Sleep staging is essential for assessing sleep quality and diagnosing sleep disorders. However, sleep staging is a labor-intensive process, making it arduous to obtain large quantities of high-quality labeled data for automatic sleep staging. Meanwhile, most of the research on automatic sleep staging pays little attention to pediatric sleep staging. To address these challenges, we propose a semi-supervised multi-scale arbitrary dilated convolution neural network (SMADNet) for pediatric sleep staging using the scalogram with a high height-to-width ratio generated by the continuous wavelet transform (CWT) as input. To extract more extended time dimensional feature representations and adapt to scalograms with a high height-to-width ratio in SMADNet, we introduce a multi-scale arbitrary dilation convolution block (MADBlock) based on our proposed arbitrary dilated convolution (ADConv). Finally, we also utilize semi-supervised learning as the training scheme for our network in order to alleviate the reliance on labeled data. Our proposed model has achieved performance comparable to state-of-the-art supervised learning methods with 30% labels. Our model is tested on a private pediatric dataset and achieved 79% accuracy, 72% kappa, and 75% MF1. Therefore, our model demonstrates a powerful feature extraction capability and has achieved performance comparable to state-of-the-art supervised learning methods with a small number of labels.