Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving {high-dimensional} ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems -- most importantly, the solution of discretised {partial} differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions.

Summarizing, combining and comparing probability distributions defined on a metric are fundamental tasks in machine learning, statistics and computer science, including multiple sensors, Bayesian inference, among others. For instance, in Bayesian inference one runs posterior sampling algorithm in parallel on different machines using small subsets of the massive data, and then aggregates subset posterior distributions via their barycenter as an approximation to the true posterior for the full data [1, 2]. Besides Bayesian inference, the average or barycenter of a collection of distributions has been successfully applied in various machine learning applications, say image processing [3] and clustering [4, 5]. The theory of optimal transport (OT) [6-9] provides a powerful framework to carry out such comparisons. OT equips the space of distributions with a distance metric known as the Wasserstein distance, which has gained substantial popularity in different fields, leading in particular to the natural consideration of barycenters. The barycenter of multiple given probability distributions under Wasserstein distance is defined as a distribution minimizing the sum of Wasserstein distances to all distributions. Due to the geometric properties of Wasserstein distance, the Wasserstein barycenter can better capture the underlying geometric structure than the barycenter with respect to other popular distances, e.g., Euclidean distance, see Figure 1. As a result, Wasserstein barycenters have a broad range of applications in text mixing [3], imaging [2, 10, 11], and model ensemble [12].

The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is not a random variable; as such, model selection to avoid overfitting becomes problematic. A method to solve this problem is to assume a prior distribution on the full rooted trees. This enables overfitting to be avoided based on the Bayes decision theory. For example, by assigning a low prior probability to a complex model, the maximum a posteriori estimator prevents overfitting. Furthermore, overfitting can be avoided by averaging all the models weighted by their posteriors. In this paper, we propose a probability distribution on a set of full rooted trees. Its parametric representation is suitable for calculating the properties of our distribution using recursive functions, such as the mode, expectation, and posterior distribution. Although such distributions have been proposed in previous studies, they are only applicable to specific applications. Therefore, we extract their mathematically essential components and derive new generalized methods to calculate the expectation, posterior distribution, etc.

Out-of-distribution (OOD) detection is critical to ensuring the reliability and safety of machine learning systems. For instance, in autonomous driving, we would like the driving system to issue an alert and hand over the control to humans when it detects unusual scenes or objects that it has never seen before and cannot make a safe decision. This problem first emerged in 2017 and since then has received increasing attention from the research community, leading to a plethora of methods developed, ranging from classification-based to density-based to distance-based ones. Meanwhile, several other problems are closely related to OOD detection in terms of motivation and methodology. These include anomaly detection (AD), novelty detection (ND), open set recognition (OSR), and outlier detection (OD). Despite having different definitions and problem settings, these problems often confuse readers and practitioners, and as a result, some existing studies misuse terms. In this survey, we first present a generic framework called generalized OOD detection, which encompasses the five aforementioned problems, i.e., AD, ND, OSR, OOD detection, and OD. Under our framework, these five problems can be seen as special cases or sub-tasks, and are easier to distinguish. Then, we conduct a thorough review of each of the five areas by summarizing their recent technical developments. We conclude this survey with open challenges and potential research directions.

We study adaptive sensing of Cox point processes, a widely used model from spatial statistics. We introduce three tasks: maximization of captured events, search for the maximum of the intensity function and learning level sets of the intensity function. We model the intensity function as a sample from a truncated Gaussian process, represented in a specially constructed positive basis. In this basis, the positivity constraint on the intensity function has a simple form. We show how an minimal description positive basis can be adapted to the covariance kernel, non-stationarity and make connections to common positive bases from prior works. Our adaptive sensing algorithms use Langevin dynamics and are based on posterior sampling (\textsc{Cox-Thompson}) and top-two posterior sampling (\textsc{Top2}) principles. With latter, the difference between samples serves as a surrogate to the uncertainty. We demonstrate the approach using examples from environmental monitoring and crime rate modeling, and compare it to the classical Bayesian experimental design approach.

Computational level explanations based on optimal feedback control with signal-dependent noise have been able to account for a vast array of phenomena in human sensorimotor behavior. However, commonly a cost function needs to be assumed for a task and the optimality of human behavior is evaluated by comparing observed and predicted trajectories. Here, we introduce inverse optimal control with signal-dependent noise, which allows inferring the cost function from observed behavior. To do so, we formalize the problem as a partially observable Markov decision process and distinguish between the agent's and the experimenter's inference problems. Specifically, we derive a probabilistic formulation of the evolution of states and belief states and an approximation to the propagation equation in the linear-quadratic Gaussian problem with signal-dependent noise. We extend the model to the case of partial observability of state variables from the point of view of the experimenter. We show the feasibility of the approach through validation on synthetic data and application to experimental data. Our approach enables recovering the costs and benefits implicit in human sequential sensorimotor behavior, thereby reconciling normative and descriptive approaches in a computational framework.

Recent advances in the field of meta-learning have tackled domains consisting of large numbers of small ("few-shot") supervised learning tasks. Meta-learning algorithms must be able to rapidly adapt to any individual few-shot task, fitting to a small support set within a task and using it to predict the labels of the task's query set. This problem setting can be extended to the Bayesian context, wherein rather than predicting a single label for each query data point, a model predicts a distribution of labels capturing its uncertainty. Successful methods in this domain include Bayesian ensembling of MAML-based models, Bayesian neural networks, and Gaussian processes with learned deep kernel and mean functions. While Gaussian processes have a robust Bayesian interpretation in the meta-learning context, they do not naturally model non-Gaussian predictive posteriors for expressing uncertainty. In this paper, we design a theoretically principled method, VMGP, extending Gaussian-process-based meta-learning to allow for high-quality, arbitrary non-Gaussian uncertainty predictions. On benchmark environments with complex non-smooth or discontinuous structure, we find our VMGP method performs significantly better than existing Bayesian meta-learning baselines.

The recent phenomenal success of language models has reinvigorated machine learning research, and large sequence models such as transformers are being applied to a variety of domains. One important problem class that has remained relatively elusive however is purposeful adaptive behavior. Currently there is a common perception that sequence models "lack the understanding of the cause and effect of their actions" leading them to draw incorrect inferences due to auto-suggestive delusions. In this report we explain where this mismatch originates, and show that it can be resolved by treating actions as causal interventions. Finally, we show that in supervised learning, one can teach a system to condition or intervene on data by training with factual and counterfactual error signals respectively.

A wide variety of methods have been developed to enable lifelong learning in conventional deep neural networks. However, to succeed, these methods require a `batch' of samples to be available and visited multiple times during training. While this works well in a static setting, these methods continue to suffer in a more realistic situation where data arrives in \emph{online streaming manner}. We empirically demonstrate that the performance of current approaches degrades if the input is obtained as a stream of data with the following restrictions: $(i)$ each instance comes one at a time and can be seen only once, and $(ii)$ the input data violates the i.i.d assumption, i.e., there can be a class-based correlation. We propose a novel approach (CIOSL) for the class-incremental learning in an \emph{online streaming setting} to address these challenges. The proposed approach leverages implicit and explicit dual weight regularization and experience replay. The implicit regularization is leveraged via the knowledge distillation, while the explicit regularization incorporates a novel approach for parameter regularization by learning the joint distribution of the buffer replay and the current sample. Also, we propose an efficient online memory replay and replacement buffer strategy that significantly boosts the model's performance. Extensive experiments and ablation on challenging datasets show the efficacy of the proposed method.

The last decade has seen the rise of Adversarial Machine Learning (AML). This discipline studies how to manipulate data to fool inference engines, and how to protect those systems against such manipulation attacks. Extensive work on attacks against regression and classification systems is available, while little attention has been paid to attacks against time series forecasting systems. In this paper, we propose a decision analysis based attacking strategy that could be utilized against Bayesian forecasting dynamic models.