[Paper Review] Nearly Optimal Approximation Rates for Deep Super ReLU Networks on Sobolev Spaces

Yahong Yang, Yue Wu+2 more|arXiv (Cornell University)|Oct 16, 2023

Asphalt Pavement Performance EvaluationEngineering3 citations

TL;DR

This paper introduces Deep Super ReLU Networks (DSRNs), which combine ReLU and squared ReLU activations in the final layers to enable high-order derivative approximation in Sobolev spaces $W^{m,p}$ for $m \geq 2$. By strategically placing only $O(\log_2 L)$ layers with squared ReLU, DSRNs achieve nearly optimal approximation rates while retaining the training advantages of ReLU networks, with theoretical guarantees on generalization and optimality via VC- and pseudo-dimension analysis of higher-order derivatives.

ABSTRACT

This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\in\mathbb{N}$ with $m\ge 2$ and $1\le p\le +\infty$. Standard ReLU deep neural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarily of ReLU DNNs, and several layers of the square of ReLU added at the end to smooth the networks output. This approach retains the advantages of ReLU DNNs, leading to the straightforward training. The paper also proves the optimality of DSRNs by estimating the VC-dimension of higher-order derivatives of DNNs, and obtains the generalization error in Sobolev spaces via an estimate of the pseudo-dimension of higher-order derivatives of DNNs.

Motivation & Objective

• To address the limitation of standard ReLU DNNs in approximating functions with high-order derivatives in Sobolev spaces $W^{m,p}$ for $m \geq 2$.
• To develop a deep network architecture that maintains the training efficiency of ReLU networks while enabling smooth outputs necessary for computing higher-order derivatives.
• To establish theoretical optimality of the approximation rate by estimating the VC-dimension and pseudo-dimension of higher-order derivatives of DNNs.
• To provide generalization error bounds in Sobolev norms via pseudo-dimension-based estimation of derivative complexity.

Proposed method

• Propose Deep Super ReLU Networks (DSRNs) that use standard ReLU in all but the last few layers, where the square of ReLU is applied to enhance smoothness.
• Limit the number of squared ReLU layers to $O(\log_2 L)$, where $L$ is the network depth, to control complexity while ensuring sufficient smoothness for $W^{m,p}$ approximation.
• Use the composition of ReLU and squared ReLU to construct networks whose higher-order derivatives are well-behaved and amenable to generalization analysis.
• Apply pseudo-dimension and VC-dimension theory to analyze the complexity of higher-order derivatives of DSRNs, enabling generalization error bounds in Sobolev norms.
• Derive generalization error bounds using the pseudo-dimension of second-order derivatives, leading to a bound of the form $\mathcal{O}\left(\frac{NL(\log_2 L \log_2 N)^{1/2}}{\sqrt{M}}\log M\right)$.
• Validate the method numerically on a 1D Poisson equation using PINNs, comparing DSRN with full-squared-ReLU networks and showing superior approximation of both solution and derivatives.

Experimental results

Research questions

• RQ1 Can ReLU-based deep neural networks be modified to achieve nearly optimal approximation rates in Sobolev spaces $W^{m,p}$ for $m \geq 2$?
• RQ2 What is the optimal placement and number of squared ReLU layers needed to ensure smoothness without sacrificing training efficiency?
• RQ3 How can the generalization error of DNNs be bounded in Sobolev norms when approximating functions with high-order derivatives?
• RQ4 What is the role of VC-dimension and pseudo-dimension in characterizing the complexity of higher-order derivatives of DNNs?
• RQ5 Can the approximation rate of DSRNs be proven optimal using derivative complexity measures like pseudo-dimension?

Key findings

• DSRNs achieve nearly optimal approximation rates for functions in $W^{m,p}$ spaces with $m \geq 2$, overcoming the inherent non-smoothness of standard ReLU DNNs.
• The number of squared ReLU layers required is $O(\log_2 L)$, which ensures sufficient smoothness while maintaining computational efficiency and training stability.
• Generalization error in Sobolev norms is bounded by $\mathcal{O}\left(\frac{NL(\log_2 L \log_2 N)^{1/2}}{\sqrt{M}}\log M\right)$, where $M$ is the sample size.
• The pseudo-dimension of second-order derivatives of DSRNs is bounded by $O(NL(\log_2 L \log_2 N)^{1/2})$, enabling tight generalization error estimates.
• Numerical experiments confirm that DSRNs accurately approximate the solution and its derivatives for a 1D Poisson equation, outperforming networks with full-squared-ReLU activation.
• The theoretical framework establishes that DSRNs are optimal in terms of approximation rate, as the derived bounds match known minimax rates up to logarithmic factors.