Virtual screening aims to find desirable compounds from chemical library by using computational methods. For this purpose with machine learning, model outputs that can be interpreted as predictive probability will be beneficial, in that a high prediction score corresponds to high probability of correctness. In this work, we present a study on the prediction performance and reliability of graph neural networks trained with the recently proposed Bayesian learning algorithms. Our work shows that Bayesian learning algorithms allow well-calibrated predictions for various GNN architectures and classification tasks. Also, we show the implications of reliable predictions on virtual screening, where Bayesian learning may lead to higher success in finding hit compounds.

Learning the joint probability of random variables (RVs) lies at the heart of statistical signal processing and machine learning. However, direct nonparametric estimation for high-dimensional joint probability is in general impossible, due to the curse of dimensionality. Recent work has proposed to recover the joint probability mass function (PMF) of an arbitrary number of RVs from three-dimensional marginals, leveraging the algebraic properties of low-rank tensor decomposition and the (unknown) dependence among the RVs. Nonetheless, accurately estimating three-dimensional marginals can still be costly in terms of sample complexity, affecting the performance of this line of work in practice in the sample-starved regime. Using three-dimensional marginals also involves challenging tensor decomposition problems whose tractability is unclear. This work puts forth a new framework for learning the joint PMF using only pairwise marginals, which naturally enjoys a lower sample complexity relative to the third-order ones. A coupled nonnegative matrix factorization (CNMF) framework is developed, and its joint PMF recovery guarantees under various conditions are analyzed. Our method also features a Gram-Schmidt (GS)-like algorithm that exhibits competitive runtime performance. The algorithm is shown to provably recover the joint PMF up to bounded error in finite iterations, under reasonable conditions. It is also shown that a recently proposed economical expectation maximization (EM) algorithm guarantees to improve upon the GS-like algorithm's output, thereby further lifting up the accuracy and efficiency. Real-data experiments are employed to showcase the effectiveness.

This article proposes a new class of Real Elliptically Skewed (RESK) distributions and associated clustering algorithms that allow for integrating robustness and skewness into a single unified cluster analysis framework. Non-symmetrically distributed and heavy-tailed data clusters have been reported in a variety of real-world applications. Robustness is essential because a few outlying observations can severely obscure the cluster structure. The RESK distributions are a generalization of the Real Elliptically Symmetric (RES) distributions. To estimate the cluster parameters and memberships, we derive an expectation maximization (EM) algorithm for arbitrary RESK distributions. Special attention is given to a new robust skew-Huber M-estimator, which is also the maximum likelihood estimator (MLE) for the skew-Huber distribution that belongs to the RESK class. Numerical experiments on simulated and real-world data confirm the usefulness of the proposed methods for skewed and heavy-tailed data sets.

The study of network formation is pervasive in economics, sociology, and many other fields. In this paper, we model network formation as a ``choice'' that is made by nodes in a network to connect to other nodes. We study these ``choices'' using discrete-choice models, in which an agent chooses between two or more discrete alternatives. One framework for studying network formation is the multinomial logit (MNL) model. We highlight limitations of the MNL model on networks that are constructed from empirical data. We employ the ``repeated choice'' (RC) model to study network formation \cite{TrainRevelt97mixedlogit}. We argue that the RC model overcomes important limitations of the MNL model and is well-suited to study network formation. We also illustrate how to use the RC model to accurately study network formation using both synthetic and real-world networks. Using synthetic networks, we also compare the performance of the MNL model and the RC model; we find that the RC model estimates the data-generation process of our synthetic networks more accurately than the MNL model. We provide examples of qualitatively interesting questions -- the presence of homophily in a teen friendship network and the fact that new patents are more likely to cite older, more cited, and similar patents -- for which the RC model allows us to achieve insights.

Evaluating the reliability of intelligent physical systems against rare catastrophic events poses a huge testing burden for real-world applications. Simulation provides a useful, if not unique, platform to evaluate the extremal risks of these AIenabled systems before their deployments. Importance Sampling (IS), while proven to be powerful for rare-event simulation, faces challenges in handling these systems due to their black-box nature that fundamentally undermines its efficiency guarantee. To overcome this challenge, we propose a framework called Deep Probabilistic Accelerated Evaluation (D-PrAE) to design IS, which leverages rare-event-set learning and and a new notion of efficiency certificate. D-PrAE combines the dominating point method with deep neural network classifiers to achieve superior estimation efficiency. We present theoretical guarantees and demonstrate the empirical effectiveness of D-PrAE via examples on the safety-testing of self-driving algorithms that are beyond the reach of classical variance reduction techniques.

The Variational Auto-Encoder (VAE) belongs to a class of models, which we will refer to as deep maximum likelihood models, that uses a deep neural network to learn a maximum likelihood model for some input data. They are perhaps the most simple and efficient deep maximum likelihood model available, and have thus gained popularity in representation learning and generative image modeling. Unfortunately, in my opinion, in some circles the term "VAE" has become somewhat synonymous with "an auto-encoder with stochastic regularization that generates useful or beautiful samples", which has led to various misconceptions about VAEs. In this tutorial, we will return to the probabilistic and information theoretic roots of VAEs, clarify common misconceptions about VAEs, and look at a toy example on 2D data that will illustrate the capabilities and limitations of VAEs. In Section 2, we will give an overview of what is a maximum likelihood model and what a VAE looks like.

We propose a unified framework for adaptive connection sampling in graph neural networks (GNNs) that generalizes existing stochastic regularization methods for training GNNs. The proposed framework not only alleviates over-smoothing and over-fitting tendencies of deep GNNs, but also enables learning with uncertainty in graph analytic tasks with GNNs. Instead of using fixed sampling rates or hand-tuning them as model hyperparameters in existing stochastic regularization methods, our adaptive connection sampling can be trained jointly with GNN model parameters in both global and local fashions. GNN training with adaptive connection sampling is shown to be mathematically equivalent to an efficient approximation of training Bayesian GNNs. Experimental results with ablation studies on benchmark datasets validate that adaptively learning the sampling rate given graph training data is the key to boost the performance of GNNs in semi-supervised node classification, less prone to over-smoothing and over-fitting with more robust prediction.

In this work, we propose a Bayesian type sparse deep learning algorithm. The algorithm utilizes a set of spike-and-slab priors for the parameters in the deep neural network. The hierarchical Bayesian mixture will be trained using an adaptive empirical method. That is, one will alternatively sample from the posterior using preconditioned stochastic gradient Langevin Dynamics (PSGLD), and optimize the latent variables via stochastic approximation. The sparsity of the network is achieved while optimizing the hyperparameters with adaptive searching and penalizing. A popular SG-MCMC approach is Stochastic gradient Langevin dynamics (SGLD). However, considering the complex geometry in the model parameter space in non-convex learning, updating parameters using a universal step size in each component as in SGLD may cause slow mixing. To address this issue, we apply a computationally manageable preconditioner in the updating rule, which provides a step-size parameter to adapt to local geometric properties. Moreover, by smoothly optimizing the hyperparameter in the preconditioning matrix, our proposed algorithm ensures a decreasing bias, which is introduced by ignoring the correction term in preconditioned SGLD. According to the existing theoretical framework, we show that the proposed algorithm can asymptotically converge to the correct distribution with a controllable bias under mild conditions. Numerical tests are performed on both synthetic regression problems and learning the solutions of elliptic PDE, which demonstrate the accuracy and efficiency of present work.

We study the use of policy gradient algorithms to optimize over a class of generalized Thompson sampling policies. Our central insight is to view the posterior parameter sampled by Thompson sampling as a kind of pseudo-action. Policy gradient methods can then be tractably applied to search over a class of sampling policies, which determine a probability distribution over pseudo-actions (i.e., sampled parameters) as a function of observed data. We also propose and compare policy gradient estimators that are specialized to Bayesian bandit problems. Numerical experiments demonstrate that direct policy search on top of Thompson sampling automatically corrects for some of the algorithm's known shortcomings and offers meaningful improvements even in long horizon problems where standard Thompson sampling is extremely effective.

Given partially observed pairwise comparison data generated by the Bradley-Terry-Luce (BTL) model, we study the problem of top-$k$ ranking. That is, to optimally identify the set of top-$k$ players. We derive the minimax rate with respect to a normalized Hamming loss. This provides the first result in the literature that characterizes the partial recovery error in terms of the proportion of mistakes for top-$k$ ranking. We also derive the optimal signal to noise ratio condition for the exact recovery of the top-$k$ set. The maximum likelihood estimator (MLE) is shown to achieve both optimal partial recovery and optimal exact recovery. On the other hand, we show another popular algorithm, the spectral method, is in general sub-optimal. Our results complement the recent work by Chen et al. (2019) that shows both the MLE and the spectral method achieve the optimal sample complexity for exact recovery. It turns out the leading constants of the sample complexity are different for the two algorithms. Another contribution that may be of independent interest is the analysis of the MLE without any penalty or regularization for the BTL model. This closes an important gap between theory and practice in the literature of ranking.