# Precise Asymptotic Analysis of Deep Random Feature Models

arXiv: https://arxiv.org/abs/2302.06210 · HTML: https://mlsanity.com/paper/2302.06210/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 53 references that are in the map.
- First posted on arXiv 13 Feb 2023.

## Builds on (most cited first, 22 of 22)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [Generalisation error in learning with random features and the hidden manifold model](https://mlsanity.com/paper/2002.09339/) (2020)
- [Learning curves of generic features maps for realistic datasets with a teacher-student model](https://mlsanity.com/paper/2102.08127/) (2021)
- [The Gaussian equivalence of generative models for learning with shallow neural networks](https://mlsanity.com/paper/2006.14709/) (2020)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [A Random Matrix Approach to Neural Networks](https://mlsanity.com/paper/1702.05419/) (2017)
- [Spectra of the Conjugate Kernel and Neural Tangent Kernel for linear-width neural networks](https://mlsanity.com/paper/2005.11879/) (2020)
- [Universality of empirical risk minimization](https://mlsanity.com/paper/2202.08832/) (2022)
- [SGD Learns the Conjugate Kernel Class of the Network](https://mlsanity.com/paper/1702.08503/) (2017)
- [Random Matrix Theory Proves that Deep Learning Representations of GAN-data Behave as Gaussian Mixtures](https://mlsanity.com/paper/2001.08370/) (2020)
- [Provable Benefits of Overparameterization in Model Compression: From Double Descent to Pruning Neural Networks](https://mlsanity.com/paper/2012.08749/) (2020)
- [Universality laws for randomized dimension reduction, with applications](https://mlsanity.com/paper/1511.09433/) (2015)
- [On the Inherent Regularization Effects of Noise Injection During Training](https://mlsanity.com/paper/2102.07379/) (2021)

## Built on by (most cited first, 22 of 22)

- [How Two-Layer Neural Networks Learn, One (Giant) Step at a Time](https://mlsanity.com/paper/2305.18270/) (2023)
- [A Theory of Non-Linear Feature Learning with One Gradient Step in Two-Layer Neural Networks](https://mlsanity.com/paper/2310.07891/) (2023)
- [Asymptotics of feature learning in two-layer networks after one gradient-step](https://mlsanity.com/paper/2402.04980/) (2024)
- [Nonlinear spiked covariance matrices and signal propagation in deep neural networks](https://mlsanity.com/paper/2402.10127/) (2024)
- [Dimension-free deterministic equivalents and scaling laws for random feature regression](https://mlsanity.com/paper/2405.15699/) (2024)
- [A Random Matrix Theory Perspective on the Spectrum of Learned Features and Asymptotic Generalization Capabilities](https://mlsanity.com/paper/2410.18938/) (2024)
- [Learning curves for deep structured Gaussian feature models](https://mlsanity.com/paper/2303.00564/) (2023)
- [Fundamental limits of overparametrized shallow neural networks for supervised learning](https://mlsanity.com/paper/2307.05635/) (2023)
- [Asymptotics of Learning with Deep Structured (Random) Features](https://mlsanity.com/paper/2402.13999/) (2024)
- [High-dimensional learning of narrow neural networks](https://mlsanity.com/paper/2409.13904/) (2024)
- [Learning dynamical systems with hit-and-run random feature maps](https://mlsanity.com/paper/2501.06661/) (2025)
- [Random features and polynomial rules](https://mlsanity.com/paper/2402.10164/) (2024)
- [Information-theoretic reduction of deep neural networks to linear models in the overparametrized proportional regime](https://mlsanity.com/paper/2505.03577/) (2025)
- [Towards Understanding the Word Sensitivity of Attention Layers: A Study via Random Features](https://mlsanity.com/paper/2402.02969/) (2024)
- [Asymptotic Analysis of Two-Layer Neural Networks after One Gradient Step under Gaussian Mixtures Data with Structure](https://mlsanity.com/paper/2503.00856/) (2025)
- [MUSO: Achieving Exact Machine Unlearning in Over-Parameterized Regimes](https://mlsanity.com/paper/2410.08557/) (2024)
- [Input-Label Correlation Governs a Linear-to-Nonlinear Transition in Random Features under Spiked Covariance](https://mlsanity.com/paper/2409.20250/) (2024)
- [One-Bit Quantization for Random Features Models](https://mlsanity.com/paper/2510.16250/) (2025)
- [A Gaussian Comparison Theorem for Training Dynamics in Machine Learning](https://mlsanity.com/paper/2603.09310/) (2026)
- [Characterizing the Generalization Error of Random Feature Regression with Arbitrary Data-Augmentation](https://mlsanity.com/paper/2605.10290/) (2026)
- [Feature Learning in Linear-Width Two-Layer Networks: Two vs. One Step of Gradient Descent](https://mlsanity.com/paper/2605.17767/) (2026)
- [Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Gaussian Matrix Product](https://mlsanity.com/paper/2606.30831/) (2026)

## Abstract (arXiv)

We provide exact asymptotic expressions for the performance of regression by an L-layer deep random feature (RF) model, where the input is mapped through multiple random embedding and non-linear activation functions. For this purpose, we establish two key steps: First, we prove a novel universality result for RF models and deterministic data, by which we demonstrate that a deep random feature model is equivalent to a deep linear Gaussian model that matches it in the first and second moments, at each layer. Second, we make use of the convex Gaussian Min-Max theorem multiple times to obtain the exact behavior of deep RF models. We further characterize the variation of the eigendistribution in different layers of the equivalent Gaussian model, demonstrating that depth has a tangible effect on model performance despite the fact that only the last layer of the model is being trained.