We study the causal bandit problem when the causal graph is unknown and develop an efficient algorithm for finding the parent node of the reward node using atomic interventions. We derive the exact equation for the expected number of interventions performed by the algorithm and show that under certain graphical conditions it could perform either logarithmically fast or, under more general assumptions, slower but still sublinearly in the number of variables. We formally show that our algorithm is optimal as it meets the universal lower bound we establish for any algorithm that performs atomic interventions. Finally, we extend our algorithm to the case when the reward node has multiple parents. Using this algorithm together with a standard algorithm from bandit literature leads to improved regret bounds.

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices \citep{lauritzen2019maximum,slawski2015estimation} and even one observation for diagonally dominant M-matrices \citep{truell2021maximum}. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted $\ell_1$-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.

Policy gradient algorithms have been widely applied to Markov decision processes and reinforcement learning problems in recent years. Regularization with various entropy functions is often used to encourage exploration and improve stability. This paper proposes an approximate Newton method for the policy gradient algorithm with entropy regularization. In the case of Shannon entropy, the resulting algorithm reproduces the natural policy gradient algorithm. For other entropy functions, this method results in brand-new policy gradient algorithms. We prove that all these algorithms enjoy Newton-type quadratic convergence and that the corresponding gradient flow converges globally to the optimal solution. We use synthetic and industrial-scale examples to demonstrate that the proposed approximate Newton method typically converges in single-digit iterations, often orders of magnitude faster than other state-of-the-art algorithms.

Recent works have applied the Proximal Policy Optimization (PPO) to the multi-agent cooperative tasks, such as Independent PPO (IPPO); and vanilla Multi-agent PPO (MAPPO) which has a centralized value function. However, previous literature shows that MAPPO may not perform as well as Independent PPO (IPPO) and the Fine-tuned QMIX on Starcraft Multi-Agent Challenge (SMAC). MAPPO-Feature-Pruned (MAPPO-FP) improves the performance of MAPPO by the carefully designed agent-specific features, which may be not friendly to algorithmic utility. By contrast, we find that MAPPO may face the problem of \textit{The Policies Overfitting in Multi-agent Cooperation(POMAC)}, as they learn policies by the sampled advantage values. Then POMAC may lead to updating the multi-agent policies in a suboptimal direction and prevent the agents from exploring better trajectories. In this paper, to mitigate the multi-agent policies overfitting, we propose a novel policy regularization method, which disturbs the advantage values via random Gaussian noise. The experimental results show that our method outperforms the Fine-tuned QMIX, MAPPO-FP, and achieves SOTA on SMAC without agent-specific features. We open-source the code at \url{https://github.com/hijkzzz/noisy-mappo}.

Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules and constitute an important step towards the modeling of distributions of many interacting molecules.

Robotic Information Gathering (RIG) is a foundational research topic that answers how a robot (team) collects informative data to efficiently build an accurate model of an unknown target function under robot embodiment constraints. RIG has many applications, including but not limited to autonomous exploration and mapping, 3D reconstruction or inspection, search and rescue, and environmental monitoring. A RIG system relies on a probabilistic model's prediction uncertainty to identify critical areas for informative data collection. Gaussian Processes (GPs) with stationary kernels have been widely adopted for spatial modeling. However, real-world spatial data is typically non-stationary -- different locations do not have the same degree of variability. As a result, the prediction uncertainty does not accurately reveal prediction error, limiting the success of RIG algorithms. We propose a family of non-stationary kernels named Attentive Kernel (AK), which is simple, robust, and can extend any existing kernel to a non-stationary one. We evaluate the new kernel in elevation mapping tasks, where AK provides better accuracy and uncertainty quantification over the commonly used stationary kernels and the leading non-stationary kernels. The improved uncertainty quantification guides the downstream informative planner to collect more valuable data around the high-error area, further increasing prediction accuracy. A field experiment demonstrates that the proposed method can guide an Autonomous Surface Vehicle (ASV) to prioritize data collection in locations with significant spatial variations, enabling the model to characterize salient environmental features.

The solution of probabilistic inverse problems for which the corresponding forward problem is constrained by physical principles is challenging. This is especially true if the dimension of the inferred vector is large and the prior information about it is in the form of a collection of samples. In this work, a novel deep learning based approach is developed and applied to solving these types of problems. The approach utilizes samples of the inferred vector drawn from the prior distribution and a physics-based forward model to generate training data for a conditional Wasserstein generative adversarial network (cWGAN). The cWGAN learns the probability distribution for the inferred vector conditioned on the measurement and produces samples from this distribution. The cWGAN developed in this work differs from earlier versions in that its critic is required to be 1-Lipschitz with respect to both the inferred and the measurement vectors and not just the former. This leads to a loss term with the full (and not partial) gradient penalty. It is shown that this rather simple change leads to a stronger notion of convergence for the conditional density learned by the cWGAN and a more robust and accurate sampling strategy. Through numerical examples it is shown that this change also translates to better accuracy when solving inverse problems. The numerical examples considered include illustrative problems where the true distribution and/or statistics are known, and a more complex inverse problem motivated by applications in biomechanics.

Bayesian optimisation is a powerful method for optimising black-box functions, popular in settings where the true function is expensive to evaluate and no gradient information is available. Bayesian optimisation can improve responses to many optimisation problems within climate change for which simulator models are unavailable or expensive to sample from. While there have been several feasibility demonstrations of Bayesian optimisation in climate-related applications, there has been no unifying review of applications and benchmarks. We provide such a review here, to encourage the use of Bayesian optimisation in important and well-suited application domains. We identify four main application domains: material discovery, wind farm layout, optimal renewable control and environmental monitoring. For each domain we identify a public benchmark or data set that is easy to use and evaluate systems against, while being representative of real-world problems. Due to the lack of a suitable benchmark for environmental monitoring, we propose LAQN-BO, based on air pollution data. Our contributions are: a) identifying a representative range of benchmarks, providing example code where necessary; b) introducing a new benchmark, LAQN-BO; and c) promoting a wider use of climate change applications among Bayesian optimisation practitioners.

The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to note that existing methods often struggle to identify the underlying equation accurately in the presence of noise. In this study, we present a new approach to discovering PDEs by combining variational Bayes and sparse linear regression. The problem of PDE discovery has been posed as a problem to learn relevant basis from a predefined dictionary of basis functions. To accelerate the overall process, a variational Bayes-based approach for discovering partial differential equations is proposed. To ensure sparsity, we employ a spike and slab prior. We illustrate the efficacy of our strategy in several examples, including Burgers, Korteweg-de Vries, Kuramoto Sivashinsky, wave equation, and heat equation (1D as well as 2D). Our method offers a promising avenue for discovering PDEs from data and has potential applications in fields such as physics, engineering, and biology.

Positron emission tomography (PET) is an important functional medical imaging technique often used in the evaluation of certain brain disorders, whose reconstruction problem is ill-posed. The vast majority of reconstruction methods in PET imaging, both iterative and deep learning, return a single estimate without quantifying the associated uncertainty. Due to ill-posedness and noise, a single solution can be misleading or inaccurate. Thus, providing a measure of uncertainty in PET image reconstruction can help medical practitioners in making critical decisions. This paper proposes a deep learning-based method for uncertainty quantification in PET image reconstruction via posterior sampling. The method is based on training a conditional generative adversarial network whose generator approximates sampling from the posterior in Bayesian inversion. The generator is conditioned on reconstruction from a low-dose PET scan obtained by a conventional reconstruction method and a high-quality magnetic resonance image and learned to estimate a corresponding standard-dose PET scan reconstruction. We show that the proposed model generates high-quality posterior samples and yields physically-meaningful uncertainty estimates.