Sparse deep learning has become a popular technique for improving the performance of deep neural networks in areas such as uncertainty quantification, variable selection, and large-scale network compression. However, most existing research has focused on problems where the observations are independent and identically distributed (i.i.d.), and there has been little work on the problems where the observations are dependent, such as time series data and sequential data in natural language processing. This paper aims to address this gap by studying the theory for sparse deep learning with dependent data. We show that sparse recurrent neural networks (RNNs) can be consistently estimated, and their predictions are asymptotically normally distributed under appropriate assumptions, enabling the prediction uncertainty to be correctly quantified. Our numerical results show that sparse deep learning outperforms state-of-the-art methods, such as conformal predictions, in prediction uncertainty quantification for time series data. Furthermore, our results indicate that the proposed method can consistently identify the autoregressive order for time series data and outperform existing methods in large-scale model compression. Our proposed method has important practical implications in fields such as finance, healthcare, and energy, where both accurate point estimates and prediction uncertainty quantification are of concern.

We propose a novel adaptive empirical Bayesian (AEB) method for sparse deep learning, where the sparsity is ensured via a class of self-adaptive spike-and-slab priors. The proposed method works by alternatively sampling from an adaptive hierarchical posterior distribution using stochastic gradient Markov Chain Monte Carlo (MCMC) and smoothly optimizing the hyperparameters using stochastic approximation (SA). The convergence of the proposed method to the asymptotically correct distribution is established under mild conditions. Empirical applications of the proposed method lead to the state-of-the-art performance on MNIST and Fashion MNIST with shallow convolutional neural networks (CNN) and the state-of-the-art compression performance on CIFAR10 with Residual Networks. The proposed method also improves resistance to adversarial attacks.

Deep learning has powered recent successes of artificial intelligence (AI). However, the deep neural network, as the basic model of deep learning, has suffered from issues such as local traps and miscalibration. In this paper, we provide a new framework for sparse deep learning, which has the above issues addressed in a coherent way. In particular, we lay down a theoretical foundation for sparse deep learning and propose prior annealing algorithms for learning sparse neural networks. The former has successfully tamed the sparse deep neural network into the framework of statistical modeling, enabling prediction uncertainty correctly quantified. The latter can be asymptotically guaranteed to converge to the global optimum, enabling the validity of the down-stream statistical inference. Numerical result indicates the superiority of the proposed method compared to the existing ones.

Sparse deep learning aims to address the challenge of huge storage consumption by deep neural networks, and to recover the sparse structure of target functions. Although tremendous empirical successes have been achieved, most sparse deep learning algorithms are lacking of theoretical supports. On the other hand, another line of works have proposed theoretical frameworks that are computationally infeasible. In this paper, we train sparse deep neural networks with a fully Bayesian treatment under spike-and-slab priors, and develop a set of computationally efficient variational inferences via continuous relaxation of Bernoulli distribution. The variational posterior contraction rate is provided, which justifies the consistency of the proposed variational Bayes method. Interestingly, our empirical results demonstrate that this variational procedure provides uncertainty quantification in terms of Bayesian predictive distribution and is also capable to accomplish consistent variable selection by training a sparse multi-layer neural network.

We introduce Spike-and-Slab Deep Learning (SS-DL), a fully Bayesian alternative to dropout for improving generalizability of deep ReLU networks. This new type of regularization enables provable recovery of smooth input-output maps with {\sl unknown} levels of smoothness. Indeed, we show that the posterior distribution concentrates at the near minimax rate for alpha-Holder smooth maps, performing as well as if we knew the smoothness level alpha ahead of time.

This is a novel combination of existing techniques that appears well-formulated with intriguing experimental results. In particular, this work leverages the strengths stochastic gradient MCMC methods with stochastic approximation to form an adaptive empirical Bayesian approach to learning the parameters and hyperparameters of a Bayesian neural network (BNN). My best understanding is that by optimizing the hyperparameters (rather than sampling them), this new method improves upon existing approaches, speeding up inference without sacrificing quality (especially in the model compression domain). Other areas of BNN literature could be cited, but I think the authors were prudent not to distract the reader from the particular area of focus. This work demonstrates considerable theoretical analysis and is supported by intriguing experimental evidence.

All reviewers acknowledged that the authors addressed their concerns well in the feedback, and agreed to accept the paper.

They serve as proper proof of concept that the approach works in principle, but do the authors have an intuition as to how well their approach scales to larger models?