Nonparametric regression using deep neural networks with ReLU activation function (1708.06633v5)

Abstract: Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to $\log n$-factors) under a general composition assumption on the regression function. The framework includes many well-studied structural constraints such as (generalized) additive models. While there is a lot of flexibility in the network architecture, the tuning parameter is the sparsity of the network. Specifically, we consider large networks with number of potential network parameters exceeding the sample size. The analysis gives some insights into why multilayer feedforward neural networks perform well in practice. Interestingly, for ReLU activation function the depth (number of layers) of the neural network architectures plays an important role and our theory suggests that for nonparametric regression, scaling the network depth with the sample size is natural. It is also shown that under the composition assumption wavelet estimators can only achieve suboptimal rates.

• The paper shows that deep ReLU networks can attain near-minimax convergence rates in nonparametric regression under compositional structure assumptions.
• It highlights a flexible network architecture where sparsity and depth scale with sample size to effectively manage model complexity.
• The findings imply that deep networks outperform traditional methods by efficiently leveraging hierarchical function structures in high dimensions.

Nonparametric Regression Using Deep Neural Networks with ReLU Activation Function: An Overview

The paper "Nonparametric Regression Using Deep Neural Networks with ReLU Activation Function" by Johannes Schmidt-Hieber investigates the application of deep neural networks (DNNs) for nonparametric regression tasks. The paper focuses on understanding how DNNs with ReLU activation functions can be utilized to achieve nearly optimal minimax rates of convergence in nonparametric regression, leveraging certain assumptions on the regression function.

Context and Motivation

Nonparametric regression is a versatile statistical tool that allows for flexible modeling of relationships without imposing strict parametric forms. Traditional methods such as kernel smoothing, wavelets, and splines have been well-studied with well-known convergence properties. However, given the recent success of deep learning in various complex tasks like image recognition and speech processing, understanding the statistical properties of neural networks, particularly deep ReLU networks, in this context is of paramount interest.

Theoretical Contributions

Minimax Rates of Convergence

The primary theoretical result of the paper is that estimators based on sparsely connected deep neural networks with ReLU activation functions can attain minimax rates of convergence under specific structural assumptions on the regression function, up to logarithmic factors. The key structural assumption is that the regression function can be represented as a composition of functions, simplifying the high-dimensional modeling challenge.

The main convergence rate derived is: $$\phi_n := \max_{i=0, \ldots, q} n^{-\frac{2\beta_i^*}{2\beta_i^* + t_i}},$$ where $\beta_i^*$ and $t_i$ are parameters related to the smoothness and dimensionality of the component functions in the hierarchical composition of the regression function.

Flexibility in Network Architecture

A notable aspect of the paper is the flexibility allowed in the network architecture. The sparsity parameter $s$, which bounds the number of non-zero parameters in the network, is identified as crucial for matching the effective model complexity with the sample size to achieve optimal rates. The results indicate that large depth $L$ of the network, scaled appropriately with sample size, plays a significant role.

The results hold under the conditions:

• The depth $L$ satisfies $L \gtrsim \log n$.
• The number of non-zero parameters $s \asymp n \phi_n \log n$, ensuring that the network has a sufficient but not excessive capacity aligned with the complexity of the underlying function.

Practical Implications

The practical implications of these findings are twofold:

1. Scalability: Deep ReLU networks can effectively be scaled to leverage large data sets common in modern applications, while maintaining rigorous statistical guarantees.
2. Adaptation to Structure: The ability of DNNs to adapt to hierarchical and compositional structures in the data allows them to circumvent the curse of dimensionality effectively in many situations, leading to improved performance over traditional nonparametric methods for suitably structured problems.

Comparison with Traditional Methods

Intriguingly, the analysis also shows that traditional methods like wavelet estimators can be significantly suboptimal when the regression function possesses a compositional structure. For instance, the paper demonstrates that wavelet series estimators fail to capitalize on these structures, resulting in slower convergence rates dominated by dimensionality.

Speculation on Future Developments

The exploration conducted in this paper paves the way for future developments in several intriguing directions:

• Refinement of Sparsity Control: Developing adaptive regularizers that dynamically adjust the sparsity parameter $s$ during training could enhance practical implementations.
• Optimization Algorithms: Investigating how different initialization, optimization techniques, and implicit regularization strategies (like SGD) impact $\Delta_n(\widehat{f}_n, f_0)$ could yield deeper insights into the practical performance of these networks.
• Function Classes: Extending the analysis to broader function classes and different types of network architectures, such as convolutional or recurrent networks, would further generalize the applicability of these results.

Conclusion

The paper by Schmidt-Hieber provides a significant theoretical advancement in understanding how deep ReLU neural networks can be effectively employed for nonparametric regression. By establishing the conditions under which these networks achieve near-minimax optimality, it opens new avenues for leveraging deep learning's flexibility and scalability in statistical applications while emphasizing the importance of network architecture and sparsity.

Given the ongoing rapid progress in both theoretical and applied deep learning, the insights from this paper are timely, providing a robust foundation for integrating DNNs into the nonparametric statistical toolkit.