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This paper introduces a new method of discretization that collocates both endpoints of the domain and enables the complete convergence of the costate variables associated with the Hamilton boundary-value problem. This is achieved through the inclusion of an \emph{exceptional sample} to the roots of the Legendre-Lobatto polynomial, thus promoting the associated differentiation matrix to be full-rank. We study the location of the new sample such that the differentiation matrix is the most robust to perturbations and we prove that this location is also the choice that mitigates the Runge phenomenon associated with polynomial interpolation. Two benchmark problems are successfully implemented in support of our theoretical findings. The new method is observed to converge exponentially with the number of discretization points used.

We present an intimate connection among the following fields: (a) distributed local algorithms: coming from the area of computer science, (b) finitary factors of iid processes: coming from the area of analysis of randomized processes, (c) descriptive combinatorics: coming from the area of combinatorics and measure theory. In particular, we study locally checkable labellings in grid graphs from all three perspectives. Most of our results are for the perspective (b) where we prove time hierarchy theorems akin to those known in the field (a) [Chang, Pettie FOCS 2017]. This approach that borrows techniques from the fields (a) and (c) implies a number of results about possible complexities of finitary factor solutions. Among others, it answers three open questions of [Holroyd et al. Annals of Prob. 2017] or the more general question of [Brandt et al. PODC 2017] who asked for a formal connection between the fields (a) and (b). In general, we hope that our treatment will help to view all three perspectives as a part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn 2020+] .

The paper applies reinforcement learning to novel Internet of Thing configurations. Our analysis of inaudible attacks on voice-activated devices confirms the alarming risk factor of 7.6 out of 10, underlining significant security vulnerabilities scored independently by NIST National Vulnerability Database (NVD). Our baseline network model showcases a scenario in which an attacker uses inaudible voice commands to gain unauthorized access to confidential information on a secured laptop. We simulated many attack scenarios on this baseline network model, revealing the potential for mass exploitation of interconnected devices to discover and own privileged information through physical access without adding new hardware or amplifying device skills. Using Microsoft's CyberBattleSim framework, we evaluated six reinforcement learning algorithms and found that Deep-Q learning with exploitation proved optimal, leading to rapid ownership of all nodes in fewer steps. Our findings underscore the critical need for understanding non-conventional networks and new cybersecurity measures in an ever-expanding digital landscape, particularly those characterized by mobile devices, voice activation, and non-linear microphones susceptible to malicious actors operating stealth attacks in the near-ultrasound or inaudible ranges. By 2024, this new attack surface might encompass more digital voice assistants than people on the planet yet offer fewer remedies than conventional patching or firmware fixes since the inaudible attacks arise inherently from the microphone design and digital signal processing.

We study the open question of how players learn to play a social optimum pure-strategy Nash equilibrium (PSNE) through repeated interactions in general-sum coordination games. A social optimum of a game is the stable Pareto-optimal state that provides a maximum return in the sum of all players' payoffs (social welfare) and always exists. We consider finite repeated games where each player only has access to its own utility (or payoff) function but is able to exchange information with other players. We develop a novel regret matching (RM) based algorithm for computing an efficient PSNE solution that could approach a desired Pareto-optimal outcome yielding the highest social welfare among all the attainable equilibria in the long run. Our proposed learning procedure follows the regret minimization framework but extends it in three major ways: (1) agents use global, instead of local, utility for calculating regrets, (2) each agent maintains a small and diminishing exploration probability in order to explore various PSNEs, and (3) agents stay with the actions that achieve the best global utility thus far, regardless of regrets. We prove that these three extensions enable the algorithm to select the stable social optimum equilibrium instead of converging to an arbitrary or cyclic equilibrium as in the conventional RM approach. We demonstrate the effectiveness of our approach through a set of applications in multi-agent distributed control, including a large-scale resource allocation game and a hard combinatorial task assignment problem for which no efficient (polynomial) solution exists.

This work focuses on equilibrium selection in no-conflict multi-agent games, where we specifically study the problem of selecting a Pareto-optimal equilibrium among several existing equilibria. It has been shown that many state-of-the-art multi-agent reinforcement learning (MARL) algorithms are prone to converging to Pareto-dominated equilibria due to the uncertainty each agent has about the policy of the other agents during training. To address sub-optimal equilibrium selection, we propose Pareto Actor-Critic (Pareto-AC), which is an actor-critic algorithm that utilises a simple property of no-conflict games (a superset of cooperative games): the Pareto-optimal equilibrium in a no-conflict game maximises the returns of all agents and therefore is the preferred outcome for all agents. We evaluate Pareto-AC in a diverse set of multi-agent games and show that it converges to higher episodic returns compared to seven state-of-the-art MARL algorithms and that it successfully converges to a Pareto-optimal equilibrium in a range of matrix games. Finally, we propose PACDCG, a graph neural network extension of Pareto-AC which is shown to efficiently scale in games with a large number of agents.

We propose a monotone discretization method for obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over a bounded Lipschitz domain. Our approach is motivated by the success of the monotone discretization of the fractional Laplacian [SIAM J. Numer. Anal. 60(6), pp. 3052-3077, 2022]. By exploiting the problem's unique structure, we establish the uniform boundedness, existence, and uniqueness of the numerical solutions. Moreover, we employ the policy iteration method to efficiently solve discrete nonlinear problems and prove its convergence after a finite number of iterations. The improved policy iteration, adapted to the regularity result, exhibits superior performance by modifying the discretization in different regions. Several numerical examples are provided to illustrate the effectiveness of our method.

Optimal Transport has sparked vivid interest in recent years, in particular thanks to the Wasserstein distance, which provides a geometrically sensible and intuitive way of comparing probability measures. For computational reasons, the Sliced Wasserstein (SW) distance was introduced as an alternative to the Wasserstein distance, and has seen uses for training generative Neural Networks (NNs). While convergence of Stochastic Gradient Descent (SGD) has been observed practically in such a setting, there is to our knowledge no theoretical guarantee for this observation. Leveraging recent works on convergence of SGD on non-smooth and non-convex functions by Bianchi et al. (2022), we aim to bridge that knowledge gap, and provide a realistic context under which fixed-step SGD trajectories for the SW loss on NN parameters converge. More precisely, we show that the trajectories approach the set of (sub)-gradient flow equations as the step decreases. Under stricter assumptions, we show a much stronger convergence result for noised and projected SGD schemes, namely that the long-run limits of the trajectories approach a set of generalised critical points of the loss function.

Convergence rate analyses of random walk Metropolis-Hastings Markov chains on general state spaces have largely focused on establishing sufficient conditions for geometric ergodicity or on analysis of mixing times. Geometric ergodicity is a key sufficient condition for the Markov chain Central Limit Theorem and allows rigorous approaches to assessing Monte Carlo error. The sufficient conditions for geometric ergodicity of the random walk Metropolis-Hastings Markov chain are refined and extended, which allows the analysis of previously inaccessible settings such as Bayesian Poisson regression. The key technical innovation is the development of explicit drift and minorization conditions for random walk Metropolis-Hastings, which allows explicit upper and lower bounds on the geometric rate of convergence. Further, lower bounds on the geometric rate of convergence are also developed using spectral theory. The existing sufficient conditions for geometric ergodicity, to date, have not provided explicit constraints on the rate of geometric rate of convergence because the method used only implies the existence of drift and minorization conditions. The theoretical results are applied to random walk Metropolis-Hastings algorithms for a class of exponential families and generalized linear models that address Bayesian Regression problems.

Musculoskeletal diseases such as sarcopenia and osteoporosis are major obstacles to health during aging. Although dual-energy X-ray absorptiometry (DXA) and computed tomography (CT) can be used to evaluate musculoskeletal conditions, frequent monitoring is difficult due to the cost and accessibility (as well as high radiation exposure in the case of CT). We propose a method (named MSKdeX) to estimate fine-grained muscle properties from a plain X-ray image, a low-cost, low-radiation, and highly accessible imaging modality, through musculoskeletal decomposition leveraging fine-grained segmentation in CT. We train a multi-channel quantitative image translation model to decompose an X-ray image into projections of CT of individual muscles to infer the lean muscle mass and muscle volume. We propose the object-wise intensity-sum loss, a simple yet surprisingly effective metric invariant to muscle deformation and projection direction, utilizing information in CT and X-ray images collected from the same patient. While our method is basically an unpaired image-to-image translation, we also exploit the nature of the bone's rigidity, which provides the paired data through 2D-3D rigid registration, adding strong pixel-wise supervision in unpaired training. Through the evaluation using a 539-patient dataset, we showed that the proposed method significantly outperformed conventional methods. The average Pearson correlation coefficient between the predicted and CT-derived ground truth metrics was increased from 0.460 to 0.863. We believe our method opened up a new musculoskeletal diagnosis method and has the potential to be extended to broader applications in multi-channel quantitative image translation tasks. Our source code will be released soon.

It has been observed by several authors that well-known periodization strategies like tent or Chebychev transforms lead to remarkable results for the recovery of multivariate functions from few samples. So far, theoretical guarantees are missing. The goal of this paper is twofold. On the one hand, we give such guarantees and briefly describe the difficulties of the involved proof. On the other hand, we combine these periodization strategies with recent novel constructive methods for the efficient subsampling of finite frames in $\mathbb{C}$. As a result we are able to reconstruct non-periodic multivariate functions from very few samples. The used sampling nodes are the result of a two-step procedure. Firstly, a random draw with respect to the Chebychev measure provides an initial node set. A further sparsification technique selects a significantly smaller subset of these nodes with equal approximation properties. This set of sampling nodes scales linearly in the dimension of the subspace on which we project and works universally for the whole class of functions. The method is based on principles developed by Batson, Spielman, and Srivastava and can be numerically implemented. Samples on these nodes are then used in a (plain) least-squares sampling recovery step on a suitable hyperbolic cross subspace of functions resulting in a near-optimal behavior of the sampling error. Numerical experiments indicate the applicability of our results.