The Erd\H{o}s-R\'enyi random graph is the simplest model for node degree distribution, and it is one of the most widely studied. In this model, pairs of $n$ vertices are selected and connected uniformly at random with probability $p$, consequently, the degrees for a given vertex follow the binomial distribution. If the number of vertices is large, the binomial can be approximated by Normal using the Central Limit Theorem, which is often allowed when $\min (np, n(1-p)) > 5$. This is true for every node independently. However, due to the fact that the degrees of nodes in a graph are not independent, we aim in this paper to test whether the degrees of per node collectively in the Erd\H{o}s-R\'enyi graph have a multivariate normal distribution MVN. A chi square goodness of fit test for the hypothesis that binomial is a distribution for the whole set of nodes is rejected because of the dependence between degrees. Before testing MVN we show that the covariance and correlation between the degrees of any pair of nodes in the graph are $p(1-p)$ and $1/(n-1)$, respectively. We test MVN considering two assumptions: independent and dependent degrees, and we obtain our results based on the percentages of rejected statistics of chi square, the $p$-values of Anderson Darling test, and a CDF comparison. We always achieve a good fit of multivariate normal distribution with large values of $n$ and $p$, and very poor fit when $n$ or $p$ are very small. The approximation seems valid when $np \geq 10$. We also compare the maximum likelihood estimate of $p$ in MVN distribution where we assume independence and dependence. The estimators are assessed using bias, variance and mean square error.

Figure 1: From top to bottom: a) naïve RL with training directly on the real world; b) RL where the policy is trained in simulation Soft robotics is a rapidly developing field that has the and tested on the real world; c) Sim-to-Real transfer with potential to revolutionize how robots interact with their domain randomization increases robustness to modelling environment [1]. Unlike their rigid counterparts, soft robots errors and enables environmental constraints exploitation; are made from materials that can deform and adapt to d) posterior distributions over simulator parameters may be their surroundings, enabling them to perform novel and automatically inferred from real-world data for use with DR. unprecedented tasks in fields such as healthcare [2] and exploration [3]. However, controlling the complex dynamics of continuous soft robots is a challenging task, as an accurate Many attempts have been made to control soft devices modelling requires infinite degrees of freedom (DoF) [4] and through model-based techniques, also pushed by the advancement nonlinear dynamics parameters that are difficult to accurately of modelling techniques [6].

Abstract: Undirected hyperbolic graph models have been extensively used as models of scale-free small-world networks with high clustering coefficient. Here we presented a simple directed hyperbolic model, where nodes randomly distributed on a hyperbolic disk are connected to a fixed number m of their nearest spatial neighbours. We introduce also a canonical version of this network (which we call network with varied connection radius''), where maximal length of outgoing bond is space-dependent and is determined by fixing the average out-degree at m. We study local bond length, in-degree and reciprocity in these networks as a function of spacial coordinates of the nodes, and show that the network has a distinct core-periphery structure. We show that for small densities of nodes the overall in-degree has a truncated power law distribution.

Abstract: We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range (2,3). In particular, we focus on the expected times for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that up to multiplicative constants: the cover time is n(logn)2, the maximum hitting time is nlogn, and the average hitting time is n. The first two results hold in expectation and a.a.s. and the last in expectation (with respect to the HRG). We prove these results by determining the effective resistance either between an average vertex and the well-connected "center" of HRGs or between an appropriately chosen collection of extremal vertices.

This paper studies the online stochastic resource allocation problem (RAP) with chance constraints. The online RAP is a 0-1 integer linear programming problem where the resource consumption coefficients are revealed column by column along with the corresponding revenue coefficients. When a column is revealed, the corresponding decision variables are determined instantaneously without future information. Moreover, in online applications, the resource consumption coefficients are often obtained by prediction. To model their uncertainties, we take the chance constraints into the consideration. To the best of our knowledge, this is the first time chance constraints are introduced in the online RAP problem. Assuming that the uncertain variables have known Gaussian distributions, the stochastic RAP can be transformed into a deterministic but nonlinear problem with integer second-order cone constraints. Next, we linearize this nonlinear problem and analyze the performance of vanilla online primal-dual algorithm for solving the linearized stochastic RAP. Under mild technical assumptions, the optimality gap and constraint violation are both on the order of $\sqrt{n}$. Then, to further improve the performance of the algorithm, several modified online primal-dual algorithms with heuristic corrections are proposed. Finally, extensive numerical experiments on both synthetic and real data demonstrate the applicability and effectiveness of our methods.

Quantum Bayesian Computation (QBC) is an emerging field that levers the computational gains available from quantum computers to provide an exponential speed-up in Bayesian computation. Our paper adds to the literature in two ways. First, we show how von Neumann quantum measurement can be used to simulate machine learning algorithms such as Markov chain Monte Carlo (MCMC) and Deep Learning (DL) that are fundamental to Bayesian learning. Second, we describe data encoding methods needed to implement quantum machine learning including the counterparts to traditional feature extraction and kernel embeddings methods. Our goal then is to show how to apply quantum algorithms directly to statistical machine learning problems. On the theoretical side, we provide quantum versions of high dimensional regression, Gaussian processes (Q-GP) and stochastic gradient descent (Q-SGD). On the empirical side, we apply a Quantum FFT model to Chicago housing data. Finally, we conclude with directions for future research.

The history of Newton's method spans over several centuries and the method has become famous for being extremely fast, and infamous for converging only from initialization that is close to a solution. Despite the latter drawback, Newton's method is a cornerstone of convex optimization and it motivated the development of numerous popular algorithms, such as quasi-Newton and trust-region procedures. Its applications and extensions are countless, so we refer to the study in [19] that lists more than 1,000 references in total. Although widely acknowledged, the extreme behaviour of Newton's method is still startling. Why does it converge so efficiently from one initialization and hopelessly diverge from a tiny perturbation of the same initialization?

Dispatching vehicle fleets to serve flights is a key task in airport ground handling (AGH). Due to the notable growth of flights, it is challenging to simultaneously schedule multiple types of operations (services) for a large number of flights, where each type of operation is performed by one specific vehicle fleet. To tackle this issue, we first represent the operation scheduling as a complex vehicle routing problem and formulate it as a mixed integer linear programming (MILP) model. Then given the graph representation of the MILP model, we propose a learning assisted large neighborhood search (LNS) method using data generated based on real scenarios, where we integrate imitation learning and graph convolutional network (GCN) to learn a destroy operator to automatically select variables, and employ an off-the-shelf solver as the repair operator to reoptimize the selected variables. Experimental results based on a real airport show that the proposed method allows for handling up to 200 flights with 10 types of operations simultaneously, and outperforms state-of-the-art methods. Moreover, the learned method performs consistently accompanying different solvers, and generalizes well on larger instances, verifying the versatility and scalability of our method.

This work initiates the systematic study of explicit distributions that are indistinguishable from a single exponential-size combinatorial object. In this we extend the work of Goldreich, Goldwasser and Nussboim (SICOMP 2010) that focused on the implementation of huge objects that are indistinguishable from the uniform distribution, satisfying some global properties (which they coined truthfulness). Indistinguishability from a single object is motivated by the study of generative models in learning theory and regularity lemmas in graph theory. Problems that are well understood in the setting of pseudorandomness present significant challenges and at times are impossible when considering generative models of huge objects. We demonstrate the versatility of this study by providing a learning algorithm for huge indistinguishable objects in several natural settings including: dense functions and graphs with a truthfulness requirement on the number of ones in the function or edges in the graphs, and a version of the weak regularity lemma for sparse graphs that satisfy some global properties. These and other results generalize basic pseudorandom objects as well as notions introduced in algorithmic fairness. The results rely on notions and techniques from a variety of areas including learning theory, complexity theory, cryptography, and game theory.

Cutting plane selection is a subroutine used in all modern mixed-integer linear programming solvers with the goal of selecting a subset of generated cuts that induce optimal solver performance. These solvers have millions of parameter combinations, and so are excellent candidates for parameter tuning. Cut selection scoring rules are usually weighted sums of different measurements, where the weights are parameters. We present a parametric family of mixed-integer linear programs together with infinitely many family-wide valid cuts. Some of these cuts can induce integer optimal solutions directly after being applied, while others fail to do so even if an infinite amount are applied. We show for a specific cut selection rule, that any finite grid search of the parameter space will always miss all parameter values, which select integer optimal inducing cuts in an infinite amount of our problems. We propose a variation on the design of existing graph convolutional neural networks, adapting them to learn cut selection rule parameters. We present a reinforcement learning framework for selecting cuts, and train our design using said framework over MIPLIB 2017 and a neural network verification data set. Our framework and design show that adaptive cut selection does substantially improve performance over a diverse set of instances, but that finding a single function describing such a rule is difficult. Code for reproducing all experiments is available at https://github.com/Opt-Mucca/Adaptive-Cutsel-MILP.