Sharp Generalization for Nonparametric Regression in Interpolation Space by Over-Parameterized Neural Networks Trained with Preconditioned Gradient Descent and Early Stopping

• URL: http://arxiv.org/abs/2407.11353v4
• Date: Sun, 05 Oct 2025 04:30:45 GMT
• Title: Sharp Generalization for Nonparametric Regression in Interpolation Space by Over-Parameterized Neural Networks Trained with Preconditioned Gradient Descent and Early Stopping
• Authors: Yingzhen Yang, Ping Li,
• Abstract summary: We study non regression using an over-parametricized two-layer neural networks trained with algorithmic guarantees.<n>We demonstrate that training the neural network with a novel Preconditioned Gradient Descent (PGD) algorithm, equipped with early stopping, achieves a sharp regression rate.
• Score: 15.975065054204753
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study nonparametric regression using an over-parameterized two-layer neural networks trained with algorithmic guarantees in this paper. We consider the setting where the training features are drawn uniformly from the unit sphere in $\RR^d$, and the target function lies in an interpolation space commonly studied in statistical learning theory. We demonstrate that training the neural network with a novel Preconditioned Gradient Descent (PGD) algorithm, equipped with early stopping, achieves a sharp regression rate of $\cO(n^{-\frac{2\alpha s'}{2\alpha s'+1}})$ when the target function is in the interpolation space $\bth{\cH_K}^{s'}$ with $s' \ge 3$. This rate is even sharper than the currently known nearly-optimal rate of $\cO(n^{-\frac{2\alpha s'}{2\alpha s'+1}})\log^2(1/\delta)$~\citep{Li2024-edr-general-domain}, where $n$ is the size of the training data and $\delta \in (0,1)$ is a small probability. This rate is also sharper than the standard kernel regression rate of $\cO(n^{-\frac{2\alpha}{2\alpha+1}})$ obtained under the regular Neural Tangent Kernel (NTK) regime when training the neural network with the vanilla gradient descent (GD), where $2\alpha = d/(d-1)$. Our analysis is based on two key technical contributions. First, we present a principled decomposition of the network output at each PGD step into a function in the reproducing kernel Hilbert space (RKHS) of a newly induced integral kernel, and a residual function with small $L^{\infty}$-norm. Second, leveraging this decomposition, we apply local Rademacher complexity theory to tightly control the complexity of the function class comprising all the neural network functions obtained in the PGD iterates. Our results further suggest that PGD enables the neural network to escape the linear NTK regime and achieve improved generalization.

Related papers

• Shallow Neural Networks Learn Low-Degree Spherical Polynomials with Learnable Channel Attention [11.227859599698588]
  We train an over-parametricized two-layer neural network (NN) with channel attention.<n>Our main result is the significantly improved sample complexity for learning such low-degrees.
  arXiv  Detail & Related papers  (2025-12-23T18:05:55Z)
• Neural Networks Learn Generic Multi-Index Models Near Information-Theoretic Limit [66.20349460098275]
  We study the gradient descent learning of a general Gaussian Multi-index model $f(boldsymbolx)=g(boldsymbolUboldsymbolx)$ with hidden subspace $boldsymbolUin mathbbRrtimes d$.<n>We prove that under generic non-degenerate assumptions on the link function, a standard two-layer neural network trained via layer-wise gradient descent can agnostically learn the target with $o_d(1)$ test error.
  arXiv  Detail & Related papers  (2025-11-19T04:46:47Z)
• Emergence and scaling laws in SGD learning of shallow neural networks [64.48316762675141]
  We study the complexity of online gradient descent (SGD) for learning a two-layer neural network with $P$ neurons on isotropic Gaussian data.<n>We provide a precise analysis of SGD dynamics for the training of a student two-layer network to minimize the mean squared error (MSE) objective.
  arXiv  Detail & Related papers  (2025-04-28T16:58:55Z)
• Gradient Descent Finds Over-Parameterized Neural Networks with Sharp Generalization for Nonparametric Regression: A Distribution-Free Analysis [19.988762532185884]
  We show that, if the neural network is trained by GD with early stopping, then the trained network renders a sharp rate of the nonparametric regression risk of $cO(eps_n2)$. It is remarked that our result does not require distributional assumptions on the training data.
  arXiv  Detail & Related papers  (2024-11-05T08:43:54Z)
• Sharper Guarantees for Learning Neural Network Classifiers with Gradient Methods [43.32546195968771]
  We study the data-dependent convergence and generalization behavior of gradient methods for neural networks with smooth activation. Our results improve upon the shortcomings of the well-established Rademacher complexity-based bounds. We show that a large step-size significantly improves upon the NTK regime's results in classifying the XOR distribution.
  arXiv  Detail & Related papers  (2024-10-13T21:49:29Z)
• Convergence Analysis of Natural Gradient Descent for Over-parameterized Physics-Informed Neural Networks [3.680127959836384]
  First-order methods, such as gradient descent (GD) and quadratic descent gradient (SGD) have been proven effective in training neural networks. However, the learning rate of GD for training two-layer neural networks exhibits poor dependence on the sample size and the Gram matrix. In this paper, we show that for the $L2$ regression problems, the learning rate can be improved from $mathcalO(1)$ to $mathcalO(1)$, which implies that GD actually enjoys a faster convergence rate.
  arXiv  Detail & Related papers  (2024-08-01T14:06:34Z)
• Enhanced Feature Learning via Regularisation: Integrating Neural Networks and Kernel Methods [0.0]
  We consider functions as expectations of Sobolev functions over all possible one-dimensional projections of the data.<n>This framework is similar to kernel ridge regression, where the kernel is $mathbbE_w ( k(B)(wtop x,wtop xprime))$, with $k(B)(a,b) := min(|a|, |b|)mathds1_ab>0$ the Brownian kernel, and the distribution of the projections $w$ is learnt
  arXiv  Detail & Related papers  (2024-07-24T13:46:50Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Generalization and Stability of Interpolating Neural Networks with Minimal Width [37.908159361149835]
  We investigate the generalization and optimization of shallow neural-networks trained by gradient in the interpolating regime. We prove the training loss number minimizations $m=Omega(log4 (n))$ neurons and neurons $Tapprox n$. With $m=Omega(log4 (n))$ neurons and $Tapprox n$, we bound the test loss training by $tildeO (1/)$.
  arXiv  Detail & Related papers  (2023-02-18T05:06:15Z)
• Gradient Descent in Neural Networks as Sequential Learning in RKBS [63.011641517977644]
  We construct an exact power-series representation of the neural network in a finite neighborhood of the initial weights. We prove that, regardless of width, the training sequence produced by gradient descent can be exactly replicated by regularized sequential learning.
  arXiv  Detail & Related papers  (2023-02-01T03:18:07Z)
• Training Overparametrized Neural Networks in Sublinear Time [14.918404733024332]
  Deep learning comes at a tremendous computational and energy cost. We present a new and a subset of binary neural networks, as a small subset of search trees, where each corresponds to a subset of search trees (Ds) We believe this view would have further applications in analysis analysis of deep networks (Ds)
  arXiv  Detail & Related papers  (2022-08-09T02:29:42Z)
• Bounding the Width of Neural Networks via Coupled Initialization -- A Worst Case Analysis [121.9821494461427]
  We show how to significantly reduce the number of neurons required for two-layer ReLU networks. We also prove new lower bounds that improve upon prior work, and that under certain assumptions, are best possible.
  arXiv  Detail & Related papers  (2022-06-26T06:51:31Z)
• On Feature Learning in Neural Networks with Global Convergence Guarantees [49.870593940818715]
  We study the optimization of wide neural networks (NNs) via gradient flow (GF) We show that when the input dimension is no less than the size of the training set, the training loss converges to zero at a linear rate under GF. We also show empirically that, unlike in the Neural Tangent Kernel (NTK) regime, our multi-layer model exhibits feature learning and can achieve better generalization performance than its NTK counterpart.
  arXiv  Detail & Related papers  (2022-04-22T15:56:43Z)
• A Convergence Analysis of Nesterov's Accelerated Gradient Method in Training Deep Linear Neural Networks [21.994004684742812]
  Momentum methods are widely used in training networks for their fast trajectory. We show that the convergence of the random number and $kappaO can converge to the global minimum. We extend our analysis to deep linear ResNets and derive a similar result.
  arXiv  Detail & Related papers  (2022-04-18T13:24:12Z)
• Why Lottery Ticket Wins? A Theoretical Perspective of Sample Complexity on Pruned Neural Networks [79.74580058178594]
  We analyze the performance of training a pruned neural network by analyzing the geometric structure of the objective function. We show that the convex region near a desirable model with guaranteed generalization enlarges as the neural network model is pruned.
  arXiv  Detail & Related papers  (2021-10-12T01:11:07Z)
• Towards an Understanding of Benign Overfitting in Neural Networks [104.2956323934544]
  Modern machine learning models often employ a huge number of parameters and are typically optimized to have zero training loss. We examine how these benign overfitting phenomena occur in a two-layer neural network setting. We show that it is possible for the two-layer ReLU network interpolator to achieve a near minimax-optimal learning rate.
  arXiv  Detail & Related papers  (2021-06-06T19:08:53Z)
• Beyond Lazy Training for Over-parameterized Tensor Decomposition [69.4699995828506]
  We show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data. Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.
  arXiv  Detail & Related papers  (2020-10-22T00:32:12Z)
• Learning Over-Parametrized Two-Layer ReLU Neural Networks beyond NTK [58.5766737343951]
  We consider the dynamic of descent for learning a two-layer neural network. We show that an over-parametrized two-layer neural network can provably learn with gradient loss at most ground with Tangent samples.
  arXiv  Detail & Related papers  (2020-07-09T07:09:28Z)
• Regularization Matters: A Nonparametric Perspective on Overparametrized Neural Network [20.132432350255087]
  Overparametrized neural networks trained by tangent descent (GD) can provably overfit any training data. This paper studies how well overparametrized neural networks can recover the true target function in the presence of random noises.
  arXiv  Detail & Related papers  (2020-07-06T01:02:23Z)
• A Revision of Neural Tangent Kernel-based Approaches for Neural Networks [34.75076385561115]
  We use the neural tangent kernel to show that networks can fit any finite training sample perfectly. A simple and analytic kernel function was derived as indeed equivalent to a fully-trained network. Our tighter analysis resolves the scaling problem and enables the validation of the original NTK-based results.
  arXiv  Detail & Related papers  (2020-07-02T05:07:55Z)
• Optimal Rates for Averaged Stochastic Gradient Descent under Neural Tangent Kernel Regime [50.510421854168065]
  We show that the averaged gradient descent can achieve the minimax optimal convergence rate. We show that the target function specified by the NTK of a ReLU network can be learned at the optimal convergence rate.
  arXiv  Detail & Related papers  (2020-06-22T14:31:37Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.