Convergence Rates of Variational Inference in Sparse Deep Learning
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:1831-1842, 2020.
Variational inference is becoming more and more popular for approximating intractable posterior distributions in Bayesian statistics and machine learning. Meanwhile, a few recent works have provided theoretical justification and new insights on deep neural networks for estimating smooth functions in usual settings such as nonparametric regression. In this paper, we show that variational inference for sparse deep learning retains precisely the same generalization properties than exact Bayesian inference. In particular, we show that a wise choice of the neural network architecture leads to near-minimax rates of convergence for Hölder smooth functions. Additionally, we show that the model selection framework over the architecture of the network via ELBO maximization does not overfit and adaptively achieves the optimal rate of convergence.