This paper studies the learning-to-control problem under process and sensing uncertainties for dynamical systems. In our previous work, we developed a data-based generalization of the iterative linear quadratic regulator (iLQR) to design closed-loop feedback control for high-dimensional dynamical systems with partial state observation. This method required perfect simulation rollouts which are not realistic in real applications. In this work, we briefly introduce this method and explore its efficacy under process and sensing uncertainties. We prove that in the fully observed case where the system dynamics are corrupted with noise but the measurements are perfect, it still converges to the global minimum. However, in the partially observed case where both process and measurement noise exist in the system, this method converges to a biased "optimum". Thus multiple rollouts need to be averaged to retrieve the true optimum. The analysis is verified in two nonlinear robotic examples simulated in the above cases.

One of the goals of causal inference is to generalize from past experiments and observational data to novel conditions. While it is in principle possible to eventually learn a mapping from a novel experimental condition to an outcome of interest, provided a sufficient variety of experiments is available in the training data, coping with a large combinatorial space of possible interventions is hard. Under a typical sparse experimental design, this mapping is ill-posed without relying on heavy regularization or prior distributions. Such assumptions may or may not be reliable, and can be hard to defend or test. In this paper, we take a close look at how to warrant a leap from past experiments to novel conditions based on minimal assumptions about the factorization of the distribution of the manipulated system, communicated in the well-understood language of factor graph models. A postulated $\textit{interventional factor model}$ (IFM) may not always be informative, but it conveniently abstracts away a need for explicitly modeling unmeasured confounding and feedback mechanisms, leading to directly testable claims. Given an IFM and datasets from a collection of experimental regimes, we derive conditions for identifiability of the expected outcomes of new regimes never observed in these training data. We implement our framework using several efficient algorithms, and apply them on a range of semi-synthetic experiments.

Real-time density estimation is ubiquitous in many applications, including computer vision and signal processing. Kernel density estimation is arguably one of the most commonly used density estimation techniques, and the use of "sliding window" mechanism adapts kernel density estimators to dynamic processes. In this paper, we derive the asymptotic mean integrated squared error (AMISE) upper bound for the "sliding window" kernel density estimator. This upper bound provides a principled guide to devise a novel estimator, which we name the temporal adaptive kernel density estimator (TAKDE). Compared to heuristic approaches for "sliding window" kernel density estimator, TAKDE is theoretically optimal in terms of the worst-case AMISE. We provide numerical experiments using synthetic and real-world datasets, showing that TAKDE outperforms other state-of-the-art dynamic density estimators (including those outside of kernel family). In particular, TAKDE achieves a superior test log-likelihood with a smaller runtime.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there were finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, many real-world data are not naturally posed in the setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions, and then convert the GPs into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is further developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications.

The dominant approach in recent literature on causal discovery from interventional data is optimization of a model score. Although the scoring is straightforward in the sense that the optimization over DAGs refers to fully specified joint models for all observational and interventional data, the optimization landscape is very high-dimensional, making score-based algorithms infeasible for graphs with hundreds/thousands of nodes that are common in biological applications. This makes causal discovery difficult, in addition to many other challenges that remain such as departing from restrictive genericity assumptions on the underlying distributions and developing methodology for high-dimensional settings. In this article, to address these challenges, we depart from a score-based strategy and leverage special properties of polytrees to obtain a highly scalable "local" approach that learns from low-dimensional marginals. Our methods yield fast and consistent algorithms to learn linear Gaussian polytrees from interventional data.

In this paper, we present new high-probability PAC-Bayes bounds for different types of losses. Firstly, for losses with a bounded range, we recover a strengthened version of Catoni's bound that holds uniformly for all parameter values. This leads to new fast rate and mixed rate bounds that are interpretable and tighter than previous bounds in the literature. In particular, the fast rate bound is equivalent to the Seeger--Langford bound. Secondly, for losses with more general tail behaviors, we introduce two new parameter-free bounds: a PAC-Bayes Chernoff analogue when the loss' cumulative generating function is bounded, and a bound when the loss' second moment is bounded. These two bounds are obtained using a new technique based on a discretization of the space of possible events for the "in probability" parameter optimization problem. This technique is both simpler and more general than previous approaches optimizing over a grid on the parameters' space. Finally, we extend all previous results to anytime-valid bounds using a simple technique applicable to any existing bound.

As a classical generative modeling approach, energy-based models have the natural advantage of flexibility in the form of the energy function. Recently, energy-based models have achieved great success in modeling high-dimensional data in computer vision and natural language processing. In line with these advancements, we build a multi-purpose energy-based probabilistic model for High Energy Physics events at the Large Hadron Collider. This framework builds on a powerful generative model and describes higher-order inter-particle interactions. It suits different encoding architectures and builds on implicit generation. As for applicational aspects, it can serve as a powerful parameterized event generator for physics simulation, a generic anomalous signal detector free from spurious correlations, and an augmented event classifier for particle identification.

Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners.

Using a fully Bayesian approach, Gaussian Process regression is extended to include marginalisation over the kernel choice and kernel hyperparameters. In addition, Bayesian model comparison via the evidence enables direct kernel comparison. The calculation of the joint posterior was implemented with a transdimensional sampler which simultaneously samples over the discrete kernel choice and their hyperparameters by embedding these in a higher-dimensional space, from which samples are taken using nested sampling. This method was explored on synthetic data from exoplanet transit light curve simulations. The true kernel was recovered in the low noise region while no kernel was preferred for larger noise. Furthermore, inference of the physical exoplanet hyperparameters was conducted. In the high noise region, either the bias in the posteriors was removed, the posteriors were broadened or the accuracy of the inference was increased. In addition, the uncertainty in mean function predictive distribution increased due to the uncertainty in the kernel choice. Subsequently, the method was extended to marginalisation over mean functions and noise models and applied to the inference of the present-day Hubble parameter, $H_0$, from real measurements of the Hubble parameter as a function of redshift, derived from the cosmologically model-independent cosmic chronometer and {\Lambda}CDM-dependent baryon acoustic oscillation observations. The inferred $H_0$ values from the cosmic chronometers, baryon acoustic oscillations and combined datasets are $H_0$ = 66$\pm$6 km/s/Mpc, $H_0$ = 67$\pm$10 km/s/Mpc and $H_0$ = 69$\pm$6 km/s/Mpc, respectively. The kernel posterior of the cosmic chronometers dataset prefers a non-stationary linear kernel. Finally, the datasets are shown to be not in tension with ln(R)=12.17$\pm$0.02.

Certifiable, adaptive uncertainty estimates for unknown quantities are an essential ingredient of sequential decision-making algorithms. Standard approaches rely on problem-dependent concentration results and are limited to a specific combination of parameterization, noise family, and estimator. In this paper, we revisit the likelihood-based inference principle and propose to use likelihood ratios to construct any-time valid confidence sequences without requiring specialized treatment in each application scenario. Our method is especially suitable for problems with well-specified likelihoods, and the resulting sets always maintain the prescribed coverage in a model-agnostic manner. The size of the sets depends on a choice of estimator sequence in the likelihood ratio. We discuss how to provably choose the best sequence of estimators and shed light on connections to online convex optimization with algorithms such as Follow-the-Regularized-Leader. To counteract the initially large bias of the estimators, we propose a reweighting scheme that also opens up deployment in non-parametric settings such as RKHS function classes. We provide a non-asymptotic analysis of the likelihood ratio confidence sets size for generalized linear models, using insights from convex duality and online learning.