Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on Hölder Class

In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $\omegaf(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $\mathcal{O}(\omegaf(\sqrt{d})\cdot2^{{-M}+\omega_{f}\left(\frac{\sqrt{d}}{N}\right))$,} where $M,N\in \mathbb{N}^{+}$ denote the hyperparameters related to widths of the networks. As a consequence, we can construct ReLU-sine-$2^x$ network with the depth $5$ and width $\max\left{\left\lceil2d^{{3/2}\left(\frac{3\mu}{\epsilon}\right){1/{\alpha}}\right\rceil,2\left\lceil\log_2\frac{3\mu} d^{{\alpha/2}}{2\epsilon}\right\rceil+2\right}$} that approximates $f\in \mathcal{H}{\mu}^{{\alpha}([0,1]d)$} within a given tolerance $\epsilon >0$ measured in $L^p$ norm $p\in[1,\infty)$, where $\mathcal{H}{\mu}^{{\alpha}([0,1]d)$} denotes the H\"older continuous function class defined on $[0,1]^d$ with order $\alpha \in (0,1]$ and constant $\mu > 0$. Therefore, the ReLU-sine-$2^x$ networks overcome the curse of dimensionality on $\mathcal{H}_{\mu}^{{\alpha}([0,1]d)$.} In addition to its supper expressive power, functions implemented by ReLU-sine-$2^x$ networks are (generalized) differentiable, enabling us to apply SGD to train.