Abstract: One-Shot Neural Architecture Search (NAS) is a promising method to significantly reduce search time without any separate training. It can be treated as a Network Compression problem on the architecture parameters from an over-parameterized network. However, there are two issues associated with most one-shot NAS methods. First, dependencies between a node and its predecessors and successors are often disregarded which result in improper treatment over \emph{zero} operations. Second, architecture parameters pruning based on their magnitude is questionable. In this paper, we employ the classic Bayesian learning approach to alleviate these two issues by modeling architecture parameters using \emph{hierarchical automatic relevance determination} (HARD) priors. Unlike other NAS methods, we train the over-parameterized network for only \emph{one} epoch then update the architecture. Impressively, this enabled us to find the architecture in both proxy and proxyless tasks on CIFAR-10 within only $0.2$ GPU days using a single GPU. As a byproduct, our approach can be transferred directly to compress convolutional neural networks by enforcing structural sparsity which achieves extremely sparse networks without accuracy deterioration.