In this work, we focus on variational Bayesian inference on the sparse Deep Neural Network (DNN) modeled under a class of spike-and-slab priors. Given a pre-specified sparse DNN structure, the corresponding variational posterior contraction rate is characterized that reveals a trade-off between the variational error and the approximation error, which are both determined by the network structural complexity (i.e., depth, width and sparsity). However, the optimal network structure, which strikes the balance of the aforementioned trade-off and yields the best rate, is generally unknown in reality. Therefore, our work further develops an {em adaptive} variational inference procedure that can automatically select a reasonably good (data-dependent) network structure that achieves the best contraction rate, without knowing the optimal network structure. In particular, when the true function is H{o}lder smooth, the adaptive variational inference is capable to attain (near-)optimal rate without the knowledge of smoothness level. The above rate still suffers from the curse of dimensionality, and thus motivates the teacher-student setup, i.e., the true function is a sparse DNN model, under which the rate only logarithmically depends on the input dimension.
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