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Implicit Regularization via Spectral Neural Networks and Non-linear Matrix Sensing

Hong T. M. Chu, Subhro Ghosh, Chi Thanh Lam, Soumendu Sundar Mukherjee

TL;DR

This work investigates implicit regularization in neural networks beyond linear settings by introducing Spectral Neural Networks (SNNs) that apply non-linear spectral activations to matrices. In the matrix sensing framework, the authors prove that gradient flow on an over-parameterized non-linear factorization converges exponentially to a nuclear-norm minimizer that fits the measurements, with the left and right singular vectors fixed by the problem data. They establish a compact, spectrally driven gradient-flow representation and show that the limiting network output $X_{\infty}$ achieves zero empirical loss and satisfies KKT conditions for nuclear-norm minimization, thereby rigorously demonstrating implicit regularization in this non-linear setting. Numerical experiments corroborate the theory, reveal robustness to assumptions, and demonstrate practical benefits in real-image reconstruction tasks, suggesting wide applicability of the spectral approach to matrix learning problems.

Abstract

The phenomenon of implicit regularization has attracted interest in recent years as a fundamental aspect of the remarkable generalizing ability of neural networks. In a nutshell, it entails that gradient descent dynamics in many neural nets, even without any explicit regularizer in the loss function, converges to the solution of a regularized learning problem. However, known results attempting to theoretically explain this phenomenon focus overwhelmingly on the setting of linear neural nets, and the simplicity of the linear structure is particularly crucial to existing arguments. In this paper, we explore this problem in the context of more realistic neural networks with a general class of non-linear activation functions, and rigorously demonstrate the implicit regularization phenomenon for such networks in the setting of matrix sensing problems, together with rigorous rate guarantees that ensure exponentially fast convergence of gradient descent.In this vein, we contribute a network architecture called Spectral Neural Networks (abbrv. SNN) that is particularly suitable for matrix learning problems. Conceptually, this entails coordinatizing the space of matrices by their singular values and singular vectors, as opposed to by their entries, a potentially fruitful perspective for matrix learning. We demonstrate that the SNN architecture is inherently much more amenable to theoretical analysis than vanilla neural nets and confirm its effectiveness in the context of matrix sensing, via both mathematical guarantees and empirical investigations. We believe that the SNN architecture has the potential to be of wide applicability in a broad class of matrix learning scenarios.

Implicit Regularization via Spectral Neural Networks and Non-linear Matrix Sensing

TL;DR

This work investigates implicit regularization in neural networks beyond linear settings by introducing Spectral Neural Networks (SNNs) that apply non-linear spectral activations to matrices. In the matrix sensing framework, the authors prove that gradient flow on an over-parameterized non-linear factorization converges exponentially to a nuclear-norm minimizer that fits the measurements, with the left and right singular vectors fixed by the problem data. They establish a compact, spectrally driven gradient-flow representation and show that the limiting network output achieves zero empirical loss and satisfies KKT conditions for nuclear-norm minimization, thereby rigorously demonstrating implicit regularization in this non-linear setting. Numerical experiments corroborate the theory, reveal robustness to assumptions, and demonstrate practical benefits in real-image reconstruction tasks, suggesting wide applicability of the spectral approach to matrix learning problems.

Abstract

The phenomenon of implicit regularization has attracted interest in recent years as a fundamental aspect of the remarkable generalizing ability of neural networks. In a nutshell, it entails that gradient descent dynamics in many neural nets, even without any explicit regularizer in the loss function, converges to the solution of a regularized learning problem. However, known results attempting to theoretically explain this phenomenon focus overwhelmingly on the setting of linear neural nets, and the simplicity of the linear structure is particularly crucial to existing arguments. In this paper, we explore this problem in the context of more realistic neural networks with a general class of non-linear activation functions, and rigorously demonstrate the implicit regularization phenomenon for such networks in the setting of matrix sensing problems, together with rigorous rate guarantees that ensure exponentially fast convergence of gradient descent.In this vein, we contribute a network architecture called Spectral Neural Networks (abbrv. SNN) that is particularly suitable for matrix learning problems. Conceptually, this entails coordinatizing the space of matrices by their singular values and singular vectors, as opposed to by their entries, a potentially fruitful perspective for matrix learning. We demonstrate that the SNN architecture is inherently much more amenable to theoretical analysis than vanilla neural nets and confirm its effectiveness in the context of matrix sensing, via both mathematical guarantees and empirical investigations. We believe that the SNN architecture has the potential to be of wide applicability in a broad class of matrix learning scenarios.
Paper Structure (15 sections, 3 theorems, 83 equations, 5 figures)

This paper contains 15 sections, 3 theorems, 83 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Gradient Flow
  4. The SNN architecture
  5. Main Results
  6. Numerical studies
  7. Numerical verification of theoretical results
  8. Experiments violating Assumptions \ref{['assump:1']} and \ref{['assump:2']}
  9. Real image reconstruction task
  10. Summary of contributions and future directions
  11. Proof of Theorem \ref{['thm:gradent-flow']}
  12. Proof of Theorem \ref{['prop:convergence-rate']}
  13. Proof of Theorem \ref{['thm:implicit-regularization']}
  14. Singular value decomposition
  15. Code availability

Key Result

Theorem 1

Suppose Assumptions assump:1, assump:2, assump:3 hold. Then the gradient flow dynamics in eq:gradient-flow are where $L_k \in \mathbb{R}^{d_1 \times d_2}$ is a diagonal matrix whose diagonal is given by

Figures (5)

  • Figure 1: Visualization of the anatomy of an SNN block and a depth-$D$ SNN architecture. Each SNN block takes as input $K$ matrices and outputs one matrix, both the input and output matrices are of size $\mathbb{R}^{d_1 \times d_2}$. In layer $i$ of the SNN, there are $L_i$ blocks, which aggregate matrices from the previous layers to produce $L_i$ output matrices as inputs for the next layer. The number of input matrices to a block equals the number of neurons in the previous layer. For example, blocks in layer 1 have $K=L_0$, blocks in layer 2 have $K=L_1$, and blocks in layer $i$ have $K=L_{i-1}$.
  • Figure 2: The evolution of training error (left panel), nuclear norm (middle panel), and 3 leading singular values (right panel) over time with learning rates (lr) of different magnitudes.
  • Figure 3: The evolution of training error (left panel), nuclear norm (middle panel), 3 leading singular values (right panel) over time with different learning rates (lr) of different models.
  • Figure 4: The reconstruction images over time with different learning rates (lr) of different models.
  • Figure 5: Top row: ground-truth $X^{\star}$, middle row & bottom row: recovering images by linear regression and our model.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof