A comparison of deep networks with ReLU activation function and linear spline-type methods (1804.02253v2)

Abstract: Deep neural networks (DNNs) generate much richer function spaces than shallow networks. Since the function spaces induced by shallow networks have several approximation theoretic drawbacks, this explains, however, not necessarily the success of deep networks. In this article we take another route by comparing the expressive power of DNNs with ReLU activation function to piecewise linear spline methods. We show that MARS (multivariate adaptive regression splines) is improper learnable by DNNs in the sense that for any given function that can be expressed as a function in MARS with $M$ parameters there exists a multilayer neural network with $O(M \log (M/\varepsilon))$ parameters that approximates this function up to sup-norm error $\varepsilon.$ We show a similar result for expansions with respect to the Faber-Schauder system. Based on this, we derive risk comparison inequalities that bound the statistical risk of fitting a neural network by the statistical risk of spline-based methods. This shows that deep networks perform better or only slightly worse than the considered spline methods. We provide a constructive proof for the function approximations.

• The paper establishes that deep networks approximate spline-based functions with O(M log(M/ε)) parameters, showcasing superior efficiency over traditional methods.
• The analysis rigorously compares risk bounds in nonparametric regression, highlighting the trade-offs between model expressivity and interpretability.
• The study proposes hybrid strategies, suggesting that leveraging DNNs to initialize spline models could enhance training effectiveness and computational efficiency.

An Analysis of Deep Networks vs Spline-Type Methods

The paper at hand explores the comparative expressive capacity of deep neural networks (DNNs) employing the ReLU activation function and well-known linear spline-type methods, specifically multivariate adaptive regression splines (MARS) and the Faber-Schauder system. The authors Konstantin Eckle and Johannes Schmidt-Hieber aim to analyze under what conditions DNNs may outperform other piecewise linear function-based methods in terms of function approximation capabilities.

Overview of Expressive Power and Approximation Results

The authors hypothesize that while DNNs are acknowledged for being more expressive than shallow networks, a thorough understanding of their relative capabilities compared to other flexible function approximation approaches remains an underexplored area. This paper focuses on the theoretical underpinnings by providing a comparative analysis of certain function spaces defined by DNNs and spline methods.

A significant finding outlined in this paper is the claim that DNNs are able to approximate any function represented in the MARS or Faber-Schauder class with a number of parameters that scales as $O(M \log (M/\epsilon))$, providing sup-norm approximation error $\epsilon$. This suggests a relatively efficient representation of spline methods via DNNs, given that such approximations leverage the full potential of distributed multilayer architectures. Interestingly, they also show that functions that are effortlessly represented within a DNN framework may require exponentially many more parameters to achieve a similar accuracy when approximated via MARS or Faber-Schauder.

Implications and Theoretical Considerations

The theoretical results bear significant implications on the understanding of risk bounds in the context of nonparametric regression. The authors establish risk comparison inequalities which demonstrate how the risk in fitting a neural network is bounded by that of a spline-based method. These results have broad relevance, especially in applications where expressivity, interpretability, and computational efficiency are important trade-offs.

For future work, these insights could influence the development of hybrid models or improved training techniques that capitalize on the strengths of both neural networks and spline-based methods. The paper suggests leveraging the approximating power of DNNs to initialize models trained using MARS, providing a pathway to potentially more effective learning algorithms.

Avenues for Future Research

The trajectory of future exploration may embellish the bridge between theoretical approximation theory and practical advantages of DNNs over classical spline methods. Further empirical studies could explore the extent of these theoretical results in real-world tasks, providing more contextual evidence. Moreover, considering other activation functions and architectures could widen the applicability of these insights.

The constructive proofs support the prospect of designing algorithmic strategies and implementations around converting MARS outputs to suitable initial configurations for DNNs. Research could extend these concepts to high-dimensional data tasks, exploring scalability and efficiency implications.

In sum, this paper situates itself as a rigorous theoretical exploration of function approximation strategies, contributing to a nuanced understanding of deep learning frameworks compared to traditional methods. While the discussion is primarily theoretical, it lays groundwork for innovative ways to think about solving complex optimization problems present in the design and training of machine learning models.