Recently it has also become increasingly important in marketing, medicine and political sciences. Process systems engineering community adopts mathematical models successfully in various endeavors such as product and plant design, control system design, operations optimization, etc. (Pantelides and Renfro, 2013; Fung et al., 2016; Safdarnejad et al., 2016). A mathematical model is usually an abstract representation of a true system via sets of equations (algebraic, ordinary differential or partial differential), inequalities (e.g., a range of model validity), and logical conditions. Model development is usually divided into three major steps a) identification of the model structure, b) design and realization of the experiments, and c) estimation of the unknown parameters. In the latter phase, one often realizes maximum-likelihood estimation via least-squares methodology as he/she assumes--knowingly or not--that the measurement error present in the measured data is statistically distributed as a white Gaussian noise. Once the parameter estimates are known, the experimenter commonly determines the quality of the obtained model. This can be done either by using some validation data--if available--or via assessing the joint-confidence regions of the estimated parameters (Beale, 1960; Bates and Watts, 1988; Rooney and Biegler, 2001; Seber and Wild, 2003).

We present a novel approach for decreasing state uncertainty in planning prior to solving the planning problem. This is done by making predictions about the state based on currently known information, using machine learning techniques. For domains where uncertainty is high, we define an active learning process for identifying which information, once sensed, will best improve the accuracy of predictions. We demonstrate that an agent is able to solve problems with uncertainties in the state with less planning effort compared to standard planning techniques. Moreover, agents can solve problems for which they could not find valid plans without using predictions. Experimental results also demonstrate that using our active learning process for identifying information to be sensed leads to gathering information that improves the prediction process.

In this comprehensive review, we describe a new mathematical problem in artificial intelligence (AI) from a mathematical modeling perspective, following the philosophy stated by Rudolf E. Kalman that "Once you get the physics right, the rest is mathematics". The new problem is called "Generalized Constraints (GCs)", and we adopt GCs as a general term to describe any type of prior information in modelings. To understand better about GCs to be a general problem, we compare them with the conventional constraints (CCs) and list their extra challenges over CCs. In the construction of AI machines, we basically encounter more often GCs for modeling, rather than CCs with well-defined forms. Furthermore, we discuss the ultimate goals of AI and redefine transparent, interpretable, and explainable AI in terms of comprehension levels about machines. We review the studies in relation to the GC problems although most of them do not take the notion of GCs. We demonstrate that if AI machines are simplified by a coupling with both knowledge-driven submodel and data-driven submodel, GCs will play a critical role in a knowledge-driven submodel as well as in the coupling form between the two submodels. Examples are given to show that the studies in view of a generalized constraint problem will help us perceive and explore novel subjects in AI, or even in mathematics, such as generalized constraint learning (GCL).

Probability trees are one of the simplest models of causal generative processes. They possess clean semantics and -- unlike causal Bayesian networks -- they can represent context-specific causal dependencies, which are necessary for e.g. causal induction. Yet, they have received little attention from the AI and ML community. Here we present concrete algorithms for causal reasoning in discrete probability trees that cover the entire causal hierarchy (association, intervention, and counterfactuals), and operate on arbitrary propositional and causal events. Our work expands the domain of causal reasoning to a very general class of discrete stochastic processes.

Safe reinforcement learning (SRL) problems are typically modeled as constrained Markov Decision Process (CMDP), in which an agent explores the environment to maximize the expected total reward and meanwhile avoids violating certain constraints on a number of expected total costs. In general, such SRL problems have nonconvex objective functions subject to multiple nonconvex constraints, and hence are very challenging to solve, particularly to provide a globally optimal policy. Many popular SRL algorithms adopt a primal-dual structure which utilizes the updating of dual variables for satisfying the constraints. In contrast, we propose a primal approach, called constraint-rectified policy optimization (CRPO), which updates the policy alternatingly between objective improvement and constraint satisfaction. CRPO provides a primal-type algorithmic framework to solve SRL problems, where each policy update can take any variant of policy optimization step. To demonstrate the theoretical performance of CRPO, we adopt natural policy gradient (NPG) for each policy update step and show that CRPO achieves an $\mathcal{O}(1/\sqrt{T})$ convergence rate to the global optimal policy in the constrained policy set and an $\mathcal{O}(1/\sqrt{T})$ error bound on constraint satisfaction. This is the first finite-time analysis of SRL algorithms with global optimality guarantee. Our empirical results demonstrate that CRPO can outperform the existing primal-dual baseline algorithms significantly.

Dynamic covariance models appear prominently in the analysis of time-series data and can provide fundamental insights into complex systems. They are important for applications ranging from computational finance and economics (Engle et al., 2019) to epidemiology (Fox and Dunson, 2015) and neuroscience (Foti and Fox, 2019). Our work is motivated by the study of dynamic functional brain network connectivity, defined as the time-indexed covariance of brain activity. Improving understanding of brain function is timely, as neurological and neuropsychiatric disorders ranging from Schizophrenia, Autism, and Alzheimer's to depression create burdens on affected individuals. To this end, understanding the variation of brain connectivity between individuals is believed to be a crucial step towards uncovering the mechanisms of neural information processing (Sakoğlu et al., 2010; Chang et al., 2016), with potentially transformative applications to understanding and treating neurological and neuropsychiatric disorders (Fox and Raichle, 2007; Calhoun et al., 2014). There are a variety of approaches for estimating dynamic covariances in the neuroscience literature. Discrete-state hidden Markov models construct simple explanations of brain connectivity in terms of recurring connectivity patterns (Vidaurre et al., 2017), and yet they fail to capture the smooth nature of brain dynamics (Shine et al., 2016a,b).

Deep neural networks (DNNs) have made a revolution in numerous fields during the last decade. However, in tasks with high safety requirements, such as medical or autonomous driving applications, providing an assessment of the models reliability can be vital. Uncertainty estimation for DNNs has been addressed using Bayesian methods, providing mathematically founded models for reliability assessment. These model are computationally expensive and generally impractical for many real-time use cases. Recently, non-Bayesian methods were proposed to tackle uncertainty estimation more efficiently. We propose an efficient method for uncertainty estimation in DNNs achieving high accuracy. We simulate the notion of multi-task learning on single-task problems by producing parallel predictions from similar models differing by their loss. This multi-loss approach allows one-phase training for single-task learning with uncertainty estimation. We keep our inference time relatively low by leveraging the advantage proposed by the Deep-Sub-Ensembles method. The novelty of this work resides in the proposed accurate variational inference with a simple and convenient training procedure, while remaining competitive in terms of computational time. We conduct experiments on SVHN, CIFAR10, CIFAR100 as well as Image-Net using different architectures. Our results show improved accuracy on the classification task and competitive results on several uncertainty measures.

Research in adversarial learning follows a cat and mouse game between attackers and defenders where attacks are proposed, they are mitigated by new defenses, and subsequently new attacks are proposed that break earlier defenses, and so on. However, it has remained unclear as to whether there are conditions under which no better attacks or defenses can be proposed. In this paper, we propose a game-theoretic framework for studying attacks and defenses which exist in equilibrium. Under a locally linear decision boundary model for the underlying binary classifier, we prove that the Fast Gradient Method attack and the Randomized Smoothing defense form a Nash Equilibrium. We then show how this equilibrium defense can be approximated given finitely many samples from a data-generating distribution, and derive a generalization bound for the performance of our approximation.

Ride-hailing platforms generally provide various service options to customers, such as solo ride services, shared ride services, etc. It is generally expected that demands for different service modes are correlated, and the prediction of demand for one service mode can benefit from historical observations of demands for other service modes. Moreover, an accurate joint prediction of demands for multiple service modes can help the platforms better allocate and dispatch vehicle resources. Although there is a large stream of literature on ride-hailing demand predictions for one specific service mode, little efforts have been paid towards joint predictions of ride-hailing demands for multiple service modes. To address this issue, we propose a deep multi-task multi-graph learning approach, which combines two components: (1) multiple multi-graph convolutional (MGC) networks for predicting demands for different service modes, and (2) multi-task learning modules that enable knowledge sharing across multiple MGC networks. More specifically, two multi-task learning structures are established. The first one is the regularized cross-task learning, which builds cross-task connections among the inputs and outputs of multiple MGC networks. The second one is the multi-linear relationship learning, which imposes a prior tensor normal distribution on the weights of various MGC networks. Although there are no concrete bridges between different MGC networks, the weights of these networks are constrained by each other and subject to a common prior distribution. Evaluated with the for-hire-vehicle datasets in Manhattan, we show that our propose approach outperforms the benchmark algorithms in prediction accuracy for different ride-hailing modes.

An algorithm for non-stationary spatial modelling using multiple secondary variables is developed. It combines Geostatistics with Quantile Random Forests to give a new interpolation and stochastic simulation algorithm. This paper introduces the method and shows that it has consistency results that are similar in nature to those applying to geostatistical modelling and to Quantile Random Forests. The method allows for embedding of simpler interpolation techniques, such as Kriging, to further condition the model. The algorithm works by estimating a conditional distribution for the target variable at each target location. The family of such distributions is called the envelope of the target variable. From this, it is possible to obtain spatial estimates, quantiles and uncertainty. An algorithm to produce conditional simulations from the envelope is also developed. As they sample from the envelope, realizations are therefore locally influenced by relative changes of importance of secondary variables, trends and variability.