Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws

• URL: http://arxiv.org/abs/2508.03688v1
• Date: Tue, 05 Aug 2025 17:57:56 GMT
• Title: Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws
• Authors: Gérard Ben Arous, Murat A. Erdogdu, N. Mert Vural, Denny Wu,
• Abstract summary: We study the optimization and sample complexity of gradient-based training of a two-layer neural network with quadratic activation function in the high-dimensional regime.<n>We present a sharp analysis of the dynamics in the feature learning regime, for both the population limit and the finite-sample discretization.
• Score: 21.18373933718468
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the optimization and sample complexity of gradient-based training of a two-layer neural network with quadratic activation function in the high-dimensional regime, where the data is generated as $y \propto \sum_{j=1}^{r}\lambda_j \sigma\left(\langle \boldsymbol{\theta_j}, \boldsymbol{x}\rangle\right), \boldsymbol{x} \sim N(0,\boldsymbol{I}_d)$, $\sigma$ is the 2nd Hermite polynomial, and $\lbrace\boldsymbol{\theta}_j \rbrace_{j=1}^{r} \subset \mathbb{R}^d$ are orthonormal signal directions. We consider the extensive-width regime $r \asymp d^\beta$ for $\beta \in [0, 1)$, and assume a power-law decay on the (non-negative) second-layer coefficients $\lambda_j\asymp j^{-\alpha}$ for $\alpha \geq 0$. We present a sharp analysis of the SGD dynamics in the feature learning regime, for both the population limit and the finite-sample (online) discretization, and derive scaling laws for the prediction risk that highlight the power-law dependencies on the optimization time, sample size, and model width. Our analysis combines a precise characterization of the associated matrix Riccati differential equation with novel matrix monotonicity arguments to establish convergence guarantees for the infinite-dimensional effective dynamics.

Related papers

• Emergence and scaling laws in SGD learning of shallow neural networks [46.632052892298375]
  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)
• Pruning is Optimal for Learning Sparse Features in High-Dimensions [15.967123173054535]
  We show that a class of statistical models can be optimally learned using pruned neural networks trained with gradient descent. We show that pruning neural networks proportional to the sparsity level of $boldsymbolV$ improves their sample complexity compared to unpruned networks.
  arXiv  Detail & Related papers  (2024-06-12T21:43:12Z)
• Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit [75.4661041626338]
  We study the problem of gradient descent learning of a single-index target function $f_*(boldsymbolx) = textstylesigma_*left(langleboldsymbolx,boldsymbolthetarangleright)$<n>We prove that a two-layer neural network optimized by an SGD-based algorithm learns $f_*$ with a complexity that is not governed by information exponents.
  arXiv  Detail & Related papers  (2024-06-03T17:56:58Z)
• Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs [56.237917407785545]
  We consider the problem of learning an $varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.
  arXiv  Detail & Related papers  (2024-05-10T09:58:47Z)
• 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)
• Neural Networks Efficiently Learn Low-Dimensional Representations with SGD [22.703825902761405]
  We show that SGD-trained ReLU NNs can learn a single-index target of the form $y=f(langleboldsymbolu,boldsymbolxrangle) + epsilon$ by recovering the principal direction. We also provide compress guarantees for NNs using the approximate low-rank structure produced by SGD.
  arXiv  Detail & Related papers  (2022-09-29T15:29:10Z)
• High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation [89.21686761957383]
  We study the first gradient descent step on the first-layer parameters $boldsymbolW$ in a two-layer network. Our results demonstrate that even one step can lead to a considerable advantage over random features.
  arXiv  Detail & Related papers  (2022-05-03T12:09:59Z)
• Nonparametric Learning of Two-Layer ReLU Residual Units [22.870658194212744]
  We describe an algorithm that learns two-layer residual units with rectified linear unit (ReLU) activation. We design layer-wise objectives as functionals whose analytic minimizers express the exact ground-truth network in terms of its parameters and nonlinearities. We prove the statistical strong consistency of our algorithm, and demonstrate the robustness and sample efficiency of our algorithm by experiments.
  arXiv  Detail & Related papers  (2020-08-17T22:11:26Z)
• Large-time asymptotics in deep learning [0.0]
  We consider the impact of the final time $T$ (which may indicate the depth of a corresponding ResNet) in training. For the classical $L2$--regularized empirical risk minimization problem, we show that the training error is at most of the order $mathcalOleft(frac1Tright)$. In the setting of $ellp$--distance losses, we prove that both the training error and the optimal parameters are at most of the order $mathcalOleft(e-mu
  arXiv  Detail & Related papers  (2020-08-06T07:33:17Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• The generalization error of max-margin linear classifiers: Benign overfitting and high dimensional asymptotics in the overparametrized regime [11.252856459394854]
  Modern machine learning classifiers often exhibit vanishing classification error on the training set. Motivated by these phenomena, we revisit high-dimensional maximum margin classification for linearly separable data.
  arXiv  Detail & Related papers  (2019-11-05T00:15:27Z)

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.