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Deep neural networks (DNNs) have become increasingly popular and achieved outstanding performance in predictive tasks. However, the DNN framework itself cannot inform the user which features are more or less relevant for making the prediction, which limits its applicability in many scientific fields. We introduce neural Gaussian mirrors (NGMs), in which mirrored features are created, via a structured perturbation based on a kernel-based conditional dependence measure, to help evaluate feature importance. We design two modifications of the DNN architecture for incorporating mirrored features and providing mirror statistics to measure feature importance. As shown in simulated and real data examples, the proposed method controls the feature selection error rate at a predefined level and maintains a high selection power even with the presence of highly correlated features.
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It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width. This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of evaluating the corresponding GP. Recently, kernel functions which mimic multi-layer random neural networks have been developed, but only outside of a Bayesian framework. As such, previous work has not identified that these kernels can be used as covariance functions for GPs and allow fully Bayesian prediction with a deep neural network. In this work, we derive the exact equivalence between infinitely wide deep networks and GPs. We further develop a computationally efficient pipeline to compute the covariance function for these GPs. We then use the resulting GPs to perform Bayesian inference for wide deep neural networks on MNIST and CIFAR-10. We observe that trained neural network accuracy approaches that of the corresponding GP with increasing layer width, and that the GP uncertainty is strongly correlated with trained network prediction error. We further find that test performance increases as finite-width trained networks are made wider and more similar to a GP, and thus that GP predictions typically outperform those of finite-width networks. Finally we connect the performance of these GPs to the recent theory of signal propagation in random neural networks.
The goal of controlled feature selection is to discover the features a response depends on while limiting the proportion of false discoveries to a predefined level. Recently, multiple methods have been proposed that use deep learning to generate knockoffs for controlled feature selection through the Model-X knockoff framework. We demonstrate, however, that these methods often fail to control the false discovery rate (FDR). There are two reasons for this shortcoming. First, these methods often learn inaccurate models of features. Second, the swap property, which is required for knockoffs to be valid, is often not well enforced. We propose a new procedure called FlowSelect that remedies both of these problems. To more accurately model the features, FlowSelect uses normalizing flows, the state-of-the-art method for density estimation. To circumvent the need to enforce the swap property, FlowSelect uses a novel MCMC-based procedure to directly compute p-values for each feature. Asymptotically, FlowSelect controls the FDR exactly. Empirically, FlowSelect controls the FDR well on both synthetic and semi-synthetic benchmarks, whereas competing knockoff-based approaches fail to do so. FlowSelect also demonstrates greater power on these benchmarks. Additionally, using data from a genome-wide association study of soybeans, FlowSelect correctly infers the genetic variants associated with specific soybean traits.
Deep Gaussian processes (DGPs) have struggled for relevance in applications due to the challenges and cost associated with Bayesian inference. In this paper we propose a sparse variational approximation for DGPs for which the approximate posterior mean has the same mathematical structure as a Deep Neural Network (DNN). We make the forward pass through a DGP equivalent to a ReLU DNN by finding an interdomain transformation that represents the GP posterior mean as a sum of ReLU basis functions. This unification enables the initialisation and training of the DGP as a neural network, leveraging the well established practice in the deep learning community, and so greatly aiding the inference task. The experiments demonstrate improved accuracy and faster training compared to current DGP methods, while retaining favourable predictive uncertainties.
Double-descent curves in neural networks describe the phenomenon that the generalisation error initially descends with increasing parameters, then grows after reaching an optimal number of parameters which is less than the number of data points, but then descends again in the overparameterised regime. Here we use a neural network Gaussian process (NNGP) which maps exactly to a fully connected network (FCN) in the infinite width limit, combined with techniques from random matrix theory, to calculate this generalisation behaviour, with a particular focus on the overparameterised regime. An advantage of our NNGP approach is that the analytical calculations are easier to interpret. We argue that neural network generalization performance improves in the overparameterised regime precisely because that is where they converge to their equivalent Gaussian process.
Mixtures-of-Experts models and their maximum likelihood estimation (MLE) via the EM algorithm have been thoroughly studied in the statistics and machine learning literature. They are subject of a growing investigation in the context of modeling with high-dimensional predictors with regularized MLE. We examine MoE with Gaussian gating network, for clustering and regression, and propose an $ell_1$-regularized MLE to encourage sparse models and deal with the high-dimensional setting. We develop an EM-Lasso algorithm to perform parameter estimation and utilize a BIC-like criterion to select the model parameters, including the sparsity tuning hyperparameters. Experiments conducted on simulated data show the good performance of the proposed regularized MLE compared to the standard MLE with the EM algorithm.
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