In this work we consider the problem of numerical integration, i.e., approximating integrals with respect to a target probability measure using only pointwise evaluations of the integrand. We focus on the setting in which the target distribution is only accessible through a set of $n$ i.i.d. observations, and the integrand belongs to a reproducing kernel Hilbert space. We propose an efficient procedure which exploits a small i.i.d. random subset of $m

We present the new Orthogonal Polynomials Approximation Algorithm (OPAA), a parallelizable algorithm that solves two problems from a functional analytic approach: first, it finds a smooth functional estimate of a density function, whether it is normalized or not; second, the algorithm provides an estimate of the normalizing weight. In the context of Bayesian inference, OPAA provides an estimate of the posterior function as well as the normalizing weight, which is also known as the evidence. A core component of OPAA is a special transform of the square root of the joint distribution into a special functional space of our construct. Through this transform, the evidence is equated with the $L^2$ norm of the transformed function, squared. Hence, the evidence can be estimated by the sum of squares of the transform coefficients. The computations can be parallelized and completed in one pass. To compute the transform coefficients, OPAA proposes a new computational scheme leveraging Gauss--Hermite quadrature in higher dimensions. Not only does it avoid the potential high variance problem associated with random sampling methods, it also enables one to speed up the computation by parallelization, and significantly reduces the complexity by a vector decomposition.

The goal of this paper is to demonstrate that common noise may serve as an exploration noise for learning the solution of a mean field game. This concept is here exemplified through a toy linear-quadratic model, for which a suitable form of common noise has already been proven to restore existence and uniqueness. We here go one step further and prove that the same form of common noise may force the convergence of the learning algorithm called `fictitious play', and this without any further potential or monotone structure. Several numerical examples are provided in order to support our theoretical analysis.

We consider multilevel low rank (MLR) matrices, defined as a row and column permutation of a sum of matrices, each one a block diagonal refinement of the previous one, with all blocks low rank given in factored form. MLR matrices extend low rank matrices but share many of their properties, such as the total storage required and complexity of matrix-vector multiplication. We address three problems that arise in fitting a given matrix by an MLR matrix in the Frobenius norm. The first problem is factor fitting, where we adjust the factors of the MLR matrix. The second is rank allocation, where we choose the ranks of the blocks in each level, subject to the total rank having a given value, which preserves the total storage needed for the MLR matrix. The final problem is to choose the hierarchical partition of rows and columns, along with the ranks and factors. This paper is accompanied by an open source package that implements the proposed methods.

Modeling and control of epidemics such as the novel Corona virus have assumed paramount importance at a global level. A natural and powerful dynamical modeling framework to use in this context is a continuous time Markov decision process (CTMDP) that encompasses classical compartmental paradigms such as the Susceptible-Infected-Recovered (SIR) model. The challenges with CTMDP based models motivate the need for a more efficient approach and the mean field approach offers an effective alternative. The mean field approach computes the collective behavior of a dynamical system comprising numerous interacting nodes (where nodes represent individuals in the population). This paper (a) presents an overview of the mean field approach to epidemic modeling and control and (b) provides a state-of-the-art update on recent advances on this topic. Our discussion in this paper proceeds along two specific threads. The first thread assumes that the individual nodes faithfully follow a socially optimal control policy prescribed by a regulatory authority. The second thread allows the individual nodes to exhibit independent, strategic behavior. In this case, the strategic interaction is modeled as a mean field game and the control is based on the associated mean field Nash equilibria. In this paper, we start with a discussion of modeling of epidemics using an extended compartmental model - SIVR and provide an illustrative example. We next provide a review of relevant literature, using a mean field approach, on optimal control of epidemics, dealing with how a regulatory authority may optimally contain epidemic spread in a population. Following this, we provide an update on the literature on the use of the mean field game based approach in the study of epidemic spread and control. We conclude the paper with relevant future research directions.

The Densest Subgraph Problem requires to find, in a given graph, a subset of vertices whose induced subgraph maximizes a measure of density. The problem has received a great deal of attention in the algorithmic literature over the last five decades, with many variants proposed and many applications built on top of this basic definition. Recent years have witnessed a revival of research interest on this problem with several interesting contributions, including some groundbreaking results, published in 2022 and 2023. This survey provides a deep overview of the fundamental results and an exhaustive coverage of the many variants proposed in the literature, with a special attention on the most recent results. The survey also presents a comprehensive overview of applications and discusses some interesting open problems for this evergreen research topic.

As humans, we perceive the three-dimensional structure of the world around us with apparent ease. Think of how vivid the three-dimensional percept is when you look at a vase of flowers sitting on the table next to you. You can tell the shape and translucency of each petal through the subtle patterns of light and shading that play across its surface and effortlessly segment each flower from the background of the scene (Figure 1.1). Looking at a framed group por- trait, you can easily count (and name) all of the people in the picture and even guess at their emotions from their facial appearance. Perceptual psychologists have spent decades trying to understand how the visual system works and, even though they can devise optical illusions1 to tease apart some of its principles (Figure 1.3), a complete solution to this puzzle remains elusive (Marr 1982; Palmer 1999; Livingstone 2008).

Optimizing over the stationary distribution of stochastic differential equations (SDEs) is computationally challenging. A new forward propagation algorithm has been recently proposed for the online optimization of SDEs. The algorithm solves an SDE, derived using forward differentiation, which provides a stochastic estimate for the gradient. The algorithm continuously updates the SDE model's parameters and the gradient estimate simultaneously. This paper studies the convergence of the forward propagation algorithm for nonlinear dissipative SDEs. We leverage the ergodicity of this class of nonlinear SDEs to characterize the convergence rate of the transition semi-group and its derivatives. Then, we prove bounds on the solution of a Poisson partial differential equation (PDE) for the expected time integral of the algorithm's stochastic fluctuations around the direction of steepest descent. We then re-write the algorithm using the PDE solution, which allows us to characterize the parameter evolution around the direction of steepest descent. Our main result is a convergence theorem for the forward propagation algorithm for nonlinear dissipative SDEs.

Randomized algorithms have propelled advances in artificial intelligence and represent a foundational research area in advancing AI for Science. Future advancements in DOE Office of Science priority areas such as climate science, astrophysics, fusion, advanced materials, combustion, and quantum computing all require randomized algorithms for surmounting challenges of complexity, robustness, and scalability. This report summarizes the outcomes of that workshop, "Randomized Algorithms for Scientific Computing (RASC)," held virtually across four days in December 2020 and January 2021.

A universal fault-tolerant quantum computer that can solve efficiently problems such as integer factorization and unstructured database search requires millions of qubits with low error rates and long coherence times. While the experimental advancement towards realizing such devices will potentially take decades of research, noisy intermediate-scale quantum (NISQ) computers already exist. These computers are composed of hundreds of noisy qubits, i.e. qubits that are not error-corrected, and therefore perform imperfect operations in a limited coherence time. In the search for quantum advantage with these devices, algorithms have been proposed for applications in various disciplines spanning physics, machine learning, quantum chemistry and combinatorial optimization. The goal of such algorithms is to leverage the limited available resources to perform classically challenging tasks. In this review, we provide a thorough summary of NISQ computational paradigms and algorithms. We discuss the key structure of these algorithms, their limitations, and advantages. We additionally provide a comprehensive overview of various benchmarking and software tools useful for programming and testing NISQ devices.