Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks (1703.02930v3)

Abstract: We prove new upper and lower bounds on the VC-dimension of deep neural networks with the ReLU activation function. These bounds are tight for almost the entire range of parameters. Letting $W$ be the number of weights and $L$ be the number of layers, we prove that the VC-dimension is $O(W L \log(W))$, and provide examples with VC-dimension $\Omega( W L \log(W/L) )$. This improves both the previously known upper bounds and lower bounds. In terms of the number $U$ of non-linear units, we prove a tight bound $\Theta(W U)$ on the VC-dimension. All of these bounds generalize to arbitrary piecewise linear activation functions, and also hold for the pseudodimensions of these function classes. Combined with previous results, this gives an intriguing range of dependencies of the VC-dimension on depth for networks with different non-linearities: there is no dependence for piecewise-constant, linear dependence for piecewise-linear, and no more than quadratic dependence for general piecewise-polynomial.

• The paper establishes sharp VC-dimension bounds with an upper bound of O(WL log(W)) and a lower bound of Ω(WL log(W/L)) for deep neural networks using ReLU activations.
• It introduces a tight pseudodimension bound of Θ(WU), linking network complexity directly to the number of non-linear units.
• The results highlight a trade-off between network depth and complexity, guiding optimal design choices for enhanced generalization.

Nearly-tight VC-dimension and Pseudodimension Bounds for Piecewise Linear Neural Networks

This paper addresses a critical theoretical aspect of neural networks: the VC-dimension and pseudodimension for networks employing piecewise linear activation functions, with a focus on ReLU functions. The findings provide tight bounds for these dimensions, considering the number of weights $W$, layers $L$, and non-linear units $U$. The research enhances understanding of the complexity and generalization capabilities of these networks within the framework of statistical learning theory.

Main Results

Key contributions of the paper include:

1. Bound Improvements: The paper establishes upper and lower bounds on the VC-dimension of deep neural networks with ReLU activations, covering almost the entire parameter range:
   • Proves an upper bound of $O(WL \log(W))$.
   • Demonstrates a lower bound of $\Omega(WL \log(W/L))$.
2. Pseudodimension Insights: A tight bound $\Theta(WU)$ is shown for the VC-dimension in terms of the non-linear units $U$. These results apply to arbitrary piecewise linear activations and related pseudodimensions.
3. Complexity Versus Depth Analysis: The paper explores how network depth affects VC-dimension, revealing a trade-off and depth's impact on complexity:
   • No dependence for piecewise-constant activations.
   • Linear dependence for piecewise-linear activations.
   • Quadratic dependence for general piecewise-polynomial activations.

Implications and Speculations

The derived bounds provide a detailed depiction of neural network complexity, enabling more informed architectural decisions when designing deep networks optimized for specific functions or computational constraints.

• Practical Implications: Understanding the tight bounds helps in designing networks with desired generalization abilities without unnecessarily increasing complexity, especially crucial in resource-constrained applications.
• Theoretical Insights: The results enhance foundational knowledge of neural network theory, especially relating to learning capacities and trade-offs between depth and breadth in network architectures.
• Future Directions: The results invite further exploration into more general activation functions, potential simplification of architectures, or exploring regimes where bounds can be relaxed or tightened further, specifically for piecewise polynomial functions where existing gaps in bounds might be addressed.

This paper successfully bridges an important gap in understanding VC-dimensions concerning architectural depth and parameterization in neural networks, laying groundwork for future advancements and applications.