Using the linear Gaussian latent variable model as a starting point we relax some of the constraints it imposes by deriving a nonparametric latent feature Gaussian variable model. This model introduces additional discrete latent variables to the original structure. The Bayesian nonparametric nature of this new model allows it to adapt complexity as more data is observed and project each data point onto a varying number of subspaces. The linear relationship between the continuous latent and observed variables make the proposed model straightforward to interpret, resembling a locally adaptive probabilistic PCA (A-PPCA). We propose two alternative Gibbs sampling procedures for inference in the new model and demonstrate its applicability on sensor data for passive health monitoring.

Complex phenomena are often modeled with computationally intensive feed-forward simulations for which a tractable analytic likelihood does not exist. In these cases, it is sometimes necessary to use an approximate likelihood or faster emulator model for efficient statistical inference. We describe a new two-sample testing framework for quantifying the quality of the fit to simulations at fixed parameter values. This framework can leverage any regression method to handle complex high-dimensional data and attain higher power in settings where well-known distance-based tests would not. We also introduce a statistically rigorous test for assessing global goodness-of-fit across simulation parameters. In cases where the fit is inadequate, our method provides valuable diagnostics by allowing one to identify regions in both feature and parameter space which the model fails to reproduce well. We provide both theoretical results and examples which illustrate the effectiveness of our approach.

In this paper, we consider a problem of failure prediction in the context of predictive maintenance applications. We present a new approach for rare failures prediction, based on a general methodology, which takes into account peculiar properties of technical systems. We illustrate the applicability of the method on the real-world test cases from aircraft operations.

There is a mismatch between the standard theoretical analyses of statistical machine learning and how learning is used in practice. The foundational assumption supporting the theory is that we can represent features and models using real-valued parameters. In practice, however, we do not use real numbers at any point during training or deployment. Instead, we rely on discrete and finite quantizations of the reals, typically floating points. In this paper, we propose a framework for reasoning about learning under arbitrary quantizations. Using this formalization, we prove the convergence of quantization-aware versions of the Perceptron and Frank-Wolfe algorithms. Finally, we report the results of an extensive empirical study of the impact of quantization using a broad spectrum of datasets.

We are interested in parallelizing the Least Angle Regression (LARS) algorithm for fitting linear regression models to high-dimensional data. We consider two parallel and communication avoiding versions of the basic LARS algorithm. The two algorithms apply to data that have different layout patterns (one is appropriate for row-partitioned data, and the other is appropriate for column-partitioned data), and they have different asymptotic costs and practical performance. The first is bLARS, a block version of LARS algorithm where we update b columns at each iteration. Assuming that the data are row-partitioned, bLARS reduces the number of arithmetic operations, latency, and bandwidth by a factor of b. The second is Tournament-bLARS (T-bLARS), a tournament version of LARS, in which case processors compete, by running several LARS computations in parallel, to choose b new columns to be added into the solution. Assuming that the data are column-partitioned, T-bLARS reduces latency by a factor of b. Similarly to LARS, our proposed methods generate a sequence of linear models. We present extensive numerical experiments that illustrate speed-ups up to 25x compared to LARS.

In distributed optimization and distributed numerical linear algebra, we often encounter an inversion bias: if we want to compute a quantity that depends on the inverse of a sum of distributed matrices, then the sum of the inverses does not equal the inverse of the sum. An example of this occurs in distributed Newton's method, where we wish to compute (or implicitly work with) the inverse Hessian multiplied by the gradient. In this case, locally computed estimates are biased, and so taking a uniform average will not recover the correct solution. To address this, we propose determinantal averaging, a new approach for correcting the inversion bias. This approach involves reweighting the local estimates of the Newton's step proportionally to the determinant of the local Hessian estimate, and then averaging them together to obtain an improved global estimate. This method provides the first known distributed Newton step that is asymptotically consistent, i.e., it recovers the exact step in the limit as the number of distributed partitions grows to infinity. To show this, we develop new expectation identities and moment bounds for the determinant and adjugate of a random matrix. Determinantal averaging can be applied not only to Newton's method, but to computing any quantity that is a linear tranformation of a matrix inverse, e.g., taking a trace of the inverse covariance matrix, which is used in data uncertainty quantification.

The United States is in the midst of an opioid epidemic with recent estimates indicating that more than 130 people die every day due to drug overdose. The over-prescription and addiction to opioid painkillers, heroin, and synthetic opioids, has led to a public health crisis and created a huge social and economic burden. Statistical learning methods that use data from multiple clinical centers across the US to detect opioid over-prescribing trends and predict possible opioid misuse are required. However, the semantic heterogeneity in the representation of clinical data across different centers makes the development and evaluation of such methods difficult and non-trivial. We create the Opioid Drug Knowledge Graph (ODKG) -- a network of opioid-related drugs, active ingredients, formulations, combinations, and brand names. We use the ODKG to normalize drug strings in a clinical data warehouse consisting of patient data from over 400 healthcare facilities in 42 different states. We showcase the use of ODKG to generate summary statistics of opioid prescription trends across US regions. These methods and resources can aid the development of advanced and scalable models to monitor the opioid epidemic and to detect illicit opioid misuse behavior. Our work is relevant to policymakers and pain researchers who wish to systematically assess factors that contribute to opioid over-prescribing and iatrogenic opioid addiction in the US.

We propose Dirichlet Simplex Nest, a class of probabilistic models suitable for a variety of data types, and develop fast and provably accurate inference algorithms by accounting for the model's convex geometry and low dimensional simplicial structure. By exploiting the connection to Voronoi tessellation and properties of Dirichlet distribution, the proposed inference algorithm is shown to achieve consistency and strong error bound guarantees on a range of model settings and data distributions. The effectiveness of our model and the learning algorithm is demonstrated by simulations and by analyses of text and financial data.

The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem, and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters, in contrast to the prediction error of the parameters. This loss function was recently introduced in Elmachtoub and Grigas (2017), which called it the Smart Predict-then-Optimize (SPO) loss. Since the SPO loss is nonconvex and noncontinuous, standard results for deriving generalization bounds do not apply. In this work, we provide an assortment of generalization bounds for the SPO loss function. In particular, we derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points, but, in the case of a general convex set, have poor dependence on the dimension. By exploiting the structure of the SPO loss function and an additional strong convexity assumption on the feasible region, we can dramatically improve the dependence on the dimension via an analysis and corresponding bounds that are akin to the margin guarantees in classification problems.

Over-parameterized models, such as DeepNets and ConvNets, form a class of models that are routinely adopted in a wide variety of applications, and for which Bayesian inference is desirable but extremely challenging. Variational inference offers the tools to tackle this challenge in a scalable way and with some degree of flexibility on the approximation, but for over-parameterized models this is challenging due to the over-regularization property of the variational objective. Inspired by the literature on kernel methods, and in particular on structured approximations of distributions of random matrices, this paper proposes Walsh-Hadamard Variational Inference (WHVI), which uses Walsh-Hadamard-based factorization strategies to reduce the parameterization and accelerate computations, thus avoiding over-regularization issues with the variational objective. Extensive theoretical and empirical analyses demonstrate that WHVI yields considerable speedups and model reductions compared to other techniques to carry out approximate inference for over-parameterized models, and ultimately show how advances in kernel methods can be translated into advances in approximate Bayesian inference.