When assessing the performance of wireless communication systems operating over fading channels, one often encounters the problem of computing expectations of some functional of sums of independent random variables (RVs). The outage probability (OP) at the output of Equal Gain Combining (EGC) and Maximum Ratio Combining (MRC) receivers is among the most important performance metrics that falls within this framework. In general, closed form expressions of expectations of functionals applied to sums of RVs are out of reach. A naive Monte Carlo (MC) simulation is of course an alternative approach. However, this method requires a large number of samples for rare event problems (small OP values for instance). Therefore, it is of paramount importance to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), being known for its efficiency in requiring less computations for achieving the same accuracy requirement. In this line, we propose a state-dependent IS scheme based on a stochastic optimal control (SOC) formulation to calculate rare events quantities that could be written in a form of an expectation of some functional of sums of independent RVs. Our proposed algorithm is generic and can be applicable without any restriction on the univariate distributions of the different fading envelops/gains or on the functional that is applied to the sum. We apply our approach to the Log-Normal distribution to compute the OP at the output of diversity receivers with and without co-channel interference. For each case, we show numerically that the proposed state-dependent IS algorithm compares favorably to most of the well-known estimators dealing with similar problems.
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Given a set $P$ of $n$ points in the plane, the $k$-center problem is to find $k$ congruent disks of minimum possible radius such that their union covers all the points in $P$. The $2$-center problem is a special case of the $k$-center problem that has been extensively studied in the recent past \cite{CAHN,HT,SH}. In this paper, we consider a generalized version of the $2$-center problem called \textit{proximity connected} $2$-center (PCTC) problem. In this problem, we are also given a parameter $\delta\geq 0$ and we have the additional constraint that the distance between the centers of the disks should be at most $\delta$. Note that when $\delta=0$, the PCTC problem is reduced to the $1$-center(minimum enclosing disk) problem and when $\delta$ tends to infinity, it is reduced to the $2$-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 \cite{ACN0}, but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic $O(n^2\log n)$ time algorithm for the problem.
The naive importance sampling (IS) estimator generally does not work well in examples involving simultaneous inference on several targets, as the importance weights can take arbitrarily large values, making the estimator highly unstable. In such situations, alternative multiple IS estimators involving samples from multiple proposal distributions are preferred. Just like the naive IS, the success of these multiple IS estimators crucially depends on the choice of the proposal distributions. The selection of these proposal distributions is the focus of this article. We propose three methods: (i) a geometric space filling approach, (ii) a minimax variance approach, and (iii) a maximum entropy approach. The first two methods are applicable to any IS estimator, whereas the third approach is described in the context of Doss's (2010) two-stage IS estimator. For the first method, we propose a suitable measure of 'closeness' based on the symmetric Kullback-Leibler divergence, while the second and third approaches use estimates of asymptotic variances of Doss's (2010) IS estimator and Geyer's (1994) reverse logistic regression estimator, respectively. Thus, when samples from the proposal distributions are obtained by running Markov chains, we provide consistent spectral variance estimators for these asymptotic variances. The proposed methods for selecting proposal densities are illustrated using various detailed examples.
We study the distributed minimum spanning tree (MST) problem, a fundamental problem in distributed computing. It is well-known that distributed MST can be solved in $\tilde{O}(D+\sqrt{n})$ rounds in the standard CONGEST model (where $n$ is the network size and $D$ is the network diameter) and this is essentially the best possible round complexity (up to logarithmic factors). However, in resource-constrained networks such as ad hoc wireless and sensor networks, nodes spending so much time can lead to significant spending of resources such as energy. Motivated by the above consideration, we study distributed algorithms for MST under the \emph{sleeping model} [Chatterjee et al., PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. In the sleeping model, a node can be in one of two modes in any round -- \emph{sleeping} or \emph{awake} (unlike the traditional model where nodes are always awake). Only the rounds in which a node is \emph{awake} are counted, while \emph{sleeping} rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the \emph{awake complexity} of a distributed algorithm, the worst-case number of rounds any node is awake. We present deterministic and randomized distributed MST algorithms that have an \emph{optimal} awake complexity of $O(\log n)$ time with a matching lower bound. We also show that our randomized awake-optimal algorithm has essentially the best possible round complexity by presenting a lower bound of $\tilde{\Omega}(n)$ on the product of the awake and round complexity of any distributed algorithm (including randomized) that outputs an MST, where $\tilde{\Omega}$ hides a $1/(\text{polylog } n)$ factor.
This paper considers the problem of inference in cluster randomized experiments when cluster sizes are non-ignorable. Here, by a cluster randomized experiment, we mean one in which treatment is assigned at the level of the cluster; by non-ignorable cluster sizes we mean that "large" clusters and "small" clusters may be heterogeneous, and, in particular, the effects of the treatment may vary across clusters of differing sizes. In order to permit this sort of flexibility, we consider a sampling framework in which cluster sizes themselves are random. In this way, our analysis departs from earlier analyses of cluster randomized experiments in which cluster sizes are treated as non-random. We distinguish between two different parameters of interest: the equally-weighted cluster-level average treatment effect, and the size-weighted cluster-level average treatment effect. For each parameter, we provide methods for inference in an asymptotic framework where the number of clusters tends to infinity and treatment is assigned using simple random sampling. We additionally permit the experimenter to sample only a subset of the units within each cluster rather than the entire cluster and demonstrate the implications of such sampling for some commonly used estimators. A small simulation study shows the practical relevance of our theoretical results.
Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.
In the storied Colonel Blotto game, two colonels allocate $a$ and $b$ troops, respectively, to $k$ distinct battlefields. A colonel wins a battle if they assign more troops to that particular battle, and each colonel seeks to maximize their total number of victories. Despite the problem's formulation in 1921, the first polynomial-time algorithm to compute Nash equilibrium (NE) strategies for this game was discovered only quite recently. In 2016, \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} formulated a breakthrough algorithm to compute NE strategies for the Colonel Blotto game\footnote{To the best of our knowledge, the algorithm from \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} has computational complexity $O(k^{14}\max\{a,b\}^{13})$}, receiving substantial media coverage (e.g. \citep{Insider}, \citep{NSF}, \citep{ScienceDaily}). In this work, we present the first known $\epsilon$-approximation algorithm to compute NE strategies in the two-player Colonel Blotto game in runtime $\widetilde{O}(\epsilon^{-4} k^8 \max\{a,b\}^2)$ for arbitrary settings of these parameters. Moreover, this algorithm computes approximate coarse correlated equilibrium strategies in the multiplayer (continuous and discrete) Colonel Blotto game (when there are $\ell > 2$ colonels) with runtime $\widetilde{O}(\ell \epsilon^{-4} k^8 n^2 + \ell^2 \epsilon^{-2} k^3 n (n+k))$, where $n$ is the maximum troop count. Before this work, no polynomial-time algorithm was known to compute exact or approximate equilibrium (in any sense) strategies for multiplayer Colonel Blotto with arbitrary parameters. Our algorithm computes these approximate equilibria by a novel (to the author's knowledge) sampling technique with which we implicitly perform multiplicative weights update over the exponentially many strategies available to each player.
To simulate noisy boson sampling approximating it by only the lower-order multi-boson interferences (e.g., by a smaller number of interfering bosons and classical particles) is very popular idea. I show that the output data from any such classical simulations can be efficiently distinguished from that of the quantum device they try to simulate, even with finite noise in the latter. The distinguishing datasets can be the experimental estimates of some large probabilities, a wide class of such is presented. This is a sequel of \textit{Quantum} \textbf{5}, 423 (2021), where I present more accessible account of the main result enhanced by additional insight on the contribution from the higher-order multi-boson interferences in presence of noise.
The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.
We introduce a novel methodology for particle filtering in dynamical systems where the evolution of the signal of interest is described by a SDE and observations are collected instantaneously at prescribed time instants. The new approach includes the discretisation of the SDE and the design of efficient particle filters for the resulting discrete-time state-space model. The discretisation scheme converges with weak order 1 and it is devised to create a sequential dependence structure along the coordinates of the discrete-time state vector. We introduce a class of space-sequential particle filters that exploits this structure to improve performance when the system dimension is large. This is numerically illustrated by a set of computer simulations for a stochastic Lorenz 96 system with additive noise. The new space-sequential particle filters attain approximately constant estimation errors as the dimension of the Lorenz 96 system is increased, with a computational cost that increases polynomially, rather than exponentially, with the system dimension. Besides the new numerical scheme and particle filters, we provide in this paper a general framework for discrete-time filtering in continuous-time dynamical systems described by a SDE and instantaneous observations. Provided that the SDE is discretised using a weakly-convergent scheme, we prove that the marginal posterior laws of the resulting discrete-time state-space model converge to the posterior marginal posterior laws of the original continuous-time state-space model under a suitably defined metric. This result is general and not restricted to the numerical scheme or particle filters specifically studied in this manuscript.
Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (model evidence), which is computationally challenging, prohibiting its use in many high-dimensional Bayesian inverse problems. With Bayesian imaging applications in mind, in this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models for applications that use images to inform decisions under uncertainty. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving l_1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10^6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices of dictionary and measurement model.
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