We consider a seller offering a large network of $N$ products over a time horizon of $T$ periods. The seller does not know the parameters of the products' linear demand model, and can dynamically adjust product prices to learn the demand model based on sales observations. The seller aims to minimize its pseudo-regret, i.e., the expected revenue loss relative to a clairvoyant who knows the underlying demand model. We consider a sparse set of demand relationships between products to characterize various connectivity properties of the product network. In particular, we study three different sparsity frameworks: (1) $L_0$ sparsity, which constrains the number of connections in the network, and (2) off-diagonal sparsity, which constrains the magnitude of cross-product price sensitivities, and (3) a new notion of spectral sparsity, which constrains the asymptotic decay of a similarity metric on network nodes. We propose a dynamic pricing-and-learning policy that combines the optimism-in-the-face-of-uncertainty and PAC-Bayesian approaches, and show that this policy achieves asymptotically optimal performance in terms of $N$ and $T$. We also show that in the case of spectral and off-diagonal sparsity, the seller can have a pseudo-regret linear in $N$, even when the network is dense.

This paper addresses and solves some challenges in the adoption of machine learning in insurance with the democratization of model deployment. The first challenge is reducing the labelling effort (hence focusing on the data quality) with the help of active learning, a feedback loop between the model inference and an oracle: as in insurance the unlabeled data is usually abundant, active learning can become a significant asset in reducing the labelling cost. For that purpose, this paper sketches out various classical active learning methodologies before studying their empirical impact on both synthetic and real datasets. Another key challenge in insurance is the fairness issue in model inferences. We will introduce and integrate a post-processing fairness for multi-class tasks in this active learning framework to solve these two issues. Finally numerical experiments on unfair datasets highlight that the proposed setup presents a good compromise between model precision and fairness.

We formally map the problem of sampling from an unknown distribution with density $p_X$ in $\mathbb{R}^d$ to the problem of learning and sampling $p_\mathbf{Y}$ in $\mathbb{R}^{Md}$ obtained by convolving $p_X$ with a fixed factorial kernel: $p_\mathbf{Y}$ is referred to as M-density and the factorial kernel as multimeasurement noise model (MNM). The M-density is smoother than $p_X$, easier to learn and sample from, yet for large $M$ the two problems are mathematically equivalent since $X$ can be estimated exactly given $\mathbf{Y}=\mathbf{y}$ using the Bayes estimator $\widehat{x}(\mathbf{y})=\mathbb{E}[X\vert\mathbf{Y}=\mathbf{y}]$. To formulate the problem, we derive $\widehat{x}(\mathbf{y})$ for Poisson and Gaussian MNMs expressed in closed form in terms of unnormalized $p_\mathbf{Y}$. This leads to a simple least-squares objective for learning parametric energy and score functions. We present various parametrization schemes of interest, including one in which studying Gaussian M-densities directly leads to multidenoising autoencoders--this is the first theoretical connection made between denoising autoencoders and empirical Bayes in the literature. Samples from $p_X$ are obtained by walk-jump sampling (Saremi & Hyvarinen, 2019) via underdamped Langevin MCMC (walk) to sample from $p_\mathbf{Y}$ and the multimeasurement Bayes estimation of $X$ (jump). We study permutation invariant Gaussian M-densities on MNIST, CIFAR-10, and FFHQ-256 datasets, and demonstrate the effectiveness of this framework for realizing fast-mixing stable Markov chains in high dimensions.

Optimal transport (OT) formalizes the problem of finding an optimal coupling between probability measures given a cost matrix. The inverse problem of inferring the cost given a coupling is Inverse Optimal Transport (IOT). IOT is less well understood than OT. We formalize and systematically analyze the properties of IOT using tools from the study of entropy-regularized OT. Theoretical contributions include characterization of the manifold of cross-ratio equivalent costs, the implications of model priors, and derivation of an MCMC sampler. Empirical contributions include visualizations of cross-ratio equivalent effect on basic examples and simulations validating theoretical results.

A natural strategy for mitigating the curse of dimensionality in estimating probability distributions is to employ lowcomplexity family of approximation distributions. For discrete distributions, Chow and Liu [5] suggested a family of tree-based approximations and gave an efficient maximum-likelihood estimator based on Kruskal's optimal spanning tree algorithm [14]. We stress that this approach makes no structural assumptions about the sampling distribution, but rather constitutes a modeling choice. Consequently, in this paradigm, the goal is to approximate the optimal-tree distribution from the data, without any guarantees on how well the latter approximates the true sampling distribution. Extensions of the Chow-Liu approach to continuous distributions were studied by Bach and Jordan [1] and by Liu et al. [16] under various assumptions.

In this paper, we focus on exploiting neural networks for the analysis and planning stage in self-adaptive architectures. The studied motivating cases in the paper involve existing (legacy) self-adaptive architectures and their adaptation logic, which has been specified by logical rules. We further assume that there is a need to endow these systems with the ability to learn based on examples of inputs and expected outputs. One simple option to address such a need is to replace the reasoning based on logical rules with a neural network. However, this step brings several problems that often create at least a temporary regress. The reason is the logical rules typically represent a large and tested body of domain knowledge, which may be lost if the logical rules are replaced by a neural network. Further, the black-box nature of generic neural networks obfuscates how the systems work inside and consequently introduces more uncertainty. In this paper, we present a method that makes it possible to endow an existing self-adaptive architectures with the ability to learn using neural networks, while preserving domain knowledge existing in the logical rules. We introduce a continuum between the existing rule-based system and a system based on a generic neural network. We show how to navigate in this continuum and create a neural network architecture that naturally embeds the original logical rules and how to gradually scale the learning potential of the network, thus controlling the uncertainty inherent to all soft computing models. We showcase and evaluate the approach on representative excerpts from two larger real-life use cases.

Let the OSS Enterprise newsletter guide your open source journey! Meta (formerly Facebook) this week announced the release of Bean Machine, a probabilistic programming system that ostensibly makes it easier to represent and learn about uncertainties in AI models. Available in early beta, Bean Machine can be used to discover unobserved properties of a model via automatic, "uncertainty-aware" learning algorithms. "[Bean Machine is] inspired from a physical device for visualizing probability distributions, a pre-computing example of a probabilistic system," the Meta researchers behind Bean Machine explained in a blog post. "We on the Bean Machine development team believe that the usability of a system forms the bedrock for its success, and we've taken care to center Bean Machine's design around a declarative philosophy within the PyTorch ecosystem." It's commonly understood that deep learning models are overconfident -- even when they make mistakes.

Policy Search and Model Predictive Control~(MPC) are two different paradigms for robot control: policy search has the strength of automatically learning complex policies using experienced data, while MPC can offer optimal control performance using models and trajectory optimization. An open research question is how to leverage and combine the advantages of both approaches. In this work, we provide an answer by using policy search for automatically choosing high-level decision variables for MPC, which leads to a novel policy-search-for-model-predictive-control framework. Specifically, we formulate the MPC as a parameterized controller, where the hard-to-optimize decision variables are represented as high-level policies. Such a formulation allows optimizing policies in a self-supervised fashion. We validate this framework by focusing on a challenging problem in agile drone flight: flying a quadrotor through fast-moving gates. Experiments show that our controller achieves robust and real-time control performance in both simulation and the real world. The proposed framework offers a new perspective for merging learning and control.

Bayesian Networks are probabilistic graphical models that can compactly represent dependencies among random variables. Missing data and hidden variables require calculating the marginal probability distribution of a subset of the variables. While knowledge of the marginal probability distribution is crucial for various problems in statistics and machine learning, its exact computation is generally not feasible for categorical variables due to the NP-hardness of this task. We develop a divide-and-conquer approach using the graphical properties of Bayesian networks to split the computation of the marginal probability distribution into sub-calculations of lower dimensionality, reducing the overall computational complexity. Exploiting this property, we present an efficient and scalable algorithm for estimating the marginal probability distribution for categorical variables. The novel method is compared against state-of-the-art approximate inference methods in a benchmarking study, where it displays superior performance. As an immediate application, we demonstrate how the marginal probability distribution can be used to classify incomplete data against Bayesian networks and use this approach for identifying the cancer subtype of kidney cancer patient samples.

While learning the graphical structure of Bayesian networks from observational data is key to describing and helping understand data generating processes in complex applications, the task poses considerable challenges due to its computational complexity. The directed acyclic graph (DAG) representing a Bayesian network model is generally not identifiable from observational data, and a variety of methods exist to estimate its equivalence class instead. Under certain assumptions, the popular PC algorithm can consistently recover the correct equivalence class by testing for conditional independence (CI), starting from marginal independence relationships and progressively expanding the conditioning set. Here, we propose the dual PC algorithm, a novel scheme to carry out the CI tests within the PC algorithm by leveraging the inverse relationship between covariance and precision matrices. Notably, the elements of the precision matrix coincide with partial correlations for Gaussian data. Our algorithm then exploits block matrix inversions on the covariance and precision matrices to simultaneously perform tests on partial correlations of complementary (or dual) conditioning sets. The multiple CI tests of the dual PC algorithm, therefore, proceed by first considering marginal and full-order CI relationships and progressively moving to central-order ones. Simulation studies indicate that the dual PC algorithm outperforms the classical PC algorithm both in terms of run time and in recovering the underlying network structure.