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This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.

The main topic of this paper are algorithms for computing Nash equilibria. We cast our particular methods as instances of a general algorithmic abstraction, namely, a method we call {\em algorithmic boosting}, which is also relevant to other fixed-point computation problems. Algorithmic boosting is the principle of computing fixed points by taking (long-run) averages of iterated maps and it is a generalization of exponentiation. We first define our method in the setting of nonlinear maps. Secondly, we restrict attention to convergent linear maps (for computing dominant eigenvectors, for example, in the PageRank algorithm) and show that our algorithmic boosting method can set in motion {\em exponential speedups in the convergence rate}. Thirdly, we show that algorithmic boosting can convert a (weak) non-convergent iterator to a (strong) convergent one. We also consider a {\em variational approach} to algorithmic boosting providing tools to convert a non-convergent continuous flow to a convergent one. Then, by embedding the construction of averages in the design of the iterated map, we constructively prove the existence of Nash equilibria (and, therefore, Brouwer fixed points). We then discuss implementations of averaging and exponentiation, an important matter even for the scalar case. We finally discuss a relationship between dominant (PageRank) eigenvectors and Nash equilibria.

A linear-time algorithm for generating auxiliary subgraphs for the 3-edge-connected components of a connected multigraph is presented. The algorithm uses an innovative graph contraction operation and makes only one pass over the graph. By contrast, the previously best-known algorithms make multiple passes over the graph to decompose it into its 2-edge-connected components or 2-vertex-connected components, then its 3-edge-connected components or 3-vertex-connected components, and then construct a cactus representation for the 2-cuts to generate the auxiliary subgraphs for the 3-edge-connected components.

In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional ones. We do so in the spirit of quasi reversibility, replacing a classically severely ill-posed PDE problem by a nearby well-posed or only mildly ill-posed one. In order to be able to make use of the known stabilising effect of one-dimensional fractional derivatives of Abel type we work in a particular rectangular (in higher space dimensions cylindrical) geometry. We start with the plain Cauchy problem of reconstructing the values of a harmonic function inside this domain from its Dirichlet and Neumann trace on part of the boundary (the cylinder base) and explore three options for doing this with fractional operators. The two other related problems are the recovery of a free boundary and then this together with simultaneous recovery of the impedance function in the boundary condition. Our main technique here will be Newton's method. The paper contains numerical reconstructions and convergence results for the devised methods.

This survey explores modern approaches for computing low-rank approximations of high-dimensional matrices by means of the randomized SVD, randomized subspace iteration, and randomized block Krylov iteration. The paper compares the procedures via theoretical analyses and numerical studies to highlight how the best choice of algorithm depends on spectral properties of the matrix and the computational resources available. Despite superior performance for many problems, randomized block Krylov iteration has not been widely adopted in computational science. The paper strengthens the case for this method in three ways. First, it presents new pseudocode that can significantly reduce computational costs. Second, it provides a new analysis that yields simple, precise, and informative error bounds. Last, it showcases applications to challenging scientific problems, including principal component analysis for genetic data and spectral clustering for molecular dynamics data.

We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.

The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.

In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.

While deep learning strategies achieve outstanding results in computer vision tasks, one issue remains. The current strategies rely heavily on a huge amount of labeled data. In many real-world problems it is not feasible to create such an amount of labeled training data. Therefore, researchers try to incorporate unlabeled data into the training process to reach equal results with fewer labels. Due to a lot of concurrent research, it is difficult to keep track of recent developments. In this survey we provide an overview of often used techniques and methods in image classification with fewer labels. We compare 21 methods. In our analysis we identify three major trends. 1. State-of-the-art methods are scaleable to real world applications based on their accuracy. 2. The degree of supervision which is needed to achieve comparable results to the usage of all labels is decreasing. 3. All methods share common techniques while only few methods combine these techniques to achieve better performance. Based on all of these three trends we discover future research opportunities.