involves two stages of sampling, and aims at regressing from probability measures to real-valued responses over a reproducing kernel Hilbert space (RKHS). Recently, theoretical analysis on DR has been carried out via kernel ridge regression and several learning behaviors have been observed. However, the topic has not been explored and understood beyond the least square based DR. By introducing a robust loss function $l_σ$ for two-stage sampling problems, we present a novel robust distribution regression (RDR) scheme. With a windowing function $V$ and a scaling parameter $σ$ which can be appropriately chosen, $l_σ$ can include a wide range of popular used loss functions that enrich the theme of DR. Moreover, the loss $l_σ$ is not necessarily convex, hence largely improving the former regression class (least square) in the literature of DR. The learning rates under different regularity ranges of the regression function $f_ρ$ are comprehensively studied and derived via integral operator techniques. The scaling parameter $σ$ is shown to be crucial in providing robustness and satisfactory learning rates of RDR.

Zhiying Fang, Han Feng, Shuo Huang et al.

Deep learning based on deep neural networks of various structures and architectures has been powerful in many practical applications, but it lacks enough theoretical verifications. In this paper, we consider a family of deep convolutional neural networks applied to approximate functions on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$. Our analysis presents rates of uniform approximation when the approximated function lies in the Sobolev space $W^r_\infty (\mathbb{S}^{d-1})$ with $r>0$ or takes an additive ridge form. Our work verifies theoretically the modelling and approximation ability of deep convolutional neural networks followed by downsampling and one fully connected layer or two. The key idea of our spherical analysis is to use the inner product form of the reproducing kernels of the spaces of spherical harmonics and then to apply convolutional factorizations of filters to realize the generated linear features.

Zhi Han, Siquan Yu, Shao-Bo Lin et al.

Deep learning is recognized to be capable of discovering deep features for representation learning and pattern recognition without requiring elegant feature engineering techniques by taking advantage of human ingenuity and prior knowledge. Thus it has triggered enormous research activities in machine learning and pattern recognition. One of the most important challenge of deep learning is to figure out relations between a feature and the depth of deep neural networks (deep nets for short) to reflect the necessity of depth. Our purpose is to quantify this feature-depth correspondence in feature extraction and generalization. We present the adaptivity of features to depths and vice-verse via showing a depth-parameter trade-off in extracting both single feature and composite features. Based on these results, we prove that implementing the classical empirical risk minimization on deep nets can achieve the optimal generalization performance for numerous learning tasks. Our theoretical results are verified by a series of numerical experiments including toy simulations and a real application of earthquake seismic intensity prediction.

Shao-Bo Lin, Di Wang, Ding-Xuan Zhou

This paper focuses on generalization performance analysis for distributed algorithms in the framework of learning theory. Taking distributed kernel ridge regression (DKRR) for example, we succeed in deriving its optimal learning rates in expectation and providing theoretically optimal ranges of the number of local processors. Due to the gap between theory and experiments, we also deduce optimal learning rates for DKRR in probability to essentially reflect the generalization performance and limitations of DKRR. Furthermore, we propose a communication strategy to improve the learning performance of DKRR and demonstrate the power of communications in DKRR via both theoretical assessments and numerical experiments.

Charles K. Chui, Shao-Bo Lin, Bo Zhang et al.

The great success of deep learning poses urgent challenges for understanding its working mechanism and rationality. The depth, structure, and massive size of the data are recognized to be three key ingredients for deep learning. Most of the recent theoretical studies for deep learning focus on the necessity and advantages of depth and structures of neural networks. In this paper, we aim at rigorous verification of the importance of massive data in embodying the out-performance of deep learning. To approximate and learn spatially sparse and smooth functions, we establish a novel sampling theorem in learning theory to show the necessity of massive data. We then prove that implementing the classical empirical risk minimization on some deep nets facilitates in realization of the optimal learning rates derived in the sampling theorem. This perhaps explains why deep learning performs so well in the era of big data.

Yuan Cao, Zhiying Fang, Yue Wu et al.

An intriguing phenomenon observed during training neural networks is the spectral bias, which states that neural networks are biased towards learning less complex functions. The priority of learning functions with low complexity might be at the core of explaining generalization ability of neural network, and certain efforts have been made to provide theoretical explanation for spectral bias. However, there is still no satisfying theoretical result justifying the underlying mechanism of spectral bias. In this paper, we give a comprehensive and rigorous explanation for spectral bias and relate it with the neural tangent kernel function proposed in recent work. We prove that the training process of neural networks can be decomposed along different directions defined by the eigenfunctions of the neural tangent kernel, where each direction has its own convergence rate and the rate is determined by the corresponding eigenvalue. We then provide a case study when the input data is uniformly distributed over the unit sphere, and show that lower degree spherical harmonics are easier to be learned by over-parameterized neural networks. Finally, we provide numerical experiments to demonstrate the correctness of our theory. Our experimental results also show that our theory can tolerate certain model misspecification in terms of the input data distribution.

Jinshan Zeng, Minrun Wu, Shao-Bo Lin et al.

In the era of big data, it is desired to develop efficient machine learning algorithms to tackle massive data challenges such as storage bottleneck, algorithmic scalability, and interpretability. In this paper, we develop a novel efficient classification algorithm, called fast polynomial kernel classification (FPC), to conquer the scalability and storage challenges. Our main tools are a suitable selected feature mapping based on polynomial kernels and an alternating direction method of multipliers (ADMM) algorithm for a related non-smooth convex optimization problem. Fast learning rates as well as feasibility verifications including the efficiency of an ADMM solver with convergence guarantees and the selection of center points are established to justify theoretical behaviors of FPC. Our theoretical assertions are verified by a series of simulations and real data applications. Numerical results demonstrate that FPC significantly reduces the computational burden and storage memory of existing learning schemes such as support vector machines, Nyström and random feature methods, without sacrificing their generalization abilities much.

Shao-Bo Lin, Yu Guang Wang, Ding-Xuan Zhou

Problems in astrophysics, space weather research and geophysics usually need to analyze noisy big data on the sphere. This paper develops distributed filtered hyperinterpolation for noisy data on the sphere, which assigns the data fitting task to multiple servers to find a good approximation of the mapping of input and output data. For each server, the approximation is a filtered hyperinterpolation on the sphere by a small proportion of quadrature nodes. The distributed strategy allows parallel computing for data processing and model selection and thus reduces computational cost for each server while preserves the approximation capability compared to the filtered hyperinterpolation. We prove quantitative relation between the approximation capability of distributed filtered hyperinterpolation and the numbers of input data and servers. Numerical examples show the efficiency and accuracy of the proposed method.

Charles K. Chui, Shao-Bo Lin, Ding-Xuan Zhou

Based on the tree architecture, the objective of this paper is to design deep neural networks with two or more hidden layers (called deep nets) for realization of radial functions so as to enable rotational invariance for near-optimal function approximation in an arbitrarily high dimensional Euclidian space. It is shown that deep nets have much better performance than shallow nets (with only one hidden layer) in terms of approximation accuracy and learning capabilities. In particular, for learning radial functions, it is shown that near-optimal rate can be achieved by deep nets but not by shallow nets. Our results illustrate the necessity of depth in neural network design for realization of rotation-invariance target functions.

Jinshan Zeng, Shao-Bo Lin, Yuan Yao et al.

In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called \textit{sigmoid-ADMM pair}), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of deep neural networks with sigmoid-type activations (called deep sigmoid nets) over their rectified linear unit (ReLU) counterparts (called deep ReLU nets) in terms of approximation. In particular, we prove that the approximation capability of deep sigmoid nets is not worse than that of deep ReLU nets by showing that ReLU activation function can be well approximated by deep sigmoid nets with two hidden layers and finitely many free parameters but not vice-verse. We also establish the global convergence of the proposed ADMM for the nonlinearly constrained formulation of the deep sigmoid nets training from arbitrary initial points to a Karush-Kuhn-Tucker (KKT) point at a rate of order ${\cal O}(1/k)$. Besides sigmoid activation, such a convergence theorem holds for a general class of smooth activations. Compared with the widely used stochastic gradient descent (SGD) algorithm for the deep ReLU nets training (called ReLU-SGD pair), the proposed sigmoid-ADMM pair is practically stable with respect to the algorithmic hyperparameters including the learning rate, initial schemes and the pro-processing of the input data. Moreover, we find that to approximate and learn simple but important functions the proposed sigmoid-ADMM pair numerically outperforms the ReLU-SGD pair.

Ding-Xuan Zhou

Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoretical foundation for understanding the approximation or generalization ability of deep learning methods generated by the network architectures such as deep convolutional neural networks having convolutional structures. Here we show that a deep convolutional neural network (CNN) is universal, meaning that it can be used to approximate any continuous function to an arbitrary accuracy when the depth of the neural network is large enough. This answers an open question in learning theory. Our quantitative estimate, given tightly in terms of the number of free parameters to be computed, verifies the efficiency of deep CNNs in dealing with large dimensional data. Our study also demonstrates the role of convolutions in deep CNNs.

Charles K. Chui, Shao-Bo Lin, Ding-Xuan Zhou

The subject of deep learning has recently attracted users of machine learning from various disciplines, including: medical diagnosis and bioinformatics, financial market analysis and online advertisement, speech and handwriting recognition, computer vision and natural language processing, time series forecasting, and search engines. However, theoretical development of deep learning is still at its infancy. The objective of this paper is to introduce a deep neural network (also called deep-net) approach to localized manifold learning, with each hidden layer endowed with a specific learning task. For the purpose of illustrations, we only focus on deep-nets with three hidden layers, with the first layer for dimensionality reduction, the second layer for bias reduction, and the third layer for variance reduction. A feedback component also designed to eliminate outliers. The main theoretical result in this paper is the order $\mathcal O\left(m^{-2s/(2s+d)}\right)$ of approximation of the regression function with regularity $s$, in terms of the number $m$ of sample points, where the (unknown) manifold dimension $d$ replaces the dimension $D$ of the sampling (Euclidean) space for shallow nets.

Yunwen Lei, Ding-Xuan Zhou

In this paper we consider online mirror descent (OMD) algorithms, a class of scalable online learning algorithms exploiting data geometric structures through mirror maps. Necessary and sufficient conditions are presented in terms of the step size sequence $\{η_t\}_{t}$ for the convergence of an OMD algorithm with respect to the expected Bregman distance induced by the mirror map. The condition is $\lim_{t\to\infty}η_t=0, \sum_{t=1}^{\infty}η_t=\infty$ in the case of positive variances. It is reduced to $\sum_{t=1}^{\infty}η_t=\infty$ in the case of zero variances for which the linear convergence may be achieved by taking a constant step size sequence. A sufficient condition on the almost sure convergence is also given. We establish tight error bounds under mild conditions on the mirror map, the loss function, and the regularizer. Our results are achieved by some novel analysis on the one-step progress of the OMD algorithm using smoothness and strong convexity of the mirror map and the loss function.

Andreas Christmann, Daohong Xiang, Ding-Xuan Zhou

Regularized empirical risk minimization using kernels and their corresponding reproducing kernel Hilbert spaces (RKHSs) plays an important role in machine learning. However, the actually used kernel often depends on one or on a few hyperparameters or the kernel is even data dependent in a much more complicated manner. Examples are Gaussian RBF kernels, kernel learning, and hierarchical Gaussian kernels which were recently proposed for deep learning. Therefore, the actually used kernel is often computed by a grid search or in an iterative manner and can often only be considered as an approximation to the "ideal" or "optimal" kernel. The paper gives conditions under which classical kernel based methods based on a convex Lipschitz loss function and on a bounded and smooth kernel are stable, if the probability measure $P$, the regularization parameter $λ$, and the kernel $k$ may slightly change in a simultaneous manner. Similar results are also given for pairwise learning. Therefore, the topic of this paper is somewhat more general than in classical robust statistics, where usually only the influence of small perturbations of the probability measure $P$ on the estimated function is considered.