[Paper Review] Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives

Yahong Yang, Haizhao Yang+1 more|arXiv (Cornell University)|May 15, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper establishes nearly optimal VC-dimension and pseudo-dimension bounds for derivatives of deep neural networks (DNNs) with ReLU activation, leveraging novel analysis of chain rule dependencies. The key contribution is a tight bound of $\mathcal{O}(N^2L^2\log L\log N)$ on the VC-dimension of DNN derivatives, enabling optimal Sobolev-space approximation and tighter generalization error bounds for physics-informed and derivative-based learning models.

ABSTRACT

This paper addresses the problem of nearly optimal Vapnik--Chervonenkis dimension (VC-dimension) and pseudo-dimension estimations of the derivative functions of deep neural networks (DNNs). Two important applications of these estimations include: 1) Establishing a nearly tight approximation result of DNNs in the Sobolev space; 2) Characterizing the generalization error of machine learning methods with loss functions involving function derivatives. This theoretical investigation fills the gap of learning error estimations for a wide range of physics-informed machine learning models and applications including generative models, solving partial differential equations, operator learning, network compression, distillation, regularization, etc.

Motivation & Objective

• To address the lack of tight theoretical bounds on the complexity of deep neural network derivatives, which are critical for Sobolev training and physics-informed machine learning.
• To close the gap in generalization error estimation for models using loss functions involving function derivatives.
• To establish nearly optimal bounds on the VC-dimension and pseudo-dimension of DNN derivatives, overcoming limitations of prior suboptimal analyses.

Proposed method

• Propose a novel method to analyze the compositional structure of DNN derivatives by explicitly modeling the interdependencies introduced by the chain rule.
• Use a refined analysis of the weak derivatives $D_i\phi$ of ReLU-based DNNs $\phi$ to derive complexity bounds.
• Establish a nearly optimal VC-dimension bound of $\text{VCdim}(D\Phi) \leq \bar{C}N^2L^2\log_2 L\log_2 N$ for DNNs with width $\leq N$, depth $\leq L$, and ReLU activation.
• Leverage Dudley’s theorem and covering number bounds via pseudo-dimension to estimate Rademacher complexity and generalization error.
• Apply the pseudo-dimension-based covering number estimate $\mathcal{N}(\varepsilon, \mathcal{F}, n) \leq \left(\frac{2enB}{\varepsilon \cdot \text{Pdim}(\mathcal{F})}\right)^{\text{Pdim}(\mathcal{F})}$ to control generalization error.
• Derive a generalization error bound of $\mathcal{O}\left(\frac{NL(\log L \log N)^{1/2}}{\sqrt{M}} \log M\right)$ for the derivative class, with $M$ training samples.

Experimental results

Research questions

• RQ1 What is the optimal VC-dimension of the derivative of a ReLU-based deep neural network with bounded width and depth?
• RQ2 How can the pseudo-dimension of DNN derivatives be bounded in a way that enables tight generalization error estimation for Sobolev-norm-based loss functions?
• RQ3 Can the complex interdependence of terms in DNN derivatives—arising from the chain rule—be exploited to derive tighter complexity bounds than previous methods?
• RQ4 To what extent do the derived bounds improve the approximation error in Sobolev spaces compared to existing results?
• RQ5 How does the derived complexity bound translate into a tighter generalization error for machine learning models using derivative-based loss functions?

Key findings

• The VC-dimension of the derivative class $D\Phi$ for ReLU-based DNNs is bounded by $\mathcal{O}(N^2L^2\log L\log N)$, which is nearly optimal.
• The pseudo-dimension of the derivative class $D\widetilde{\Phi}$ is bounded by $\mathcal{O}(N^2L^2\log L\log N)$, matching the VC-dimension bound in order of magnitude.
• The Rademacher complexity of the derivative class is bounded by $\mathcal{O}\left(\frac{NL(\log L \log N)^{1/2}}{\sqrt{M}} \log M\right)$, where $M$ is the number of training samples.
• The generalization error for derivative-based loss functions is bounded by $\mathcal{O}\left(\frac{NL(\log L \log N)^{1/2}}{\sqrt{M}} \log M\right)$, which is tighter than previous results.
• The derived bounds enable a nearly optimal approximation rate in Sobolev spaces, confirming the theoretical optimality of DNNs in this setting.
• The analysis resolves a key limitation in prior work by accounting for the multiplicative structure of chain rule terms in DNN derivatives, leading to significantly improved bounds.