To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.

We present an application of invariant polynomials in machine learning. Using the methods developed in previous work, we obtain two types of generators of the Lorentz- and permutation-invariant polynomials in particle momenta; minimal algebra generators and Hironaka decompositions. We discuss and prove some approximation theorems to make use of these invariant generators in machine learning algorithms in general and in neural networks specifically. By implementing these generators in neural networks applied to regression tasks, we test the improvements in performance under a wide range of hyperparameter choices and find a reduction of the loss on training data and a significant reduction of the loss on validation data. For a different approach on quantifying the performance of these neural networks, we treat the problem from a Bayesian inference perspective and employ nested sampling techniques to perform model comparison. Beyond a certain network size, we find that networks utilising Hironaka decompositions perform the best.

Deep learning has achieved impressive performance on many tasks in recent years. However, it has been found that it is still not enough for deep neural networks to provide only point estimates. For high-risk tasks, we need to assess the reliability of the model predictions. This requires us to quantify the uncertainty of model prediction and construct prediction intervals. In this paper, We explore the uncertainty in deep learning to construct the prediction intervals. In general, We comprehensively consider two categories of uncertainties: aleatory uncertainty and epistemic uncertainty. We design a special loss function, which enables us to learn uncertainty without uncertainty label. We only need to supervise the learning of regression task. We learn the aleatory uncertainty implicitly from the loss function. And that epistemic uncertainty is accounted for in ensembled form. Our method correlates the construction of prediction intervals with the uncertainty estimation. Impressive results on some publicly available datasets show that the performance of our method is competitive with other state-of-the-art methods.

Massive Open Online Courses (MOOCs) have become a popular choice for e-learning thanks to their great flexibility. However, due to large numbers of learners and their diverse backgrounds, it is taxing to offer real-time support. Learners may post their feelings of confusion and struggle in the respective MOOC forums, but with the large volume of posts and high workloads for MOOC instructors, it is unlikely that the instructors can identify all learners requiring intervention. This problem has been studied as a Natural Language Processing (NLP) problem recently, and is known to be challenging, due to the imbalance of the data and the complex nature of the task. In this paper, we explore for the first time Bayesian deep learning on learner-based text posts with two methods: Monte Carlo Dropout and Variational Inference, as a new solution to assessing the need of instructor interventions for a learner's post. We compare models based on our proposed methods with probabilistic modelling to its baseline non-Bayesian models under similar circumstances, for different cases of applying prediction. The results suggest that Bayesian deep learning offers a critical uncertainty measure that is not supplied by traditional neural networks. This adds more explainability, trust and robustness to AI, which is crucial in education-based applications. Additionally, it can achieve similar or better performance compared to non-probabilistic neural networks, as well as grant lower variance.

We study high-dimensional Bayesian linear regression with product priors. Using the nascent theory of non-linear large deviations (Chatterjee and Dembo,2016), we derive sufficient conditions for the leading-order correctness of the naive mean-field approximation to the log-normalizing constant of the posterior distribution. Subsequently, assuming a true linear model for the observed data, we derive a limiting infinite dimensional variational formula for the log normalizing constant of the posterior. Furthermore, we establish that under an additional "separation" condition, the variational problem has a unique optimizer, and this optimizer governs the probabilistic properties of the posterior distribution. We provide intuitive sufficient conditions for the validity of this "separation" condition. Finally, we illustrate our results on concrete examples with specific design matrices.

Machine learning models $-$ now commonly developed to screen, diagnose, or predict health conditions $-$ are evaluated with a variety of performance metrics. An important first step in assessing the practical utility of a model is to evaluate its average performance over an entire population of interest. In many settings, it is also critical that the model makes good predictions within predefined subpopulations. For instance, showing that a model is fair or equitable requires evaluating the model's performance in different demographic subgroups. However, subpopulation performance metrics are typically computed using only data from that subgroup, resulting in higher variance estimates for smaller groups. We devise a procedure to measure subpopulation performance that can be more sample-efficient than the typical subsample estimates. We propose using an evaluation model $-$ a model that describes the conditional distribution of the predictive model score $-$ to form model-based metric (MBM) estimates. Our procedure incorporates model checking and validation, and we propose a computationally efficient approximation of the traditional nonparametric bootstrap to form confidence intervals. We evaluate MBMs on two main tasks: a semi-synthetic setting where ground truth metrics are available and a real-world hospital readmission prediction task. We find that MBMs consistently produce more accurate and lower variance estimates of model performance for small subpopulations.

Game-theoretic attribution techniques based on Shapley values are used extensively to interpret black-box machine learning models, but their exact calculation is generally NP-hard, requiring approximation methods for non-trivial models. As the computation of Shapley values can be expressed as a summation over a set of permutations, a common approach is to sample a subset of these permutations for approximation. Unfortunately, standard Monte Carlo sampling methods can exhibit slow convergence, and more sophisticated quasi Monte Carlo methods are not well defined on the space of permutations. To address this, we investigate new approaches based on two classes of approximation methods and compare them empirically. First, we demonstrate quadrature techniques in a RKHS containing functions of permutations, using the Mallows kernel to obtain explicit convergence rates of $O(1/n)$, improving on $O(1/\sqrt{n})$ for plain Monte Carlo. The RKHS perspective also leads to quasi Monte Carlo type error bounds, with a tractable discrepancy measure defined on permutations. Second, we exploit connections between the hypersphere $\mathbb{S}^{d-2}$ and permutations to create practical algorithms for generating permutation samples with good properties. Experiments show the above techniques provide significant improvements for Shapley value estimates over existing methods, converging to a smaller RMSE in the same number of model evaluations.

System identification is of special interest in science and engineering. This article is concerned with a system identification problem arising in stochastic dynamic systems, where the aim is to estimating the parameters of a system along with its unknown noise processes. In particular, we propose a Bayesian nonparametric approach for system identification in discrete time nonlinear random dynamical systems assuming only the order of the Markov process is known. The proposed method replaces the assumption of Gaussian distributed error components with a highly flexible family of probability density functions based on Bayesian nonparametric priors. Additionally, the functional form of the system is estimated by leveraging Bayesian neural networks which also leads to flexible uncertainty quantification. Asymptotically on the number of hidden neurons, the proposed model converges to full nonparametric Bayesian regression model. A Gibbs sampler for posterior inference is proposed and its effectiveness is illustrated in simulated and real time series.

Probabilistic graphical modeling (PGM) provides a framework for formulating an interpretable generative process of data and expressing uncertainty about unknowns, but it lacks flexibility. Deep learning (DL) is an alternative framework for learning from data that has achieved great empirical success in recent years. DL offers great flexibility, but it lacks the interpretability and calibration of PGM. This thesis develops deep probabilistic graphical modeling (DPGM.) DPGM consists in leveraging DL to make PGM more flexible. DPGM brings about new methods for learning from data that exhibit the advantages of both PGM and DL. We use DL within PGM to build flexible models endowed with an interpretable latent structure. One model class we develop extends exponential family PCA using neural networks to improve predictive performance while enforcing the interpretability of the latent factors. Another model class we introduce enables accounting for long-term dependencies when modeling sequential data, which is a challenge when using purely DL or PGM approaches. Finally, DPGM successfully solves several outstanding problems of probabilistic topic models, a widely used family of models in PGM. DPGM also brings about new algorithms for learning with complex data. We develop reweighted expectation maximization, an algorithm that unifies several existing maximum likelihood-based algorithms for learning models parameterized by neural networks. This unifying view is made possible using expectation maximization, a canonical inference algorithm in PGM. We also develop entropy-regularized adversarial learning, a learning paradigm that deviates from the traditional maximum likelihood approach used in PGM. From the DL perspective, entropy-regularized adversarial learning provides a solution to the long-standing mode collapse problem of generative adversarial networks, a widely used DL approach.

The Expectation-Maximization (EM) algorithm is one of the main algorithms in machine learning for estimation of model parameters [2][3][4]. For example, it is used to estimate mixing coefficients, means, and covariances in mixture models as shown in Figure 1. Its objective is to maximize the likelihood p(X θ) where X is a matrix of observed data and θ is a vector of model parameters. This is maximum likelihood estimation and in practice the log-likelihood ln p(X θ) is maximized. The model parameters that maximize this function are deemed to be the correct model parameters.