In this paper, we improve the PAC-Bayesian error bound for linear regression derived in Germain et al. [10]. The improvements are twofold. First, the proposed error bound is tighter, and converges to the generalization loss with a well-chosen temperature parameter. Second, the error bound also holds for training data that are not independently sampled. In particular, the error bound applies to certain time series generated by well-known classes of dynamical models, such as ARX models.

Bayesian learning of model parameters in neural networks is important in scenarios where estimates with well-calibrated uncertainty are important. In this paper, we propose Bayesian quantized networks (BQNs), quantized neural networks (QNNs) for which we learn a posterior distribution over their discrete parameters. We provide a set of efficient algorithms for learning and prediction in BQNs without the need to sample from their parameters or activations, which not only allows for differentiable learning in QNNs, but also reduces the variance in gradients. We demonstrate BQNs achieve both lower predictive errors and better-calibrated uncertainties than E-QNN (with less than 20% of the negative log-likelihood). A Bayesian approach to deep learning considers the network's parameters to be random variables and seeks to infer their posterior distribution given the training data. Models trained this way, called Bayesian neural networks (BNNs) (Wang & Y eung, 2016), in principle have well-calibrated uncertainties when they make predictions, which is important in scenarios such as active learning and reinforcement learning (Gal, 2016). Furthermore, the posterior distribution over the model parameters provides valuable information for evaluation and compression of neural networks. There are three main challenges in using BNNs: (1) Intractable posterior: Computing and storing the exact posterior distribution over the network weights is intractable due to the complexity and high-dimensionality of deep networks. These challenges are typically addressed either by making simplifying assumptions about the distributions of the parameters and activations, or by using sampling-based approaches, which are expensive and unreliable (likely to overestimate the uncertainties in predictions). Our goal is to propose a sampling-free method which uses probabilistic propagation to deterministically learn BNNs. A seemingly unrelated area of deep learning research is that of quantized neural networks (QNNs), which offer advantages of computational and memory efficiency compared to continuous-valued models.

This paper presents a variational Bayesian kernel selection (VBKS) algorithm for sparse Gaussian process regression (SGPR) models. In contrast to existing GP kernel selection algorithms that aim to select only one kernel with the highest model evidence, our proposed VBKS algorithm considers the kernel as a random variable and learns its belief from data such that the uncertainty of the kernel can be interpreted and exploited to avoid overconfident GP predictions. To achieve this, we represent the probabilistic kernel as an additional variational variable in a variational inference (VI) framework for SGPR models where its posterior belief is learned together with that of the other variational variables (i.e., inducing variables and kernel hyperparameters). In particular, we transform the discrete kernel belief into a continuous parametric distribution via reparameterization in order to apply VI. Though it is computationally challenging to jointly optimize a large number of hyperparameters due to many kernels being evaluated simultaneously by our VBKS algorithm, we show that the variational lower bound of the log-marginal likelihood can be decomposed into an additive form such that each additive term depends only on a disjoint subset of the variational variables and can thus be optimized independently. Stochastic optimization is then used to maximize the variational lower bound by iteratively improving the variational approximation of the exact posterior belief via stochastic gradient ascent, which incurs constant time per iteration and hence scales to big data. We empirically evaluate the performance of our VBKS algorithm on synthetic and massive real-world datasets.

Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundamental principles of flow design, and discuss foundational topics such as expressive power and computational trade-offs. We also broaden the conceptual framing of flows by relating them to more general probability transformations. Lastly, we summarize the use of flows for tasks such as generative modeling, approximate inference, and supervised learning.

Deep ensembles have been empirically shown to be a promising approach for improving accuracy, uncertainty and out-of-distribution robustness of deep learning models. While deep ensembles were theoretically motivated by the bootstrap, non-bootstrap ensembles trained with just random initialization also perform well in practice, which suggests that there could be other explanations for why deep ensembles work well. Bayesian neural networks, which learn distributions over the parameters of the network, are theoretically well-motivated by Bayesian principles, but do not perform as well as deep ensembles in practice, particularly under dataset shift. One possible explanation for this gap between theory and practice is that popular scalable approximate Bayesian methods tend to focus on a single mode, whereas deep ensembles tend to explore diverse modes in function space. We investigate this hypothesis by building on recent work on understanding the loss landscape of neural networks and adding our own exploration to measure the similarity of functions in the space of predictions. Our results show that random initializations explore entirely different modes, while functions along an optimization trajectory or sampled from the subspace thereof cluster within a single mode predictions-wise, while often deviating significantly in the weight space. We demonstrate that while low-loss connectors between modes exist, they are not connected in the space of predictions. Developing the concept of the diversity--accuracy plane, we show that the decorrelation power of random initializations is unmatched by popular subspace sampling methods.

Clustering with variable selection is a challenging but critical task for modern small-n-large-p data. Existing methods based on Gaussian mixture models or sparse K-means provide solutions to continuous data. With the prevalence of RNA-seq technology and lack of count data modeling for clustering, the current practice is to normalize count expression data into continuous measures and apply existing models with Gaussian assumption. In this paper, we develop a negative binomial mixture model with gene regularization to cluster samples (small $n$) with high-dimensional gene features (large $p$). EM algorithm and Bayesian information criterion are used for inference and determining tuning parameters. The method is compared with sparse Gaussian mixture model and sparse K-means using extensive simulations and two real transcriptomic applications in breast cancer and rat brain studies. The result shows superior performance of the proposed count data model in clustering accuracy, feature selection and biological interpretation by pathway enrichment analysis.

We study the phenomenon that some modules of deep neural networks (DNNs) are more critical than others. Meaning that rewinding their parameter values back to initialization, while keeping other modules fixed at the trained parameters, results in a large drop in the network's performance. Our analysis reveals interesting properties of the loss landscape which leads us to propose a complexity measure, called module criticality, based on the shape of the valleys that connects the initial and final values of the module parameters. We formulate how generalization relates to the module criticality, and show that this measure is able to explain the superior generalization performance of some architectures over others, whereas earlier measures fail to do so. 1 Introduction Neural networks have had tremendous practical impact in various domains such as revolutionizing many tasks in computer vision, speech and natural language processing. However, many aspects of their design and analysis have remained mysterious to this date. One of the most important questions is "what makes an architecture work better than others given a specific task?" Extensive research in this area has led to many potential explanations on why some types of architectures have better performance; however, we lack a unified view that provides a complete and satisfactory answer. In order to attain a unified view on superiority of one architecture over another in terms of generalization performance, we need to come up with a measure that effectively captures this. Analyzing the generalization behavior of neural networks has been an active area of research since Baum and Haussler (1989). Many generalization bounds and complexity measures have been proposed so far. Bartlett (1998) emphasized on the norm of the weights in predicting the generalization error.

Item response theory (IRT) is a non-linear generative probabilistic paradigm for using exams to identify, quantify, and compare latent traits of individuals, relative to their peers, within a population of interest. In pre-existing multidimensional IRT methods, one requires a factorization of the test items. For this task, linear exploratory factor analysis is used, making IRT a posthoc model. We propose skipping the initial factor analysis by using a sparsity-promoting horseshoe prior to perform factorization directly within the IRT model so that all training occurs in a single self-consistent step. Being a hierarchical Bayesian model, we adapt the WAIC to the problem of dimensionality selection. IRT models are analogous to probabilistic autoencoders. By binding the generative IRT model to a Bayesian neural network (forming a probabilistic autoencoder), one obtains a scoring algorithm consistent with the interpretable Bayesian model. In some IRT applications the black-box nature of a neural network scoring machine is desirable. In this manuscript, we demonstrate within-IRT factorization and comment on scoring approaches.

We place an Indian Buffet Process (IBP) prior over the neural structure of a Bayesian Neural Network (BNN), thus allowing the complexity of the BNN to increase and decrease automatically. We apply this methodology to the problem of resource allocation in continual learning, where new tasks occur and the network requires extra resources. Our BNN exploits online variational inference with relaxations to the Bernoulli and Beta distributions (which constitute the IBP prior), so allowing the use of the reparameterisation trick to learn variational posteriors via gradient-based methods. As we automatically learn the number of weights in the BNN, overfitting and underfitting problems are largely overcome. We show empirically that the method offers competitive results compared to Variational Continual Learning (VCL) in some settings.

Hamiltonian Monte Carlo (HMC) is an efficient Bayesian sampling method that can make distant proposals in the parameter space by simulating a Hamiltonian dynamical system. Despite its popularity in machine learning and data science, HMC is inefficient to sample from spiky and multimodal distributions. Motivated by the energy-time uncertainty relation from quantum mechanics, we propose a Quantum-Inspired Hamiltonian Monte Carlo algorithm (QHMC). This algorithm allows a particle to have a random mass with a probability distribution rather than a fixed mass. We prove the convergence property of QHMC in the spatial domain and in the time sequence. We further show why such a random mass can improve the performance when we sample a broad class of distributions. In order to handle the big training data sets in large-scale machine learning, we develop a stochastic gradient version of QHMC using Nos\'e-Hoover thermostat called QSGNHT, and we also provide theoretical justifications about its steady-state distributions. Finally in the experiments, we demonstrate the effectiveness of QHMC and QSGNHT on synthetic examples, bridge regression, image denoising and neural network pruning. The proposed QHMC and QSGNHT can indeed achieve much more stable and accurate sampling results on the test cases.