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Generalization Guarantees for Neural Networks via Harnessing the Low-rank Structure of the Jacobian

Samet Oymak, Zalan Fabian, Mingchen Li, Mahdi Soltanolkotabi

TL;DR

This work develops a data-dependent framework to explain generalization in overparameterized neural nets trained by gradient descent by exploiting the low-rank structure of the network Jacobian. By decomposing learning dynamics into an information space spanned by top Jacobian singular vectors and a nuisance space, the authors derive explicit bias–variance bounds and show that fast learning and good generalization occur when labels lie largely in the information space and the Jacobian is approximately low-rank. They establish results for both random and arbitrary initializations, including pretraining, and provide case studies (e.g., Gaussian mixture models) and numerical experiments on CIFAR-10 validating the theory. The approach yields data-dependent guarantees that allow small-width networks to generalize on well-structured data, highlights the impact of label corruption, and offers a principled lens to understand gradient-descent dynamics beyond the NTK regime.

Abstract

Modern neural network architectures often generalize well despite containing many more parameters than the size of the training dataset. This paper explores the generalization capabilities of neural networks trained via gradient descent. We develop a data-dependent optimization and generalization theory which leverages the low-rank structure of the Jacobian matrix associated with the network. Our results help demystify why training and generalization is easier on clean and structured datasets and harder on noisy and unstructured datasets as well as how the network size affects the evolution of the train and test errors during training. Specifically, we use a control knob to split the Jacobian spectum into "information" and "nuisance" spaces associated with the large and small singular values. We show that over the information space learning is fast and one can quickly train a model with zero training loss that can also generalize well. Over the nuisance space training is slower and early stopping can help with generalization at the expense of some bias. We also show that the overall generalization capability of the network is controlled by how well the label vector is aligned with the information space. A key feature of our results is that even constant width neural nets can provably generalize for sufficiently nice datasets. We conduct various numerical experiments on deep networks that corroborate our theoretical findings and demonstrate that: (i) the Jacobian of typical neural networks exhibit low-rank structure with a few large singular values and many small ones leading to a low-dimensional information space, (ii) over the information space learning is fast and most of the label vector falls on this space, and (iii) label noise falls on the nuisance space and impedes optimization/generalization.

Generalization Guarantees for Neural Networks via Harnessing the Low-rank Structure of the Jacobian

TL;DR

This work develops a data-dependent framework to explain generalization in overparameterized neural nets trained by gradient descent by exploiting the low-rank structure of the network Jacobian. By decomposing learning dynamics into an information space spanned by top Jacobian singular vectors and a nuisance space, the authors derive explicit bias–variance bounds and show that fast learning and good generalization occur when labels lie largely in the information space and the Jacobian is approximately low-rank. They establish results for both random and arbitrary initializations, including pretraining, and provide case studies (e.g., Gaussian mixture models) and numerical experiments on CIFAR-10 validating the theory. The approach yields data-dependent guarantees that allow small-width networks to generalize on well-structured data, highlights the impact of label corruption, and offers a principled lens to understand gradient-descent dynamics beyond the NTK regime.

Abstract

Modern neural network architectures often generalize well despite containing many more parameters than the size of the training dataset. This paper explores the generalization capabilities of neural networks trained via gradient descent. We develop a data-dependent optimization and generalization theory which leverages the low-rank structure of the Jacobian matrix associated with the network. Our results help demystify why training and generalization is easier on clean and structured datasets and harder on noisy and unstructured datasets as well as how the network size affects the evolution of the train and test errors during training. Specifically, we use a control knob to split the Jacobian spectum into "information" and "nuisance" spaces associated with the large and small singular values. We show that over the information space learning is fast and one can quickly train a model with zero training loss that can also generalize well. Over the nuisance space training is slower and early stopping can help with generalization at the expense of some bias. We also show that the overall generalization capability of the network is controlled by how well the label vector is aligned with the information space. A key feature of our results is that even constant width neural nets can provably generalize for sufficiently nice datasets. We conduct various numerical experiments on deep networks that corroborate our theoretical findings and demonstrate that: (i) the Jacobian of typical neural networks exhibit low-rank structure with a few large singular values and many small ones leading to a low-dimensional information space, (ii) over the information space learning is fast and most of the label vector falls on this space, and (iii) label noise falls on the nuisance space and impedes optimization/generalization.

Paper Structure

This paper contains 33 sections, 30 theorems, 273 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation and contributions
  3. Model and training
  4. Components of a Jacobian-based theory of generalization
  5. Prelude: fitting a linear model
  6. Information and nuisance spaces of the Jacobian
  7. Main results
  8. Results for random initialization
  9. Generalization guarantees with arbitrary initialization
  10. Case Study: Gaussian mixture model
  11. Prior Art
  12. Numerical experiments
  13. Technical approach and General Theory
  14. Proofs
  15. Proofs for General Theory (Proof of Theorem \ref{['many step thm']})
  16. ...and 18 more sections

Key Result

Theorem 3.2

Let $\zeta, \Gamma, \bar{\alpha}$ be scalars obeying $\zeta\le 1/2$, $\Gamma\ge 1$, and $\bar{\alpha}\ge 0$ which determine the overall precision, cut-off and learning duration, respectively.Note that this theorem and its conclusions hold for any choice of these parameters in the specified range. Co with $\Gamma\ge 1$. We run gradient descent iterations of the form grad dec me with a learning rate

Figures (9)

  • Figure 1: Illustration of a one-hidden layer neural net with $d$ inputs, $k$ hidden units and ${K}$ outputs along with a one-hot encoded label.
  • Figure 2: Plots of the (a) total test error and (b) the test error components for the model in Section \ref{['linmodel']}. The test error decreases rapidly over the information subspace but slowly increases over the nuisance subspace.
  • Figure 3: Depiction of the training and generalization dynamics of gradient methods based on the information and nuisance spaces associated with the neural net Jacobian.
  • Figure 4: The singular values of the normalized Jacobian spectrum $\sqrt{\frac{KC}{n}}{\cal{J}}(\bm{W}_0)$ of a one-hidden layer neural network with $K=3$ outputs. Here, the data set is generated according to the Gaussian mixture model in Definition \ref{['GMM']} with $K=3$ classes and $\sigma=0.1$. We pick the cluster center so that the distance between any two is at least $0.5$. We consider two cases: $n=30C$ (solid line) and $n=60C$ (dashed line). These plots demonstrate that the top $KC$ singular values grow with the square root of the size of the data set ($\sqrt{n}$).
  • Figure 5: Histogram of the singular values of the initial and final Jacobian of the neural network during training.
  • ...and 4 more figures

Theorems & Definitions (36)

  • Definition 2.1: Information & Nuisance Spaces
  • Definition 3.1: Multiclass Neural Tangent Kernel (M-NTK) jacot2018neural
  • Theorem 3.2
  • Theorem 3.3
  • Definition 3.4: Gaussian mixture model
  • Theorem 3.5: Generalization for Gaussian Mixture Models-simplified
  • Definition 5.1: Reference Jacobian and its SVD
  • Definition 5.2: Information/Nuisance Subspaces
  • Theorem 5.3: Meta Theorem
  • Definition 6.1: early stopping value and distance
  • ...and 26 more