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This research aims to provide the geometrical foundation of the uncertainty principle within a new causal structure of spacetime so-called Symmetrical Special Relativity (SSR), where there emerges a Lorentz violation due to the presence of an invariant minimum speed $V$ related to the vacuum energy. SSR predicts that a dS-scenario occurs only for a certain regime of speeds $v$, where $v0$. For $v=v_0$, Minkowski (pseudo-Euclidian) space is recovered for representing the flat space ($\Lambda=0$), and for $v>v_0$ ($\Phi>0$), Anti-de Sitter (AdS) scenario prevails ($\Lambda<0$). The fact that the current universe is flat as its average density of matter distribution ($\rho_m$ given for a slightly negative curvature $R$) coincides with its vacuum energy density ($\rho_{\Lambda}$ given for a slightly positive curvature $\Lambda$), i.e., the {\it cosmic coincidence problem}, is now addressed by SSR. SSR provides its energy-momentum tensor of perfect fluid, leading to the EOS of vacuum ($p=-\rho_{\Lambda}$). Einstein equation for vacuum given by such SSR approach allows us to obtain $\rho_{\Lambda}$ associated with a scalar curvature $\Lambda$, whereas the solution of Einstein equation only in the presence of a homogeneous distribution of matter $\rho_m$ for the whole universe presents a scalar curvature $R$, in such a way that the presence of the background field $\Lambda$ opposes the Riemannian curvature $R$, thus leading to a current effective curvature $R_{eff}=R+\Lambda\approx 0$ according to observations., Comment: 19 pages, 2 figures

Virtual scenario-based testing methods to validate autonomous driving systems are predominantly centred around collision avoidance, and lack a comprehensive approach to evaluate optimal driving behaviour holistically. Furthermore, current validation approaches do not align with authorisation and monitoring requirements put forth by regulatory bodies. We address these validation gaps by outlining a universal evaluation framework that: incorporates the notion of careful and competent driving, unifies behavioural competencies and evaluation criteria, and is amenable at a scenario-specific and aggregate behaviour level. This framework can be leveraged to evaluate optimal driving in scenario-based testing, and for post-deployment monitoring to ensure continual compliance with regulation and safety standards., Comment: arXiv admin comment: This version has been removed by arXiv administrators as the submitter did not have the rights to agree to the license at the time of submission

We propose a toy model of a spherical universe made up of an exotic dark gas with temperature $T$ in thermal equilibrium with a black-body in adiabatic expansion. Each particle of this exotic gas mimics a kind of particle of dark energy represented by the vacuum energy, being quantized into virtual particles with extremely small masses that form such gas representng the own tissue of the expanding space-time governed by a negative pressure whose origin is the equation of state (EOS) of vacuum, i.e., $p=-\rho$, where $\rho$ is the vacuum energy density. So, each vacuum particle occupies a tiny area of space so-called Planck area $L_p^{2}$, which represents the minimum area of the whole space-time given by the spherical surface with area $4\pi R_H^2$, where $R_H$ is the Hubble radius. We realize that such spherical surface is the surface of the black-body for representing the dark universe as if it were the surface of an expanding balloon. Thus, we are able to derive the law of universal gravitation, thus leading us to understand the cosmological anti-gravity. We estimate the very small order of magnitude of the cosmological constant and the acceleration of expansion of the dark sphere. In this toy model, as the dark universe can be thought of as a large black body, when we obtain its power and frequency of emission of radiation, we find very low values. We conclude that such radiation and frequency of the black body made up of dark energy is a background gravitational wave with very low frequency in the order of $10^{-17}$Hz due to the slight stretching of the fabric of space-time., Comment: 8 pages, 1 figure

Communication-constrained algorithms for decentralized learning and optimization rely on local updates coupled with the exchange of compressed signals. In this context, differential quantization is an effective technique to mitigate the negative impact of compression by leveraging correlations between successive iterates. In addition, the use of error feedback, which consists of incorporating the compression error into subsequent steps, is a powerful mechanism to compensate for the bias caused by the compression. Under error feedback, performance guarantees in the literature have so far focused on algorithms employing a fusion center or a special class of contractive compressors that cannot be implemented with a finite number of bits. In this work, we propose a new decentralized communication-efficient learning approach that blends differential quantization with error feedback. The approach is specifically tailored for decentralized learning problems where agents have individual risk functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. We show that, under some general conditions on the compression noise, and for sufficiently small step-sizes $\mu$, the resulting communication-efficient strategy is stable both in terms of mean-square error and average bit rate: by reducing $\mu$, it is possible to keep the estimation errors small (on the order of $\mu$) without increasing indefinitely the bit rate as $\mu\rightarrow 0$. The results establish that, in the small step-size regime and with a finite number of bits, it is possible to attain the performance achievable in the absence of compression., Comment: arXiv admin note: text overlap with arXiv:2209.07821

Adaptive experimental design (AED) methods are increasingly being used in industry as a tool to boost testing throughput or reduce experimentation cost relative to traditional A/B/N testing methods. However, the behavior and guarantees of such methods are not well-understood beyond idealized stationary settings. This paper shares lessons learned regarding the challenges of naively using AED systems in industrial settings where non-stationarity is prevalent, while also providing perspectives on the proper objectives and system specifications in such settings. We developed an AED framework for counterfactual inference based on these experiences, and tested it in a commercial environment.

In policy learning, the goal is typically to optimize a primary performance metric, but other subsidiary metrics often also warrant attention. This paper presents two strategies for evaluating these subsidiary metrics under a policy that is optimal for the primary one. The first relies on a novel margin condition that facilitates Wald-type inference. Under this and other regularity conditions, we show that the one-step corrected estimator is efficient. Despite the utility of this margin condition, it places strong restrictions on how the subsidiary metric behaves for nearly optimal policies, which may not hold in practice. We therefore introduce alternative, two-stage strategies that do not require a margin condition. The first stage constructs a set of candidate policies and the second builds a uniform confidence interval over this set. We provide numerical simulations to evaluate the performance of these methods in different scenarios.

Kantorovich operators are non-linear extensions of Markov operators and are omnipresent in several branches of mathematical analysis. The asymptotic behaviour of their iterates plays an important role even in classical ergodic, potential and probability theories, which are normally concerned with linear Markovian operators, semi-groups, and resolvents. The Kantorovich operators that appear implicitly in these cases, though non-linear, are all positively 1-homogenous. General Kantorovich operators amount to assigning "a cost" to most operations on measures and functions normally conducted "for free" in these classical settings. Motivated by extensions of the Monge-Kantorovich duality in mass transport, the stochastic counterpart of Aubry-Mather theory for Lagrangian systems, weak KAM theory \`a la Fathi-Mather, and ergodic optimization of dynamical systems, we study the asymptotic properties of general Kantorovich operators., Comment: 49 pages. Updated version - if any - can be downloaded at https://www.birs.ca/~nassif/

The present review aims to show that a modified space-time with an invariant minimum speed provides a relation with Weyl geometry in the Newtonian approximation of weak-field. The deformed Special Relativity so-called Symmetrical Special Relativity (SSR) has an invariant minimum speed $V$, which is associated with a preferred reference frame $S_V$ for representing the vacuum energy, thus leading to the cosmological constant ($\Lambda$). The equation of state (EOS) of vacuum energy for $\Lambda$, i.e., $\rho_{\Lambda}=\epsilon=-p$ emerges naturally from such space-time, where $p$ is the pressure and $\rho_{\Lambda}=\epsilon$ is the vacuum energy density. With the aim of establishing a relationship between $V$ and $\Lambda$ in the modified metric of the space-time, we should consider a dark spherical universe with Hubble radius $R_H$, having a very low value of $\epsilon$ that governs the accelerated expansion of universe. In doing this, we aim to show that SSR-metric has an equivalence with a de-Sitter (dS)-metric ($\Lambda>0$). On the other hand, according to the Boomerang experiment that reveals a slightly accelerated expansion of the universe, SSR leads to a dS-metric with an approximation for $\Lambda<<1$ close to a flat space-time, which is in the $\Lambda CDM$ scenario where the space is quasi-flat, so that $\Omega_{m}+\Omega_{\Lambda}\approx 1$. We have $\Omega{cdm}\approx 23\%$ by representing dark cold matter, $\Omega_m\approx 27\%$ for matter and $\Omega_{\Lambda}\approx 73\%$ for the vacuum energy. Thus, the theory is adjusted for the redshift $z=1$. This corresponds to the time $\tau_0$ of transition between gravity and anti-gravity, leading to a slight acceleration of expansion related to a tiny value of $\Lambda$, i.e., we find $\Lambda_0=1.934\times 10^{-35}s^{-2}$. This result is in agreement with observations., Comment: 15 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2303.14120

Most linear experimental design problems assume homogeneous variance although heteroskedastic noise is present in many realistic settings. Let a learner have access to a finite set of measurement vectors $\mathcal{X}\subset \mathbb{R}^d$ that can be probed to receive noisy linear responses of the form $y=x^{\top}\theta^{\ast}+\eta$. Here $\theta^{\ast}\in \mathbb{R}^d$ is an unknown parameter vector, and $\eta$ is independent mean-zero $\sigma_x^2$-sub-Gaussian noise defined by a flexible heteroskedastic variance model, $\sigma_x^2 = x^{\top}\Sigma^{\ast}x$. Assuming that $\Sigma^{\ast}\in \mathbb{R}^{d\times d}$ is an unknown matrix, we propose, analyze and empirically evaluate a novel design for uniformly bounding estimation error of the variance parameters, $\sigma_x^2$. We demonstrate the benefits of this method with two adaptive experimental design problems under heteroskedastic noise, fixed confidence transductive best-arm identification and level-set identification and prove the first instance-dependent lower bounds in these settings. Lastly, we construct near-optimal algorithms and demonstrate the large improvements in sample complexity gained from accounting for heteroskedastic variance in these designs empirically.

The magnetic and structural properties of the recently discovered pnictogen/chalcogen-free superconductor LaFeSiH ($T_c\simeq10$~K) have been investigated by $^{57}$Fe synchrotron M{\"o}ssbauer source (SMS) spectroscopy, x-ray and neutron powder diffraction and $^{29}$Si nuclear magnetic resonance spectroscopy (NMR). No sign of long range magnetic order or local moments has been detected in any of the measurements and LaFeSiH remains tetragonal down to 2 K. The activated temperature dependence of both the NMR Knight shift and the relaxation rate $1/T_1$ is analogous to that observed in strongly overdoped Fe-based superconductors. These results, together with the temperature-independent NMR linewidth, show that LaFeSiH is an homogeneous metal, far from any magnetic or nematic instability, and with similar Fermi surface properties as strongly overdoped iron pnictides. This raises the prospect of enhancing the $T_c$ of LaFeSiH by reducing its carrier concentration through appropriate chemical substitutions. Additional SMS spectroscopy measurements under hydrostatic pressure up to 18.8~GPa found no measurable hyperfine field.

Temporal graphs represent interactions between entities over time. Deciding whether entities can reach each other through temporal paths is useful for various applications such as in communication networks and epidemiology. Previous works have studied the scenario in which addition of new interactions can happen at any point in time. A known strategy maintains, incrementally, a Timed Transitive Closure by using a dynamic data structure composed of $O(n^2)$ binary search trees containing non-nested time intervals. However, space usage for storing these trees grows rapidly as more interactions are inserted. In this paper, we present a compact data structures that represent each tree as two dynamic bit-vectors. In our experiments, we observed that our data structure improves space usage while having similar time performance for incremental updates when comparing with the previous strategy in temporally dense temporal graphs., Comment: 17 pages, 6 figures

Classical paradigms for distributed learning, such as federated or decentralized gradient descent, employ consensus mechanisms to enforce homogeneity among agents. While these strategies have proven effective in i.i.d. scenarios, they can result in significant performance degradation when agents follow heterogeneous objectives or data. Distributed strategies for multitask learning, on the other hand, induce relationships between agents in a more nuanced manner, and encourage collaboration without enforcing consensus. We develop a generalization of the exact diffusion algorithm for subspace constrained multitask learning over networks, and derive an accurate expression for its mean-squared deviation when utilizing noisy gradient approximations. We verify numerically the accuracy of the predicted performance expressions, as well as the improved performance of the proposed approach over alternatives based on approximate projections.

We study dynamic discrete choice models, where a commonly studied problem involves estimating parameters of agent reward functions (also known as "structural" parameters), using agent behavioral data. Maximum likelihood estimation for such models requires dynamic programming, which is limited by the curse of dimensionality. In this work, we present a novel algorithm that provides a data-driven method for selecting and aggregating states, which lowers the computational and sample complexity of estimation. Our method works in two stages. In the first stage, we use a flexible inverse reinforcement learning approach to estimate agent Q-functions. We use these estimated Q-functions, along with a clustering algorithm, to select a subset of states that are the most pivotal for driving changes in Q-functions. In the second stage, with these selected "aggregated" states, we conduct maximum likelihood estimation using a commonly used nested fixed-point algorithm. The proposed two-stage approach mitigates the curse of dimensionality by reducing the problem dimension. Theoretically, we derive finite-sample bounds on the associated estimation error, which also characterize the trade-off of computational complexity, estimation error, and sample complexity. We demonstrate the empirical performance of the algorithm in two classic dynamic discrete choice estimation applications.