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تسجيل مستخدم جديدDeep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality on Holder Class
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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 $omega_f(cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $mathcal{O}(omega_f(sqrt{d})cdot2^{-M}+omega_{f}left(frac{sqrt{d}}{N}right))$, where $M,Nin mathbb{N}^{+}$ 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 $maxleft{leftlceil2d^{3/2}left(frac{3mu}{epsilon}right)^{1/{alpha}}rightrceil,2leftlceillog_2frac{3mu d^{alpha/2}}{2epsilon}rightrceil+2right}$ that approximates $fin mathcal{H}_{mu}^{alpha}([0,1]^d)$ within a given tolerance $epsilon >0$ measured in $L^p$ norm $pin[1,infty)$, where $mathcal{H}_{mu}^{alpha}([0,1]^d)$ denotes the Holder 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.
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