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1. Epistemic Uncertainty Quantification in State-space LPV Model Identification Using Bayesian Neural Networks

   Bao, Yajie; Mohammadpour Velni, Javad; Shahbakhti, Mahdi (January 2020, IEEE control systems letters)

   This paper presents a variational Bayesian inference Neural Network (BNN) approach to quantify uncertainties in matrix function estimation for the state-space linear parameter-varying (LPV) model identification problem using only inputs/outputs data. The proposed method simultaneously estimates states and posteriors of matrix functions given data. In particular, states are estimated by reaching a consensus between an estimator based on past system trajectory and an estimator by recurrent equations of states; posteriors are approximated by minimizing the Kullback–Leibler (KL) divergence between the parameterized posterior distribution and the true posterior of the LPV model parameters. Furthermore, techniques such as transfer learning are explored in this work to reduce computational cost and prevent convergence failure of Bayesian inference. The proposed data-driven method is validated using experimental data for identification of a control-oriented reactivity controlled compression ignition (RCCI) engine model. 

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2. Asymptotic Consistency of α-Rényi-Approximate Posteriors

   Jaiswal, Prateek; Rao, Vinayak; Honnappa, Harsha (May 2020, Journal of machine learning research)

   Nguyen, XuanLong (Ed.)

   We study the asymptotic consistency properties of α-Rényi approximate posteriors, a class of variational Bayesian methods that approximate an intractable Bayesian posterior with a member of a tractable family of distributions, the member chosen to minimize the α-Rényi divergence from the true posterior. Unique to our work is that we consider settings with α > 1, resulting in approximations that upperbound the log-likelihood, and consequently have wider spread than traditional variational approaches that minimize the Kullback-Liebler (KL) divergence from the posterior. Our primary result identifies sufficient conditions under which consistency holds, centering around the existence of a ‘good’ sequence of distributions in the approximating family that possesses, among other properties, the right rate of convergence to a limit distribution. We further characterize the good sequence by demonstrating that a sequence of distributions that converges too quickly cannot be a good sequence. We also extend our analysis to the setting where α equals one, corresponding to the minimizer of the reverse KL divergence, and to models with local latent variables. We also illustrate the existence of good sequence with a number of examples. Our results complement a growing body of work focused on the frequentist properties of variational Bayesian methods. 

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3. A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flows

   https://doi.org/10.1016/j.cma.2023.116682  

   Dasgupta, Agnimitra; Patel, Dhruv V; Ray, Deep; Johnson, Erik A; Oberai, Assad A (February 2024, Computer Methods in Applied Mechanics and Engineering)

   We propose a novel modular inference approach combining two different generative models — generative adversarial networks (GAN) and normalizing flows — to approximate the posterior distribution of physics-based Bayesian inverse problems framed in high-dimensional ambient spaces. We dub the proposed framework GAN-Flow. The proposed method leverages the intrinsic dimension reduction and superior sample generation capabilities of GANs to define a low-dimensional data-driven prior distribution. Once a trained GAN-prior is available, the inverse problem is solved entirely in the latent space of the GAN using variational Bayesian inference with normalizing flow-based variational distribution, which approximates low-dimensional posterior distribution by transforming realizations from the low-dimensional latent prior (Gaussian) to corresponding realizations of a low-dimensional variational posterior distribution. The trained GAN generator then maps realizations from this approximate posterior distribution in the latent space back to the high-dimensional ambient space. We also propose a two-stage training strategy for GAN-Flow wherein we train the two generative models sequentially. Thereafter, GAN-Flow can estimate the statistics of posterior-predictive quantities of interest at virtually no additional computational cost. The synergy between the two types of generative models allows us to overcome many challenges associated with the application of Bayesian inference to large-scale inverse problems, chief among which are describing an informative prior and sampling from the high-dimensional posterior. GAN-Flow does not involve Markov chain Monte Carlo simulation, making it particularly suitable for solving large-scale inverse problems. We demonstrate the efficacy and flexibility of GAN-Flow on various physics-based inverse problems of varying ambient dimensionality and prior knowledge using different types of GANs and normalizing flows. Notably, one of the applications we consider involves a 65,536-dimensional inverse problem of phase retrieval wherein an object is reconstructed from sparse noisy measurements of the magnitude of its Fourier transform. 

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4. On the Universality of the Logistic Loss Function

   https://doi.org/10.1109/ISIT.2018.8437786  

   Painsky, Amichai; Wornell, Gregory (June 2018, Proceedings of International Symposium on Information Theory)

   A loss function measures the discrepancy between the true values (observations) and their estimated fits, for a given instance of data. A loss function is said to be proper (unbiased, Fisher consistent) if the fits are defined over a unit simplex, and the minimizer of the expected loss is the true underlying probability of the data. Typical examples are the zero-one loss, the quadratic loss and the Bernoulli log-likelihood loss (log-loss). In this work we show that for binary classification problems, the divergence associated with smooth, proper and convex loss functions is bounded from above by the Kullback-Leibler (KL) divergence, up to a multiplicative normalization constant. It implies that by minimizing the log-loss (associated with the KL divergence), we minimize an upper bound to any choice of loss functions from this set. This property justifies the broad use of log-loss in regression, decision trees, deep neural networks and many other applications. In addition, we show that the KL divergence bounds from above any separable Bregman divergence that is convex in its second argument (up to a multiplicative normalization constant). This result introduces a new set of divergence inequalities, similar to the well-known Pinsker inequality. 

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5. Amortized Population Gibbs Samplers with Neural Sufficient Statistics

   Wu, H.; Zimmermann, H.; Sennesh, E. and; van de Meent, Jan-Willem (July 2021, Proceedings of the 37th International Conference on Machine Learning)

   III, H.D.; Singh, A. (Ed.)

   We develop amortized population Gibbs (APG) samplers, a class of scalable methods that frame structured variational inference as adaptive importance sampling. APG samplers construct high-dimensional proposals by iterating over updates to lower-dimensional blocks of variables. We train each conditional proposal by minimizing the inclusive KL divergence with respect to the conditional posterior. To appropriately account for the size of the input data, we develop a new parameterization in terms of neural sufficient statistics. Experiments show that APG samplers can be used to train highly-structured deep generative models in an unsupervised manner, and achieve substantial improvements in inference accuracy relative to standard autoencoding variational methods. 

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