Most scientific challenges can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. While most Computational Science and Engineering and Scientific Machine Learning challenges can be framed as Type 1 and Type 2 problems, many scientific problems can only be categorized as Type 3. Despite their prevalence, these Type 3 challenges have been largely overlooked due to their inherent complexity. Although Gaussian Process (GP) methods are sometimes perceived as well-founded but old technology limited to Type 1 curve fitting, their scope has recently been expanded to Type 2 problems. In this paper, we introduce an interpretable GP framework for Type 3 problems, targeting the data-driven discovery and completion of computational hypergraphs. Our approach is based on a kernel generalization of Row Echelon Form reduction from linear systems to nonlinear ones and variance-based analysis. Here, variables are linked via GPs and those contributing to the highest data variance unveil the hypergraph's structure. We illustrate the scope and efficiency of the proposed approach with applications to (algebraic) equation discovery, network discovery (gene pathways, chemical, and mechanical) and raw data analysis.

A Bayesian approach to machine learning is attractive when we need to quantify uncertainty, deal with missing observations, when samples are scarce, or when the data is sparse. All of these commonly apply when analysing healthcare data. To address these analytical requirements, we propose a deep generative model for multinomial count data where both the weights and hidden units of the network are Dirichlet distributed. A Gibbs sampling procedure is formulated that takes advantage of a series of augmentation relations, analogous to the Zhou-Cong-Chen model. We apply the model on small handwritten digits, and a large experimental dataset of DNA mutations in cancer, and we show how the model is able to extract biologically meaningful meta-signatures in a fully data-driven way.

Nonparametric estimation of nonlocal interaction kernels is crucial in various applications involving interacting particle systems. The inference challenge, situated at the nexus of statistical learning and inverse problems, comes from the nonlocal dependency. A central question is whether the optimal minimax rate of convergence for this problem aligns with the rate of $M^{-\frac{2\beta}{2\beta+1}}$ in classical nonparametric regression, where $M$ is the sample size and $\beta$ represents the smoothness exponent of the radial kernel. Our study confirms this alignment for systems with a finite number of particles. We introduce a tamed least squares estimator (tLSE) that attains the optimal convergence rate for a broad class of exchangeable distributions. The tLSE bridges the smallest eigenvalue of random matrices and Sobolev embedding. This estimator relies on nonasymptotic estimates for the left tail probability of the smallest eigenvalue of the normal matrix. The lower minimax rate is derived using the Fano-Tsybakov hypothesis testing method. Our findings reveal that provided the inverse problem in the large sample limit satisfies a coercivity condition, the left tail probability does not alter the bias-variance tradeoff, and the optimal minimax rate remains intact. Our tLSE method offers a straightforward approach for establishing the optimal minimax rate for models with either local or nonlocal dependency.

Simulation-Based Inference (SBI) is a common name for an emerging family of approaches that infer the model parameters when the likelihood is intractable. Existing SBI methods either approximate the likelihood, such as Approximate Bayesian Computation (ABC), or directly model the posterior, such as Sequential Neural Posterior Estimation (SNPE). While ABC is efficient on low-dimensional problems, on higher-dimensional tasks, it is generally outperformed by SNPE which leverages function approximation. In this paper, we propose Pseudo-Likelihood Inference (PLI), a new method that brings neural approximation into ABC, making it competitive on challenging Bayesian system identification tasks. By utilizing integral probability metrics, we introduce a smooth likelihood kernel with an adaptive bandwidth that is updated based on information-theoretic trust regions. Thanks to this formulation, our method (i) allows for optimizing neural posteriors via gradient descent, (ii) does not rely on summary statistics, and (iii) enables multiple observations as input. In comparison to SNPE, it leads to improved performance when more data is available. The effectiveness of PLI is evaluated on four classical SBI benchmark tasks and on a highly dynamic physical system, showing particular advantages on stochastic simulations and multi-modal posterior landscapes.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as \emph{properness}, which asserts that Bayes' rule is optimal. Recent works have sought to \emph{learn losses} and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps $\mathbb{R}$ to $[0,1]$ to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between $\mathbb{R}^{C-1}$ and the projected probability simplex $\tilde{\Delta}^{C-1}$ by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns \emph{proper canonical losses} and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a $t$-test at 99% significance on all datasets with greater than 10 classes.

Changes in the data distribution at test time can have deleterious effects on the performance of predictive models $p(y|x)$. We consider situations where there are additional meta-data labels (such as group labels), denoted by $z$, that can account for such changes in the distribution. In particular, we assume that the prior distribution $p(y, z)$, which models the dependence between the class label $y$ and the "nuisance" factors $z$, may change across domains, either due to a change in the correlation between these terms, or a change in one of their marginals. However, we assume that the generative model for features $p(x|y,z)$ is invariant across domains. We note that this corresponds to an expanded version of the widely used "label shift" assumption, where the labels now also include the nuisance factors $z$. Based on this observation, we propose a test-time label shift correction that adapts to changes in the joint distribution $p(y, z)$ using EM applied to unlabeled samples from the target domain distribution, $p_t(x)$. Importantly, we are able to avoid fitting a generative model $p(x|y, z)$, and merely need to reweight the outputs of a discriminative model $p_s(y, z|x)$ trained on the source distribution. We evaluate our method, which we call "Test-Time Label-Shift Adaptation" (TTLSA), on several standard image and text datasets, as well as the CheXpert chest X-ray dataset, and show that it improves performance over methods that target invariance to changes in the distribution, as well as baseline empirical risk minimization methods. Code for reproducing experiments is available at https://github.com/nalzok/test-time-label-shift .

Car-following (CF) algorithms are crucial components of traffic simulations and have been integrated into many production vehicles equipped with Advanced Driving Assistance Systems (ADAS). Insights from the model of car-following behavior help us understand the causes of various macro phenomena that arise from interactions between pairs of vehicles. Car-following models encompass multiple disciplines, including traffic engineering, physics, dynamic system control, cognitive science, machine learning, and reinforcement learning. This paper presents an extensive survey that highlights the differences, complementarities, and overlaps among microscopic traffic flow and control models based on their underlying principles and design logic. It reviews representative algorithms, ranging from theory-based kinematic models, Psycho-Physical Models, and Adaptive cruise control models to data-driven algorithms like Reinforcement Learning and Imitation Learning (IL). The manuscript discusses the strengths and limitations of these models and explores their applications in different contexts. This review synthesizes existing researches across different domains to fill knowledge gaps and offer guidance for future research by identifying the latest trends in car following models and their applications.

Virtually all machine learning tasks are characterized using some form of loss function, and "good performance" is typically stated in terms of a sufficiently small average loss, taken over the random draw of test data. While optimizing for performance on average is intuitive, convenient to analyze in theory, and easy to implement in practice, such a choice brings about trade-offs. In this work, we survey and introduce a wide variety of non-traditional criteria used to design and evaluate machine learning algorithms, place the classical paradigm within the proper historical context, and propose a view of learning problems which emphasizes the question of "what makes for a desirable loss distribution?" in place of tacit use of the expected loss.

It is common to model a deterministic response function, such as the output of a computer experiment, as a Gaussian process with a Mat\'ern covariance kernel. The smoothness parameter of a Mat\'ern kernel determines many important properties of the model in the large data limit, including the rate of convergence of the conditional mean to the response function. We prove that the maximum likelihood estimate of the smoothness parameter cannot asymptotically undersmooth the truth when the data are obtained on a fixed bounded subset of $\mathbb{R}^d$. That is, if the data-generating response function has Sobolev smoothness $\nu_0 > d/2$, then the smoothness parameter estimate cannot be asymptotically less than $\nu_0$. The lower bound is sharp. Additionally, we show that maximum likelihood estimation recovers the true smoothness for a class of compactly supported self-similar functions. For cross-validation we prove an asymptotic lower bound $\nu_0 - d/2$, which however is unlikely to be sharp. The results are based on approximation theory in Sobolev spaces and some general theorems that restrict the set of values that the parameter estimators can take.

A key challenge of modern machine learning systems is to achieve Out-of-Distribution (OOD) generalization -- generalizing to target data whose distribution differs from that of source data. Despite its significant importance, the fundamental question of ``what are the most effective algorithms for OOD generalization'' remains open even under the standard setting of covariate shift. This paper addresses this fundamental question by proving that, surprisingly, classical Maximum Likelihood Estimation (MLE) purely using source data (without any modification) achieves the minimax optimality for covariate shift under the well-specified setting. That is, no algorithm performs better than MLE in this setting (up to a constant factor), justifying MLE is all you need. Our result holds for a very rich class of parametric models, and does not require any boundedness condition on the density ratio. We illustrate the wide applicability of our framework by instantiating it to three concrete examples -- linear regression, logistic regression, and phase retrieval. This paper further complement the study by proving that, under the misspecified setting, MLE is no longer the optimal choice, whereas Maximum Weighted Likelihood Estimator (MWLE) emerges as minimax optimal in certain scenarios.