Depth-Width Trade-offs for Neural Networks via Topological Entropy
This provides theoretical insights into depth-width trade-offs for neural networks, which is foundational for deep learning theory but incremental in nature.
The paper tackles the problem of understanding how depth and width affect neural network expressivity by connecting it to topological entropy from dynamical systems, establishing an upper bound of O(l log m) for ReLU networks and showing that approximating functions with high topological entropy requires exponentially large networks.
One of the central problems in the study of deep learning theory is to understand how the structure properties, such as depth, width and the number of nodes, affect the expressivity of deep neural networks. In this work, we show a new connection between the expressivity of deep neural networks and topological entropy from dynamical system, which can be used to characterize depth-width trade-offs of neural networks. We provide an upper bound on the topological entropy of neural networks with continuous semi-algebraic units by the structure parameters. Specifically, the topological entropy of ReLU network with $l$ layers and $m$ nodes per layer is upper bounded by $O(l\log m)$. Besides, if the neural network is a good approximation of some function $f$, then the size of the neural network has an exponential lower bound with respect to the topological entropy of $f$. Moreover, we discuss the relationship between topological entropy, the number of oscillations, periods and Lipschitz constant.