Consistent Sparse Deep Learning: Theory and Computation

• 成果类型: Article
• 署名作者: Sun, Yan; Song, Qifan; Liang, Faming
• 署名单位: Purdue University System; Purdue University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2021.1895175
• 发表日期: 2022
• 页码: 1981-1995

摘要

Deep learning has been the engine powering many successes of data science. However, the deep neural network (DNN), as the basic model of deep learning, is often excessively over-parameterized, causing many difficulties in training, prediction and interpretation. We propose a frequentist-like method for learning sparse DNNs and justify its consistency under the Bayesian framework: the proposed method could learn a sparse DNN with at most O(n/ log(n)) connections and nice theoretical guarantees such as posterior consistency, variable selection consistency and asymptotically optimal generalization bounds. In particular, we establish posterior consistency for the sparseDNNwith amixture Gaussian prior, showthat the structure of the sparse DNN can be consistently determined using a Laplace approximation-basedmarginal posterior inclusion probability approach, and use Bayesian evidence to elicit sparse DNNs learned by an optimization method such as stochastic gradient descent in multiple runs with different initializations. The proposed method is computationally more efficient than standard Bayesian methods for large-scale sparse DNNs. The numerical results indicate that the proposed method can perform very well for large-scale network compression and high-dimensional nonlinear variable selection, both advancing interpretable machine learning.