Consistent Sparse Deep Learning: Theory and Computation
提出一种类似频率学派的方法来学习稀疏深度神经网络，在贝叶斯框架下证明其一致性，能学习最多O(n/log(n))个连接的稀疏网络，并给出后验一致性、变量选择一致性和渐近最优泛化界等理论保证，计算上比标准贝叶斯方法更高效，适用于大规模网络压缩和高维非线性变量选择。
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 sparse DNN with a mixture Gaussian prior, show that the structure of the sparse DNN can be consistently determined using a Laplace approximation-based marginal 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.