arXiv subjects

Gennady Samorodnitsky

Publications and source records attributed to Gennady Samorodnitsky.

At least 37 records · Page 2Linked to original sources

arXiv2022

On Penalization in Stochastic Multi-armed Bandits

We study an important variant of the stochastic multi-armed bandit (MAB) problem, which takes penalization into consideration. Instead of directly maximizing cumulative expected reward, we need to balance between the total reward and fairness level. In this paper, we present some new insights in MAB and formulate the problem in the penalization framework, where rigorous penalized regret can be well defined and more sophisticated regret analysis is possible. Under such a framework, we propose a hard-threshold UCB-like algorithm, which enjoys many merits including asymptotic fairness, nearly optimal regret, better tradeoff between reward and fairness. Both gap-dependent and gap-independent regret bounds have been established. Multiple insightful comments are given to illustrate the soundness of our theoretical analysis. Numerous experimental results corroborate the theory and show the superiority of our method over other existing methods.

stat.ML↗

arXiv2022

Catoni-style Confidence Sequences under Infinite Variance

In this paper, we provide an extension of confidence sequences for settings where the variance of the data-generating distribution does not exist or is infinite. Confidence sequences furnish confidence intervals that are valid at arbitrary data-dependent stopping times, naturally having a wide range of applications. We first establish a lower bound for the width of the Catoni-style confidence sequences for the finite variance case to highlight the looseness of the existing results. Next, we derive tight Catoni-style confidence sequences for data distributions having a relaxed bounded~$p^{th}-$moment, where~$p \in (1,2]$, and strengthen the results for the finite variance case of~$p =2$. The derived results are shown to better than confidence sequences obtained using Dubins-Savage inequality.

math.ST↗

arXiv2022

Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes

We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erdös-Rényi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices at the critical dimension and below. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

math.PR↗

arXiv2022

Cauchy, normal and correlations versus heavy tails

A surprising result of Pillai and Meng (2016) showed that a transformation $\sum_{j=1}^n w_j X_j/Y_j$ of two iid centered normal random vectors, $(X_1,\ldots, X_n)$ and $(Y_1,\ldots, Y_n)$, $n>1$, for any weights $0\leq w_j\leq 1$, $ j=1,\ldots, n$, $\sum_{j=1}^n w_j=1$, has a Cauchy distribution regardless of any correlations within the normal vectors. The correlations appear to lose out in the competition with the heavy tails. To clarify how extensive this phenomenon is, we analyze two other transformations of two iid centered normal random vectors. These transformations are similar in spirit to the transformation considered by Pillai and Meng (2016). One transformation involves absolute values: $\sum_{j=1}^n w_j X_j/|Y_j|$. The second involves randomly stopped Brownian motions: $\sum_{j=1}^n w_j X_j\bigl(Y_j^{-2}\bigr)$, where $\bigl\{\bigl( X_1(t),\ldots, X_n(t)\bigr), \, t\geq 0\bigr\},\ n>1,$ is a Brownian motion with positive variances; $(Y_1,\ldots, Y_n)$ is a centered normal random vector with the same law as $( X_1(1),\ldots, X_n(1))$ and independent of it; and $X(Y^{-2})$ is the value of the Brownian motion $X(t)$ evaluated at the random time $t=Y^{-2}$. All three transformations result in a Cauchy distribution if the covariance matrix of the normal components is diagonal, or if all the correlations implied by the covariance matrix equal 1. However, while the transformation Pillai and Meng (2016) considered produces a Cauchy distribution regardless of the normal covariance matrix. the transformations we consider here do not always produce a Cauchy distribution. The correlations between jointly normal random variables are not always overwhelmed by the heaviness of the marginal tails. The mysteries of the connections between normal and Cauchy laws remain to be understood.

math.PR↗

arXiv2021

Extreme Bandits using Robust Statistics

We consider a multi-armed bandit problem motivated by situations where only the extreme values, as opposed to expected values in the classical bandit setting, are of interest. We propose distribution free algorithms using robust statistics and characterize the statistical properties. We show that the provided algorithms achieve vanishing extremal regret under weaker conditions than existing algorithms. Performance of the algorithms is demonstrated for the finite-sample setting using numerical experiments. The results show superior performance of the proposed algorithms compared to the well known algorithms.

stat.ML↗

arXiv2021

A New Shape of Extremal Clusters For Certain Stationary Semi-Exponential Processes With Moderate Long Range Dependence

Extremal clusters of stationary processes with long memory can be quite intricate. For certain stationary infinitely divisible processes with subexponential tails, including both power-like tails and certain lighter tails, e.g. lognormal-like tails, such clusters may take the shape of stable regenerative sets. In this paper we show that for semi-exponential tails, which are even lighter, a new shape of extremal clusters arises. In this case each stable regenerative set supports a random panoply of varying extremes.

math.PR↗

arXiv2021

Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex

Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually governed by a single parameter. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex and, more importantly, of its dynamic analogue that we introduce here. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. The dynamic versions of the clique complex and the Linial-Meshulum complex are special cases of our setup. Our key result concerns the regime where face-counts of a particular dimension dominate. We show that the Betti numbers corresponding to this dimension and the Euler characteristic satisfy functional strong law of large numbers and functional central limit theorems. Surprisingly, in the latter result, the limiting Gaussian process depends only upon the dynamics in the smallest non-trivial dimension.

math.PR↗

arXiv2020

Extremal clustering under moderate long range dependence and moderately heavy tails

We study clustering of the extremes in a stationary sequence with subexponential tails in the maximum domain of attraction of the Gumbel We obtain functional limit theorems in the space of random sup-measures and in the space $D(0,\infty)$. The limits have the Gumbel distribution if the memory is only moderately long. However, as our results demonstrate rather strikingly, the "heuristic of a single big jump" could fail even in a moderately long range dependence setting. As the tails become lighter, the extremal behavior of a stationary process may depend on multiple large values of the driving noise.

math.PR↗

arXiv2019

High minima of non-smooth Gaussian processes

In this short note we study the asymptotic behaviour of the minima over compact intervals of Gaussian processes, whose paths are not necessarily smooth. We show that, beyond the logarithmic large deviation Gaussian estimates, this problem is closely related to the classical small-ball problem. Under certain conditions we estimate the term describing the correction to the large deviation behaviour. In addition, the asymptotic distribution of the location of the minimum, conditionally on the minimum exceeding a high threshold, is also studied.

math.PR↗

arXiv2019

From infinite urn schemes to self-similar stable processes

We investigate the randomized Karlin model with parameter $β\in(0,1)$, which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index $β/2\in(0,1/2)$. We show here that when the randomization is heavy-tailed with index $α\in(0,2)$, then the odd-occupancy process scales to a $(β/α)$-self-similar symmetric $α$-stable process with stationary increments.

math.PR↗

arXiv2018

Extreme Value Analysis Without the Largest Values: What Can Be Done?

In this paper we are concerned with the analysis of heavy-tailed data when a portion of the extreme values is unavailable. This research was motivated by an analysis of the degree distributions in a large social network. The degree distributions of such networks tend to have power law behavior in the tails. We focus on the Hill estimator, which plays a starring role in heavy-tailed modeling. The Hill estimator for this data exhibited a smooth and increasing "sample path" as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation we introduce a new version of the Hill estimator. It is a function of the number of the upper order statistics used in the estimation, but also depends on the number of unavailable extreme values. We establish functional convergence of the normalized Hill estimator to a Gaussian process. An estimation procedure is developed based on the limit theory to estimate the number of missing extremes and extreme value parameters including the tail index and the bias of Hill's estimator. We illustrate how this approach works in both simulations and real data examples.

math.ST↗

arXiv2018

Distance covariance for discretized stochastic processes

Given an iid sequence of pairs of stochastic processes on the unit interval we construct a measure of independence for the components of the pairs. We define distance covariance and distance correlation based on approximations of the component processes at finitely many discretization points. Assuming that the mesh of the discretization converges to zero as a suitable function of the sample size, we show that the sample distance covariance and correlation converge to limits which are zero if and only if the component processes are independent. To construct a test for independence of the discretized component processes we show consistency of the bootstrap for the corresponding sample distance covariance/correlation.

math.ST↗

arXiv2018

Extremal Value Theory for Long Range Dependent Stable Random Fields

We study the extremes for a class of a symmetric stable random fields with long range dependence. We prove functional extremal theorems both in the space of sup measures and in the space of cadlag functions of several variables. The limits in both types of theorems are of a new kind, and only in a certain range of parameters these limits have the Fréchet distribution.

math.PR↗

arXiv2018

Regularly Varying Random Fields

We study the extremes of multivariate regularly varying random fields. The crucial tools in our study are the tail field and the spectral field, notions that extend the tail and spectral processes of Basrak and Segers (2009). The spatial context requires multiple notions of extremal index, and the tail and spectral fields are applied to clarify these notions and other aspects of extremal clusters. An important application of the techniques we develop is to the Brown-Resnick random fields.

math.PR↗

arXiv2018

Extremal theory for long range dependent infinitely divisible processes

We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $α$-Fréchet distribution and the skewed $α$-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters $α\in(0,\infty)$ and $β\in(0,1)$, with representations based on intersections of independent $β$-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index $-α$. The intriguing structure of these random sup-measures is due to intersections of independent $β$-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as $β$ increases to one. The results in this paper extend substantially previous investigations where only $α\in(0,2)$ and $β\in(0,1/2)$ have been considered.

math.PR↗

arXiv2017

Asymptotic behaviour of high Gaussian minima

We investigate what happens when an entire sample path of a smooth Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high level is exceeded, the conditional shape of the process above the high level, and the location of the minimum of the process given that the sample path is above a high level.

math.PR↗

arXiv2017

Discrete Extremes

Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data. Extreme values of a random variable $X$ are commonly modeled using the generalized Pareto distribution, a method that often gives good results in practice. When $X$ is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively. Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births.

math.ST↗

arXiv2017

Distance covariance for stochastic processes

The distance covariance of two random vectors is a measure of their dependence. The empirical distance covariance and correlation can be used as statistical tools for testing whether two random vectors are independent. We propose an analogs of the distance covariance for two stochastic processes defined on some interval. Their empirical analogs can be used to test the independence of two processes.

math.ST↗