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1. Conditional Deep Gaussian Processes: Empirical Bayes Hyperdata Learning

   https://doi.org/10.3390/e23111387  

   Lu, Chi-Ken ; Shafto, Patrick ( November 2021 , Entropy)

   null (Ed.)

   It is desirable to combine the expressive power of deep learning with Gaussian Process (GP) in one expressive Bayesian learning model. Deep kernel learning showed success as a deep network used for feature extraction. Then, a GP was used as the function model. Recently, it was suggested that, albeit training with marginal likelihood, the deterministic nature of a feature extractor might lead to overfitting, and replacement with a Bayesian network seemed to cure it. Here, we propose the conditional deep Gaussian process (DGP) in which the intermediate GPs in hierarchical composition are supported by the hyperdata and the exposed GP remains zero mean. Motivated by the inducing points in sparse GP, the hyperdata also play the role of function supports, but are hyperparameters rather than random variables. It follows our previous moment matching approach to approximate the marginal prior for conditional DGP with a GP carrying an effective kernel. Thus, as in empirical Bayes, the hyperdata are learned by optimizing the approximate marginal likelihood which implicitly depends on the hyperdata via the kernel. We show the equivalence with the deep kernel learning in the limit of dense hyperdata in latent space. However, the conditional DGP and the corresponding approximate inference enjoy the benefit of being more Bayesian than deep kernel learning. Preliminary extrapolation results demonstrate expressive power from the depth of hierarchy by exploiting the exact covariance and hyperdata learning, in comparison with GP kernel composition, DGP variational inference and deep kernel learning. We also address the non-Gaussian aspect of our model as well as way of upgrading to a full Bayes inference. 

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2. Uncertainty-aware mixed-variable machine learning for materials design

   https://doi.org/10.1038/s41598-022-23431-2  

   Zhang, Hengrui ; Chen, Wei ; Iyer, Akshay ; Apley, Daniel W. ; Chen, Wei ( December 2022 , Scientific Reports)

   Abstract Data-driven design shows the promise of accelerating materials discovery but is challenging due to the prohibitive cost of searching the vast design space of chemistry, structure, and synthesis methods. Bayesian optimization (BO) employs uncertainty-aware machine learning models to select promising designs to evaluate, hence reducing the cost. However, BO with mixed numerical and categorical variables, which is of particular interest in materials design, has not been well studied. In this work, we survey frequentist and Bayesian approaches to uncertainty quantification of machine learning with mixed variables. We then conduct a systematic comparative study of their performances in BO using a popular representative model from each group, the random forest-based Lolo model (frequentist) and the latent variable Gaussian process model (Bayesian). We examine the efficacy of the two models in the optimization of mathematical functions, as well as properties of structural and functional materials, where we observe performance differences as related to problem dimensionality and complexity. By investigating the machine learning models’ predictive and uncertainty estimation capabilities, we provide interpretations of the observed performance differences. Our results provide practical guidance on choosing between frequentist and Bayesian uncertainty-aware machine learning models for mixed-variable BO in materials design. 

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3. Statistical Guarantees for Transformation Based Models with applications to Implicit Variational Inference

   Plummer, Sean ; Zhou, Shuang ; Bhattacharya, Anirban ; Dunson, David ; Pati, Debdeep ( April 2021 , Proceedings of Machine Learning Research)

   null (Ed.)

   Transformation-based methods have been an attractive approach in non-parametric inference for problems such as unconditional and conditional density estimation due to their unique hierarchical structure that models the data as flexible transformation of a set of common latent variables. More recently, transformation-based models have been used in variational inference (VI) to construct flexible implicit families of variational distributions. However, their use in both nonparametric inference and variational inference lacks theoretical justification. We provide theoretical justification for the use of non-linear latent variable models (NL-LVMs) in non-parametric inference by showing that the support of the transformation induced prior in the space of densities is sufficiently large in the L1 sense. We also show that, when a Gaussian process (GP) prior is placed on the transformation function, the posterior concentrates at the optimal rate up to a logarithmic factor. Adopting the flexibility demonstrated in the non-parametric setting, we use the NL-LVM to construct an implicit family of variational distributions, deemed GP-IVI. We delineate sufficient conditions under which GP-IVI achieves optimal risk bounds and approximates the true posterior in the sense of the Kullback–Leibler divergence. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference. 

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4. A Gaussian latent variable model for incomplete mixed type data

   M. Ajirak and P. M Djurić ( January 2023 , Conference record IEEE International Conference on Acoustics Speech and Signal Processing)

   In many machine learning problems, one has to work with data of different types, including continuous, discrete, and categorical data. Further, it is often the case that many of these data are missing from the database. This paper proposes a Gaussian process framework that efficiently captures the information from mixed numerical and categorical data that effectively incorporates missing variables. First, we propose a generative model for the mixed-type data. The generative model exploits Gaussian processes with kernels constructed from the latent vectors. We also propose a method for inference of the unknowns, and in its implementation, we rely on a sparse spectrum approximation of the Gaussian processes and variational inference. We demonstrate the performance of the method for both supervised and unsupervised tasks. First, we investigate the imputation of missing variables in an unsupervised setting, and then we show the results of joint imputation and classification on IBM employee data. 

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5. Bayesian Network Structure Optimization for Improved Design Space Mapping for Design Exploration With Materials Design Applications

   https://doi.org/10.1115/DETC2017-67643  

   Sharpe, Conner ; Morris, Clinton ; Goldsberry, Benjamin ; Seepersad, Carolyn Conner ; Haberman, Michael R. ( August 2017 , Proceedings of the ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference)

   Modern design problems present both opportunities and challenges, including multifunctionality, high dimensionality, highly nonlinear multimodal responses, and multiple levels or scales. These factors are particularly important in materials design problems and make it difficult for traditional optimization algorithms to search the space effectively, and designer intuition is often insufficient in problems of this complexity. Efficient machine learning algorithms can map complex design spaces to help designers quickly identify promising regions of the design space. In particular, Bayesian network classifiers (BNCs) have been demonstrated as effective tools for top-down design of complex multilevel problems. The most common instantiations of BNCs assume that all design variables are independent. This assumption reduces computational cost, but can limit accuracy especially in engineering problems with interacting factors. The ability to learn representative network structures from data could provide accurate maps of the design space with limited computational expense. Population-based stochastic optimization techniques such as genetic algorithms (GAs) are ideal for optimizing networks because they accommodate discrete, combinatorial, and multimodal problems. Our approach utilizes GAs to identify optimal networks based on limited training sets so that future test points can be classified as accurately and efficiently as possible. This method is first tested on a common machine learning data set, and then demonstrated on a sample design problem of a composite material subjected to a planar sound wave. 

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