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

We study the problem of learning a low-degree spherical polynomial of degree $k_0 = Θ(1) \ge 1$ defined on the unit sphere in $\RR^d$ by training an over-parameterized two-layer neural network with augmented feature in this paper. Our main result is the significantly improved sample complexity for learning such low-degree polynomials. We show that, for any regression risk $\eps \in (0, Θ(d^{-k_0})]$, an over-parameterized two-layer neural network trained by a novel Gradient Descent with Projection (GDP) requires a sample complexity of $n \asymp Θ( \log(4/δ) \cdot d^{k_0}/\eps)$ with probability $1-δ$ for $δ\in (0,1)$, in contrast with the representative sample complexity Θ(d^{k_0} \max\set{\eps^{-2},\log d}). Moreover, such sample complexity is nearly unimprovable since the trained network renders a nearly optimal rate of the nonparametric regression risk of the order $\log({4}/δ) \cdot Θ(d^{k_0}/{n})$ with probability at least $1-δ$. On the other hand, the minimax optimal rate for the regression risk with a kernel of rank $Θ(d^{k_0})$ is $Θ(d^{k_0}/{n})$, so that the rate of the nonparametric regression risk of the network trained by GDP is nearly minimax optimal. In the case that the ground truth degree $k_0$ is unknown, we present a novel and provable adaptive degree selection algorithm which identifies the true degree and achieves the same nearly optimal regression rate. To the best of our knowledge, this is the first time that a nearly optimal risk bound is obtained by training an over-parameterized neural network with a popular activation function (ReLU) and algorithmic guarantee for learning low-degree spherical polynomials. Due to the feature learning capability of GDP, our results are beyond the regular Neural Tangent Kernel (NTK) limit.

A convergence analysis is developed for the regularized Newton method for training neural networks (NNs) in the overparameterized limit. As the number of hidden units tends to infinity, the NN training dynamics converge in probability to the solution of a deterministic limit equation involving a ``Newton neural tangent kernel'' (NNTK). Explicit rates characterizing this convergence are provided and, in the infinite-width limit, we prove that the NN converges exponentially fast to the target data (i.e., a global minimizer with zero loss). We show that this convergence is uniform across the frequency spectrum, addressing the spectral bias inherent in gradient descent. The eigenvalues of the NTK for gradient descent accumulate at zero, leading to slow convergence for target data with high-frequency components. In contrast, the NNTK has uniformly lower bounded eigenvalues if the regularization parameter is selected appropriately, allowing Newton's method to converge more quickly for data with high-frequency components. Mathematical challenges that need to be addressed in our analysis include the implicit parameter update of the Newton method with a potentially indefinite Hessian matrix and the fact that the dimension of this linear system of equations tends to infinity as the NN width grows. This complicates deriving the training dynamics in the overparameterized limit as well as proving the convergence of the finite-width dynamics thereto. The analysis identifies a scaling formula for selecting the regularization parameter, which we show can vanish at a suitable rate as the number of hidden units becomes larger. We prove that, for sufficiently large numbers of hidden units, the regularized Hessian remains positive definite during training and the Newton updates for individual NN parameters converge to zero, showing that the model behaves as a linearization around the initialization.

Grokking shows that fitting the training data and learning the underlying signal can occur at very different stages. However, existing theories offer limited quantitative insight into how this delayed generalization depends on inductive bias and signal structure. Our work addresses the gap by developing a statistical theory that characterizes how regularization geometry and signal sparsity govern generalization near interpolation. In particular, we focus on the prototypical setting of high-dimensional regression and identify regimes in which sparsity-promoting regularization makes exact interpolation much more accurate than approximate fitting. In strongly overparameterized noiseless problems, we prove a zero--one generalization law and construct a family of convex norms whose interpolators transition from the trivial risk of the all-zero predictor to exact recovery, while keeping the training error equal to $0$. Furthermore, when feature dimension and sample size are proportional, we provide a precise characterization of training and generalization errors along $\ell_r$-regularization paths. This in turn allows us to quantify the generalization gain that remains near interpolation: we show that this gain increases as the norm becomes more sparsity-promoting and as the target becomes sparser, with a sharp drop in generalization reached for noiseless data and $\ell_1$ regularization. Experiments on diagonal linear networks and transformers trained on modular arithmetic demonstrate the generality of our theoretical predictions. Finally, beyond grokking, our work reveals a statistical instability in minimum-norm interpolation: small perturbations in the regularization strength can lead to drastically different generalization, while preserving small training error.