arXiv subjects

Jianfa Lai

Publications and source records attributed to Jianfa Lai.

6 recordsLinked to original sources

arXiv2026

Branch Scaling Manifests as Implicit Architectural Regularization for Improving Generalization in Overparameterized ResNets

Scaling factors in residual branches have emerged as a prevalent method for boosting neural network performance, especially in normalization-free architectures. While prior work has primarily examined scaling effects from an optimization perspective, this paper investigates their role in residual architectures through the lens of generalization theory. Specifically, we establish that wide residual networks (ResNets) with constant scaling factors become asymptotically unlearnable as depth increases. In contrast, when the scaling factor exhibits rapid depth-wise decay combined with early stopping, over-parameterized ResNets achieve minimax-optimal generalization rates. To establish this, we demonstrate that the generalization capability of wide ResNets can be approximated by kernel regression associated with the Neural Tangent Kernel (NTK). Our theoretical findings are validated through experiments on synthetic data and real-world classification tasks, including MNIST and CIFAR-100.

cs.LG↗

arXiv2026

Towards a Statistical Understanding of Neural Networks: Beyond the Neural Tangent Kernel Theories

A primary advantage of neural networks lies in their feature learning characteristics, which is challenging to theoretically analyze due to the complexity of their training dynamics. We examine feature learning and its potential benefits for generalization from a statistical perspective. After reviewing the neural tangent kernel (NTK) theory and recent results in kernel regression, which address the generalization issue of sufficiently wide neural networks, we examine limitations and implications of the fixed kernel theory (as the NTK theory) and review recent theoretical advancements in feature learning. Moving beyond theories with fixed features, we consider neural networks as adaptive feature models. Finally, we propose an over-parameterized Gaussian sequence model as a prototype for the adaptive feature model to study feature learning characteristics and motivate their future analysis for neural networks.

cs.LG↗

arXiv2024

The Optimality of Kernel Classifiers in Sobolev Space

Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability $η(x)=\mathbb{P}(Y=1\mid X=x)$, we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of $2η(x)-1$ and apply the method to real datasets.

math.ST↗

arXiv2023

Generalization Ability of Wide Residual Networks

In this paper, we study the generalization ability of the wide residual network on $\mathbb{S}^{d-1}$ with the ReLU activation function. We first show that as the width $m\rightarrow\infty$, the residual network kernel (RNK) uniformly converges to the residual neural tangent kernel (RNTK). This uniform convergence further guarantees that the generalization error of the residual network converges to that of the kernel regression with respect to the RNTK. As direct corollaries, we then show $i)$ the wide residual network with the early stopping strategy can achieve the minimax rate provided that the target regression function falls in the reproducing kernel Hilbert space (RKHS) associated with the RNTK; $ii)$ the wide residual network can not generalize well if it is trained till overfitting the data. We finally illustrate some experiments to reconcile the contradiction between our theoretical result and the widely observed ``benign overfitting phenomenon''

stat.ML↗

arXiv2023

Generalization Ability of Wide Neural Networks on $\mathbb{R}$

We perform a study on the generalization ability of the wide two-layer ReLU neural network on $\mathbb{R}$. We first establish some spectral properties of the neural tangent kernel (NTK): $a)$ $K_{d}$, the NTK defined on $\mathbb{R}^{d}$, is positive definite; $b)$ $λ_{i}(K_{1})$, the $i$-th largest eigenvalue of $K_{1}$, is proportional to $i^{-2}$. We then show that: $i)$ when the width $m\rightarrow\infty$, the neural network kernel (NNK) uniformly converges to the NTK; $ii)$ the minimax rate of regression over the RKHS associated to $K_{1}$ is $n^{-2/3}$; $iii)$ if one adopts the early stopping strategy in training a wide neural network, the resulting neural network achieves the minimax rate; $iv)$ if one trains the neural network till it overfits the data, the resulting neural network can not generalize well. Finally, we provide an explanation to reconcile our theory and the widely observed ``benign overfitting phenomenon''.

stat.ML↗

arXiv2023

Construction of Uniform Designs over Continuous Domain in Computer Experiments

The construction of uniform designs (UDs) has received much attention in computer experiments over the past decades, but most of the previous works obtain uniform designs over a U-type by lattice domain. Due to increasing demands for continuous factors, UDs over the continuous domain is in lack. Moreover, the uniformity can be further improved over a continuous domain. In this paper, we use coordinate descent methods with an initialization derived by threshold accepting (TA) algorithm to construct UDs over a continuous domain with Centered $L_2$-discrepancy. The new UDs perform better than the recorded ones in several computer experiments by the Kriging modeling.

stat.CO↗