Minimum width for universal approximation using ReLU networks on compact domain

The universal approximation property of width-bounded networks has been studied as a dual of the classical universal approximation theorem for depth-bounded ones. There were several attempts to characterize the minimum width $w_{\min}$ enabling the universal approximation property; however, only a few of them found the exact values. In this work, we show that the minimum width for the universal approximation of $L^p$ functions from $[0,1]^{d_x}$ to $\mathbb R^{d_y}$ is exactly $\max\{d_x,d_y,2\}$ if an activation function is ReLU-Like (e.g., ReLU, GELU, Softplus). Compared to the known result $w_{\min}=\max\{d_x+1,d_y\}$ when the domain is ${\mathbb R^{d_x}}$, our result first shows that approximation on a compact domain requires smaller width than on ${\mathbb R^{d_x}}$. We next prove a lower bound on $w_{\min}$ for uniform approximation using general activation functions including ReLU: $w_{\min}\ge d_y+1$ if $d_x<d_y\le2d_x$. Together with our first result, this shows a dichotomy between $L^p$ and uniform approximations for general activation functions and input/output dimensions.

Deep Collective Knowledge Distillation

Many existing studies on knowledge distillation have focused on methods in which a student model mimics a teacher model well. Simply imitating the teacher's knowledge, however, is not sufficient for the student to surpass that of the teacher. We explore a method to harness the knowledge of other students to complement the knowledge of the teacher. We propose deep collective knowledge distillation for model compression, called DCKD, which is a method for training student models with rich information to acquire knowledge from not only their teacher model but also other student models. The knowledge collected from several student models consists of a wealth of information about the correlation between classes. Our DCKD considers how to increase the correlation knowledge of classes during training. Our novel method enables us to create better performing student models for collecting knowledge. This simple yet powerful method achieves state-of-the-art performances in many experiments. For example, for ImageNet, ResNet18 trained with DCKD achieves 72.27\%, which outperforms the pretrained ResNet18 by 2.52\%. For CIFAR-100, the student model of ShuffleNetV1 with DCKD achieves 6.55\% higher top-1 accuracy than the pretrained ShuffleNetV1.