A non-parametric conditional factor regression model for high-dimensional input and response

In this paper, we propose a non-parametric conditional factor regression (NCFR)model for domains with high-dimensional input and response. NCFR enhances linear regression in two ways: a) introducing low-dimensional latent factors leading to dimensionality reduction and b) integrating an Indian Buffet Process as a prior for the latent factors to derive unlimited sparse dimensions. Experimental results comparing NCRF to several alternatives give evidence to remarkable prediction performance.Comment: 9 pages, 3 figures, NIPS submissio

Assessment of the lognormality assumption of seismic fragility curves using non-parametric representations

Fragility curves are commonly used in civil engineering to estimate the vulnerability of structures to earthquakes. The probability of failure associated with a failure criterion (e.g. the maximal inter-storey drift ratio being greater than a prescribed threshold) is represented as a function of the intensity of the earthquake ground motion (e.g. peak ground acceleration or spectral acceleration). The classical approach consists in assuming a lognormal shape of the fragility curves. In this paper, we introduce two non-parametric approaches to establish the fragility curves without making any assumption, namely the conditional Monte Carlo simulation and the kernel density estimation. As an illustration, we compute the fragility curves of a 3-storey steel structure, accounting for the nonlinear behavior of the system. The curves obtained by the proposed approaches are compared with each other and with those obtained using the classical lognormal assumption.Comment: 17 pages, 10 figures, submitted to Structural Safety journa

Stochastic Multi-objective Optimization on a Budget: Application to multi-pass wire drawing with quantified uncertainties

Design optimization of engineering systems with multiple competing objectives is a painstakingly tedious process especially when the objective functions are expensive-to-evaluate computer codes with parametric uncertainties. The effectiveness of the state-of-the-art techniques is greatly diminished because they require a large number of objective evaluations, which makes them impractical for problems of the above kind. Bayesian global optimization (BGO), has managed to deal with these challenges in solving single-objective optimization problems and has recently been extended to multi-objective optimization (MOO). BGO models the objectives via probabilistic surrogates and uses the epistemic uncertainty to define an information acquisition function (IAF) that quantifies the merit of evaluating the objective at new designs. This iterative data acquisition process continues until a stopping criterion is met. The most commonly used IAF for MOO is the expected improvement over the dominated hypervolume (EIHV) which in its original form is unable to deal with parametric uncertainties or measurement noise. In this work, we provide a systematic reformulation of EIHV to deal with stochastic MOO problems. The primary contribution of this paper lies in being able to filter out the noise and reformulate the EIHV without having to observe or estimate the stochastic parameters. An addendum of the probabilistic nature of our methodology is that it enables us to characterize our confidence about the predicted Pareto front. We verify and validate the proposed methodology by applying it to synthetic test problems with known solutions. We demonstrate our approach on an industrial problem of die pass design for a steel wire drawing process.Comment: 19 pages, 14 figure

Breiman's "Two Cultures" Revisited and Reconciled

In a landmark paper published in 2001, Leo Breiman described the tense standoff between two cultures of data modeling: parametric statistical and algorithmic machine learning. The cultural division between these two statistical learning frameworks has been growing at a steady pace in recent years. What is the way forward? It has become blatantly obvious that this widening gap between "the two cultures" cannot be averted unless we find a way to blend them into a coherent whole. This article presents a solution by establishing a link between the two cultures. Through examples, we describe the challenges and potential gains of this new integrated statistical thinking.Comment: This paper celebrates the 70th anniversary of Statistical Machine Learning--- how far we've come, and how far we have to go. Keywords: Integrated statistical learning theory, Exploratory machine learning, Uncertainty prediction machine, ML-powered modern applied statistics, Information theor

When Gaussian Process Meets Big Data: A Review of Scalable GPs

The vast quantity of information brought by big data as well as the evolving computer hardware encourages success stories in the machine learning community. In the meanwhile, it poses challenges for the Gaussian process (GP) regression, a well-known non-parametric and interpretable Bayesian model, which suffers from cubic complexity to data size. To improve the scalability while retaining desirable prediction quality, a variety of scalable GPs have been presented. But they have not yet been comprehensively reviewed and analyzed in order to be well understood by both academia and industry. The review of scalable GPs in the GP community is timely and important due to the explosion of data size. To this end, this paper is devoted to the review on state-of-the-art scalable GPs involving two main categories: global approximations which distillate the entire data and local approximations which divide the data for subspace learning. Particularly, for global approximations, we mainly focus on sparse approximations comprising prior approximations which modify the prior but perform exact inference, posterior approximations which retain exact prior but perform approximate inference, and structured sparse approximations which exploit specific structures in kernel matrix; for local approximations, we highlight the mixture/product of experts that conducts model averaging from multiple local experts to boost predictions. To present a complete review, recent advances for improving the scalability and capability of scalable GPs are reviewed. Finally, the extensions and open issues regarding the implementation of scalable GPs in various scenarios are reviewed and discussed to inspire novel ideas for future research avenues.Comment: 20 pages, 6 figure

Neural Likelihoods via Cumulative Distribution Functions

We leverage neural networks as universal approximators of monotonic functions to build a parameterization of conditional cumulative distribution functions (CDFs). By the application of automatic differentiation with respect to response variables and then to parameters of this CDF representation, we are able to build black box CDF and density estimators. A suite of families is introduced as alternative constructions for the multivariate case. At one extreme, the simplest construction is a competitive density estimator against state-of-the-art deep learning methods, although it does not provide an easily computable representation of multivariate CDFs. At the other extreme, we have a flexible construction from which multivariate CDF evaluations and marginalizations can be obtained by a simple forward pass in a deep neural net, but where the computation of the likelihood scales exponentially with dimensionality. Alternatives in between the extremes are discussed. We evaluate the different representations empirically on a variety of tasks involving tail area probabilities, tail dependence and (partial) density estimation.Comment: 10 page

Estimation Considerations in Contextual Bandits

Contextual bandit algorithms are sensitive to the estimation method of the outcome model as well as the exploration method used, particularly in the presence of rich heterogeneity or complex outcome models, which can lead to difficult estimation problems along the path of learning. We study a consideration for the exploration vs. exploitation framework that does not arise in multi-armed bandits but is crucial in contextual bandits; the way exploration and exploitation is conducted in the present affects the bias and variance in the potential outcome model estimation in subsequent stages of learning. We develop parametric and non-parametric contextual bandits that integrate balancing methods from the causal inference literature in their estimation to make it less prone to problems of estimation bias. We provide the first regret bound analyses for contextual bandits with balancing in the domain of linear contextual bandits that match the state of the art regret bounds. We demonstrate the strong practical advantage of balanced contextual bandits on a large number of supervised learning datasets and on a synthetic example that simulates model mis-specification and prejudice in the initial training data. Additionally, we develop contextual bandits with simpler assignment policies by leveraging sparse model estimation methods from the econometrics literature and demonstrate empirically that in the early stages they can improve the rate of learning and decrease regret

A Contemporary Overview of Probabilistic Latent Variable Models

In this paper we provide a conceptual overview of latent variable models within a probabilistic modeling framework, an overview that emphasizes the compositional nature and the interconnectedness of the seemingly disparate models commonly encountered in statistical practice

Sequential Design for Optimal Stopping Problems

We propose a new approach to solve optimal stopping problems via simulation. Working within the backward dynamic programming/Snell envelope framework, we augment the methodology of Longstaff-Schwartz that focuses on approximating the stopping strategy. Namely, we introduce adaptive generation of the stochastic grids anchoring the simulated sample paths of the underlying state process. This allows for active learning of the classifiers partitioning the state space into the continuation and stopping regions. To this end, we examine sequential design schemes that adaptively place new design points close to the stopping boundaries. We then discuss dynamic regression algorithms that can implement such recursive estimation and local refinement of the classifiers. The new algorithm is illustrated with a variety of numerical experiments, showing that an order of magnitude savings in terms of design size can be achieved. We also compare with existing benchmarks in the context of pricing multi-dimensional Bermudan options.Comment: 24 page