Approximating smooth functions by deep neural networks with sigmoid activation function
This provides a theoretical foundation for understanding network topology requirements to achieve specific accuracy in function approximation, though it is incremental as it extends prior results to a more general class of networks.
The paper tackles the problem of approximating smooth functions using deep neural networks with sigmoid activation, showing that networks with fixed depth and width of order M^d achieve an approximation rate of M^{-2p}, and generalizing this to a rate of W_0^{-p/d} in terms of overall weights.
We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any $d$-dimensional, smooth function on a compact set with a rate of order $W^{-p/d}$, where $W$ is the number of nonzero weights in the network and $p$ is the smoothness of the function. Unfortunately, these rates only hold for a special class of sparsely connected DNNs. We ask ourselves if we can show the same approximation rate for a simpler and more general class, i.e., DNNs which are only defined by its width and depth. In this article we show that DNNs with fixed depth and a width of order $M^d$ achieve an approximation rate of $M^{-2p}$. As a conclusion we quantitatively characterize the approximation power of DNNs in terms of the overall weights $W_0$ in the network and show an approximation rate of $W_0^{-p/d}$. This more general result finally helps us to understand which network topology guarantees a special target accuracy.