Operator learning focuses on approximating mappings $\mathcal{G}^\dagger:\mathcal{U} \rightarrow\mathcal{V}$ between infinite-dimensional spaces of functions, such as $u: \Omega_u\rightarrow\mathbb{R}$ and $v: \Omega_v\rightarrow\mathbb{R}$. This makes it particularly suitable for solving parametric nonlinear partial differential equations (PDEs). While most machine learning methods for operator learning rely on variants of deep neural networks (NNs), recent studies have shown that Gaussian Processes (GPs) are also competitive while offering interpretability and theoretical guarantees. In this paper, we introduce a hybrid GP/NN-based framework for operator learning that leverages the strengths of both methods. Instead of approximating the function-valued operator $\mathcal{G}^\dagger$, we use a GP to approximate its associated real-valued bilinear form $\widetilde{\mathcal{G}}^\dagger: \mathcal{U}\times\mathcal{V}^*\rightarrow\mathbb{R}.$ This bilinear form is defined by $\widetilde{\mathcal{G}}^\dagger(u,\varphi) := [\varphi,\mathcal{G}^\dagger(u)],$ which allows us to recover the operator $\mathcal{G}^\dagger$ through $\mathcal{G}^\dagger(u)(y)=\widetilde{\mathcal{G}}^\dagger(u,\delta_y).$ The GP mean function can be zero or parameterized by a neural operator and for each setting we develop a robust training mechanism based on maximum likelihood estimation (MLE) that can optionally leverage the physics involved. Numerical benchmarks show that (1) it improves the performance of a base neural operator by using it as the mean function of a GP, and (2) it enables zero-shot data-driven models for accurate predictions without prior training. Our framework also handles multi-output operators where $\mathcal{G}^\dagger:\mathcal{U} \rightarrow\prod_{s=1}^S\mathcal{V}^s$, and benefits from computational speed-ups via product kernel structures and Kronecker product matrix representations.

Decision making under uncertainty is challenging since the data-generating process (DGP) is often unknown. Bayesian inference proceeds by estimating the DGP through posterior beliefs about the model's parameters. However, minimising the expected risk under these posterior beliefs can lead to sub-optimal decisions due to model uncertainty or limited, noisy observations. To address this, we introduce Distributionally Robust Optimisation with Bayesian Ambiguity Sets (DRO-BAS) which hedges against uncertainty in the model by optimising the worst-case risk over a posterior-informed ambiguity set. We show that our method admits a closed-form dual representation for many exponential family members and showcase its improved out-of-sample robustness against existing Bayesian DRO methodology in the Newsvendor problem.

We introduce a general iterative procedure called risk-based calibration (RC) designed to minimize the empirical risk under the 0-1 loss (empirical error) for probabilistic classifiers. These classifiers are based on modeling probability distributions, including those constructed from the joint distribution (generative) and those based on the class conditional distribution (conditional). RC can be particularized to any probabilistic classifier provided a specific learning algorithm that computes the classifier's parameters in closed form using data statistics. RC reinforces the statistics aligned with the true class while penalizing those associated with other classes, guided by the 0-1 loss. The proposed method has been empirically tested on 30 datasets using na\"ive Bayes, quadratic discriminant analysis, and logistic regression classifiers. RC improves the empirical error of the original closed-form learning algorithms and, more notably, consistently outperforms the gradient descent approach with the three classifiers.

This book uses the modern theory of artificial intelligence (AI) to understand human suffering or mental pain. Both humans and sophisticated AI agents process information about the world in order to achieve goals and obtain rewards, which is why AI can be used as a model of the human brain and mind. This book intends to make the theory accessible to a relatively general audience, requiring only some relevant scientific background. The book starts with the assumption that suffering is mainly caused by frustration. Frustration means the failure of an agent (whether AI or human) to achieve a goal or a reward it wanted or expected. Frustration is inevitable because of the overwhelming complexity of the world, limited computational resources, and scarcity of good data. In particular, such limitations imply that an agent acting in the real world must cope with uncontrollability, unpredictability, and uncertainty, which all lead to frustration. Fundamental in such modelling is the idea of learning, or adaptation to the environment. While AI uses machine learning, humans and animals adapt by a combination of evolutionary mechanisms and ordinary learning. Even frustration is fundamentally an error signal that the system uses for learning. This book explores various aspects and limitations of learning algorithms and their implications regarding suffering. At the end of the book, the computational theory is used to derive various interventions or training methods that will reduce suffering in humans. The amount of frustration is expressed by a simple equation which indicates how it can be reduced. The ensuing interventions are very similar to those proposed by Buddhist and Stoic philosophy, and include mindfulness meditation. Therefore, this book can be interpreted as an exposition of a computational theory justifying why such philosophies and meditation reduce human suffering.

The identification theory for causal effects in directed acyclic graphs (DAGs) with hidden variables is well-developed, but methods for estimating and inferring functionals beyond the g-formula remain limited. Previous studies have proposed semiparametric estimators for identifiable functionals in a broad class of DAGs with hidden variables. While demonstrating double robustness in some models, existing estimators face challenges, particularly with density estimation and numerical integration for continuous variables, and their estimates may fall outside the parameter space of the target estimand. Their asymptotic properties are also underexplored, especially when using flexible statistical and machine learning models for nuisance estimation. This study addresses these challenges by introducing novel one-step corrected plug-in and targeted minimum loss-based estimators of causal effects for a class of DAGs that extend classical back-door and front-door criteria (known as the treatment primal fixability criterion in prior literature). These estimators leverage machine learning to minimize modeling assumptions while ensuring key statistical properties such as asymptotic linearity, double robustness, efficiency, and staying within the bounds of the target parameter space. We establish conditions for nuisance functional estimates in terms of L2(P)-norms to achieve root-n consistent causal effect estimates. To facilitate practical application, we have developed the flexCausal package in R.

Maximum Likelihood Estimation of continuous variable models can be very challenging in high dimensions, due to potentially complex probability distributions. The existence of multiple interdependencies among variables can make it very difficult to establish convergence guarantees. This leads to a wide use of brute-force methods, such as grid searching and Monte-Carlo sampling and, when applicable, complex and problem-specific algorithms. In this paper, we consider inference problems where the variables are related by multiaffine expressions. We propose a novel Alternating and Iteratively-Reweighted Least Squares (AIRLS) algorithm, and prove its convergence for problems with Generalized Normal Distributions. We also provide an efficient method to compute the variance of the estimates obtained using AIRLS. Finally, we show how the method can be applied to graphical statistical models. We perform numerical experiments on several inference problems, showing significantly better performance than state-of-the-art approaches in terms of scalability, robustness to noise, and convergence speed due to an empirically observed super-linear convergence rate.

The premise of semi-supervised learning (SSL) is that combining labeled and unlabeled data yields significantly more accurate models. Despite empirical successes, the theoretical understanding of SSL is still far from complete. In this work, we study SSL for high dimensional sparse Gaussian classification. To construct an accurate classifier a key task is feature selection, detecting the few variables that separate the two classes. % For this SSL setting, we analyze information theoretic lower bounds for accurate feature selection as well as computational lower bounds, assuming the low-degree likelihood hardness conjecture. % Our key contribution is the identification of a regime in the problem parameters (dimension, sparsity, number of labeled and unlabeled samples) where SSL is guaranteed to be advantageous for classification. Specifically, there is a regime where it is possible to construct in polynomial time an accurate SSL classifier. However, % any computationally efficient supervised or unsupervised learning schemes, that separately use only the labeled or unlabeled data would fail. Our work highlights the provable benefits of combining labeled and unlabeled data for {classification and} feature selection in high dimensions. We present simulations that complement our theoretical analysis.

This book introduces the mathematical foundations and techniques that lead to the development and analysis of many of the algorithms that are used in machine learning. It starts with an introductory chapter that describes notation used throughout the book and serve at a reminder of basic concepts in calculus, linear algebra and probability and also introduces some measure theoretic terminology, which can be used as a reading guide for the sections that use these tools. The introductory chapters also provide background material on matrix analysis and optimization. The latter chapter provides theoretical support to many algorithms that are used in the book, including stochastic gradient descent, proximal methods, etc. After discussing basic concepts for statistical prediction, the book includes an introduction to reproducing kernel theory and Hilbert space techniques, which are used in many places, before addressing the description of various algorithms for supervised statistical learning, including linear methods, support vector machines, decision trees, boosting, or neural networks. The subject then switches to generative methods, starting with a chapter that presents sampling methods and an introduction to the theory of Markov chains. The following chapter describe the theory of graphical models, an introduction to variational methods for models with latent variables, and to deep-learning based generative models. The next chapters focus on unsupervised learning methods, for clustering, factor analysis and manifold learning. The final chapter of the book is theory-oriented and discusses concentration inequalities and generalization bounds.

Multi-output Gaussian process (MGP) is commonly used as a transfer learning method to leverage information among multiple outputs. A key advantage of MGP is providing uncertainty quantification for prediction, which is highly important for subsequent decision-making tasks. However, traditional MGP may not be sufficiently flexible to handle multivariate data with dynamic characteristics, particularly when dealing with complex temporal correlations. Additionally, since some outputs may lack correlation, transferring information among them may lead to negative transfer. To address these issues, this study proposes a non-stationary MGP model that can capture both the dynamic and sparse correlation among outputs. Specifically, the covariance functions of MGP are constructed using convolutions of time-varying kernel functions. Then a dynamic spike-and-slab prior is placed on correlation parameters to automatically decide which sources are informative to the target output in the training process. An expectation-maximization (EM) algorithm is proposed for efficient model fitting. Both numerical studies and a real case demonstrate its efficacy in capturing dynamic and sparse correlation structure and mitigating negative transfer for high-dimensional time-series data. Finally, a mountain-car reinforcement learning case highlights its potential application in decision making problems.

Bayesian observer and actor models have provided normative explanations for many behavioral phenomena in perception, sensorimotor control, and other areas of cognitive science and neuroscience. They attribute behavioral variability and biases to different interpretable entities such as perceptual and motor uncertainty, prior beliefs, and behavioral costs. However, when extending these models to more complex tasks with continuous actions, solving the Bayesian decision-making problem is often analytically intractable. Moreover, inverting such models to perform inference over their parameters given behavioral data is computationally even more difficult. Therefore, researchers typically constrain their models to easily tractable components, such as Gaussian distributions or quadratic cost functions, or resort to numerical methods. To overcome these limitations, we amortize the Bayesian actor using a neural network trained on a wide range of different parameter settings in an unsupervised fashion. Using the pre-trained neural network enables performing gradient-based Bayesian inference of the Bayesian actor model's parameters. We show on synthetic data that the inferred posterior distributions are in close alignment with those obtained using analytical solutions where they exist. Where no analytical solution is available, we recover posterior distributions close to the ground truth. We then show that identifiability problems between priors and costs can arise in more complex cost functions. Finally, we apply our method to empirical data and show that it explains systematic individual differences of behavioral patterns.