For quadratic targets in d dimensions, two-layer quadratic networks achieve lower risk when fully trained than in random features or neural tangent regimes if hidden units < d.
The Annals of Statistics , volume =
8 Pith papers cite this work. Polarity classification is still indexing.
8
Pith papers citing it
abstract
Interpolators -- estimators that achieve zero training error -- have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. In this paper, we study minimum $\ell_2$ norm ("ridgeless") interpolation in high-dimensional least squares regression. We consider two different models for the feature distribution: a linear model, where the feature vectors $x_i \in {\mathbb R}^p$ are obtained by applying a linear transform to a vector of i.i.d. entries, $x_i = \Sigma^{1/2} z_i$ (with $z_i \in {\mathbb R}^p$); and a nonlinear model, where the feature vectors are obtained by passing the input through a random one-layer neural network, $x_i = \varphi(W z_i)$ (with $z_i \in {\mathbb R}^d$, $W \in {\mathbb R}^{p \times d}$ a matrix of i.i.d. entries, and $\varphi$ an activation function acting componentwise on $W z_i$). We recover -- in a precise quantitative way -- several phenomena that have been observed in large-scale neural networks and kernel machines, including the "double descent" behavior of the prediction risk, and the potential benefits of overparametrization.
representative citing papers
Bounded overestimation of the number of factors in PCA preserves √T-valid inference and consistent factor-space recovery under a random-matrix local law.
Temperature scaling of density-matrix eigenvalues from LLM semantic embeddings optimizes proper-score calibration and corrects systematic overconfidence so entropy equals risk.
Introduces bounded discrete graphical models and the BRIDGE regularized score matching estimator with nonasymptotic error bounds and exact support recovery for high-dimensional discrete data.
Establishes convergence guarantees for overparameterized 2-layer ReLU networks in flow matching, generalization bounds for the velocity-field objective, and Wasserstein guarantees for generated samples, using multi-task representation learning bounds.
SPIN lets weak LLMs become strong by self-generating training data from previous model versions and training to prefer human-annotated responses over its own outputs, outperforming DPO even with extra GPT-4 data on benchmarks.
A sparse-activation model predicts double-descent loss with distinct under- and over-parameterized scaling exponents set by sparsity, plus a compute-optimal frontier favoring dataset growth.
citing papers explorer
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Limitations of Lazy Training of Two-layers Neural Networks
For quadratic targets in d dimensions, two-layer quadratic networks achieve lower risk when fully trained than in random features or neural tangent regimes if hidden units < d.
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Fixed-order PCA: Theory for Overestimated Factor Models
Bounded overestimation of the number of factors in PCA preserves √T-valid inference and consistent factor-space recovery under a random-matrix local law.
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Eigenvalue Calibration for Semantic Embeddings of Large Language Models
Temperature scaling of density-matrix eigenvalues from LLM semantic embeddings optimizes proper-score calibration and corrects systematic overconfidence so entropy equals risk.
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Estimation of High Dimensional Bounded Discrete Graphical Models via Regularized Generalized Score Matching
Introduces bounded discrete graphical models and the BRIDGE regularized score matching estimator with nonasymptotic error bounds and exact support recovery for high-dimensional discrete data.
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A Theory on Flow Matching with Neural Networks
Establishes convergence guarantees for overparameterized 2-layer ReLU networks in flow matching, generalization bounds for the velocity-field objective, and Wasserstein guarantees for generated samples, using multi-task representation learning bounds.
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Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models
SPIN lets weak LLMs become strong by self-generating training data from previous model versions and training to prefer human-annotated responses over its own outputs, outperforming DPO even with extra GPT-4 data on benchmarks.
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Asymmetric Scaling Laws from Sparse Features
A sparse-activation model predicts double-descent loss with distinct under- and over-parameterized scaling exponents set by sparsity, plus a compute-optimal frontier favoring dataset growth.
- Proximal Estimation and Inference