ICON STARF 2024 - SANDAL "ERA Chair in Mathematical Statistics and Data Science for the University of Luxembourg"

International Conference on Statistics and Related Fields (ICON STARF) in honour of Vladimir Koltchinskii

The International Conference on Statistics and Related Fields (ICON STARF) will take place from 3 (arrival date) to 7 June (departure date) 2024 at the Centro Residenziale Universitario di Bertinoro

Programme

TUESDAY 4 JUNE

9:00 Introduction, Yannick Baraud

9:15 Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges, Ming Yuan (Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity of such data provides vast new opportunities for modeling and analysis, but efficiently extracting information content from them, both statistically and computationally, presents unique and fundamental challenges. Addressing these challenges requires an interdisciplinary approach that brings together tools and insights from statistics, optimization and numerical linear algebra among other fields. Despite these hurdles, significant progress has been made in the last decade. In this talk, I will review some of the key advancements and identify common threads among them, under several common statistical settings.)

10:00 Multi-view models and adaptive density estimation under low-rank constraints, Alexandre B Tsybakov (This talk considers the problem of estimating discrete and continuous probability densities under low-rank constraints. For discrete distributions, we assume that the two-dimensional array to estimate is a rank K probability matrix. In the continuous case, we assume that the density with respect to the Lebesgue measure satisfies a generalized multi-view model, meaning that it is HÃder smooth and can be decomposed as a sum of K components, each of which is a product of one-dimensional functions. We propose estimators that achieve, up to logarithmic factors, the minimax optimal convergence rates under such low-rank constraints. In the discrete case, the proposed estimator is adaptive to the rank K. In the continuos case, the estimator is adaptive to the unknown support as well as to the smoothness, and to the unknown number of separable components K. We propose efficient algorithms for computing the estimators. Joint work with Julien Chhor and Olga Klopp)

10:45 Break

11:00 Estimating multiple high-dimensional vector means, Gilles Blanchard (The problem of simultaneously estimating multiple means from independent samples has a long history in statistics, from the seminal works of Stein, Robbins in the 50s, Efron and Morris in the 70s and up to the present day. In this talk I will concentrate on contributions to the high-dimensional case: technical tools used for the results rely on sharp non-asymptotic concentration inequalities in high dimension, for which the contributions of V. Koltchinskii have been fundamental. We consider an aggregation scheme of empirical means of each sample, and study the possible improvement in quadratic risk over the simple empirical means. Full heterogeneity of sample sizes is allowed with zero a priori knowledge of the structure of the mean vectors, nor of the covariances. We focus on the role of the effective dimension of the data in a “dimensional asymptotics” point of view, highlighting that the risk improvement of the proposed method satisfies an oracle inequality approaching an adaptive (minimax) improvement as the effective dimension grows large. This is joint work with Jean-Baptiste Fermanian and Hannah Marienwald)

13:00 Lunch

17:00 Concentration Inequalities and Moment Bounds for Self-Adjoint Operators with Heavy tails, Stanislav Minsker (We present Fuk-Nagaev – type inequality for the sums of independent self-adjoint operators. This bound could be viewed as an extension of the well known “Matrix Bernstein” inequality to the case of operators with heavy-tailed norms. As a corollary, we deduce Rosenthal moment inequality that improves upon the previously known versions even in the scalar case. Finally, we will discuss applications of these bounds to the covariance estimation problem.)

17:45 Rates of convergence for tensor denoising, Sara van de Geer (We consider least squares estimation of a regression function on [0, 1]d with a constraint on its derivatives. Let μ be Lebesgue measure on [0, 1]d and F ⊂ L2(μ) be a class of functions f with ∥Df∥1 := R|Df|dμ ≤ 1, where D :=Qdj=1 ∂m/(∂xj)m. One calls ∥Df∥1 the m-th order Vitali total variation. Let N := {f : Df = 0}. We show that {f ∈ F : f ⊥ N} has – up to logarithmic terms – entropy of the same order as for the case d = 1. This generalizes the entropy result of [1] for the case m = 1 to m ∈ N. We apply the result to the tensor denoising regression problem. The entropy bound implies that that the rate of convergence of the least squares estimator is – up to logarithmic terms – of order n− m 2m+1 in any dimension.)

19:00 Dinner

21:00 A photographic approach to modern statistics, Lucien Birgé (Last Minute Suprise!)

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WEDNESDAY 5 JUNE

9:00 Chernoff’s Method: “Reading Notes for H. Montgomery and A. Odlyzko (1988)” “and S. Chatterjee and P. Diaconis (2014)”, Jon A. Wellner (This talk will first give a brief review of Chernoff’s method, especially the application of Chernoff’s method to randomly weighted sums of Rademacher random variables as studied by H. Montgomery and A. Odlyzko (1988). In brief, this involves two applications of Chernoff’s method. S. Chatterjee and P. Diaconis (2014) used the methods and results of Montgomery to study “Fluctuations of the Bose-Einstein condensate”. This results in (exponential) bounds for the tail probabilities of randomly weighted sums of centered standard exponential random variables. The talk will give an incomplete set of reading notes for the latter paper.)

9:45 Some history and some novel applications of the Dirichlet process, Aad van der Vaart (The Dirichlet process was introduced in the 1970s as a nonparametric prior distribution on the set of probability distributiions. The resulting posterior distribution is closely related to the empirical process and its bootstrap version. We review this connection, some properties and generalisations, and some novel applications in semiparametric estimation.)

10:30 Break

10:45 Nonparametric inference in McKean Vlasov models, Richard Nickl (We consider nonparametric statistical inference on a interaction potential W from noisy discrete space-time measurements of periodic solutions of the nonlinear McKean-Vlasov equation, describing the probability density of the mean field limit of an interacting particle system. We show how Gaussian process priors assigned to W give rise to posterior mean estimators that exhibit ‘fast’ convergence rates for the implied estimated densities towards the ground truths. We further show that if the initial condition is not too smooth and satisfies a standard deconvolvability condition, then one can consistently infer the potential W itself at convergence rates N−θ for appropriate θ>0, where N is the number of measurements. The exponent θ can be taken to approach 1/2 as the regularity of W increases corresponding to ‘near-parametric’ models.)

13:00 Lunch

17:00 Online Policy Learning and Inference by Matrix Completion, Dong Xia (Making online decisions can be challenging when features are sparse and orthogonal to historical ones, especially when the optimal policy is learned through collaborative filtering. We formulate the problem as a matrix completion bandit (MCB), where the expected reward under each arm is characterized by an unknown low-rank matrix. The ε-greedy bandit and the online gradient descent algorithm are explored. Policy learning and regret performance are studied under a specific schedule for exploration probabilities and step sizes. A faster decaying exploration probability yields smaller regret but learns the optimal policy less accurately. We investigate an online debiasing method based on inverse propensity weighting (IPW) and a general framework for online policy inference. The IPW-based estimators are asymptotically normal under mild arm-optimality conditions. Numerical simulations corroborate our theoretical findings. Our methods are applied to the San Francisco parking pricing project data, revealing intriguing discoveries and outperforming the benchmark policy.)

17:45 On the Foundations of Interactive Decision Making, Alexander Rakhlin (We present a general framework for interactive decision making that subsumes multi-armed bandits, contextual bandits, structured bandits, and reinforcement learning. In these settings, the statistician interacts with the environment to collect data and necessarily faces an exploration-exploitation dilemma. We analyze both statistical and algorithmic aspects of the problem.)

19:30 Banquet

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THURSDAY 6 JUNE

9:00 Asymptotical equivalent model for locally stationary time series, Cristina Butucea (Asymptotical equivalence in the sense of Le Cam’s theory proves that two statistical experiments (on different measurable spaces) sharing the same parameter space contain asymptotically the same amount of information on the unknown parameter. It has been established that the statistical experiment given by a sample of a stationary Gaussian process with unknown smooth spectral density is asymptotically equivalent to a Gaussian white noise model where the drift function is the log spectral density (Golubev, Nussbaum, Zhou, 2010). Locally stationary time series can be characterized by a bivariate function with positive values, known as the time-varying spectral density, see e.g. Dahlhaus 1996. We show asymptotic equivalence of a centered Gaussian locally stationary processes with unknown smooth time-varying spectral density and a bi-variate Gaussian white noise model with drift given by the log of the same function. Thus, new methods may be provided for the analysis of locally stationary time series and they can benefit from the theoretical guarantees in the asymptotically equivalent model. This is joint work with Alexander Meister and Angelika Rohde.)

9:45 Inference on the maximal rank of covariance matrices varying continuously in time, Markus Reiss (We study the rank of the instantaneous or spot covariance matrix Σ(t) of a multidimensional process X(t). Given high-frequency observations X(i/n), i = 0, . . . , n, we test the null hypothesis rank(ΣX(t)) ≤ r for all t against local alternatives where the average (r + 1)st eigenvalue is larger than an optimal detection rate vn. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and spectral gap of Σ(t). Explicit matrix perturbation and concentration results provide non-asymptotic uniform critical values and optimality of the signal detection rate vn. Time permitting, we discuss additional observation noise, which shows features of spiked covariance models, but with different non-parametric asymptotics. This is joint work with Lars Winkelmann)

10:30 Break

10:45 High-Order Statistical Expansion in Functional Estimation, Cun-hui Zhang (We study the estimation of a given function of an unknown high-dimensional mean vector based on independent observations. The key element of our approach is a new method which we call High-Order Degenerate Statistical Expansion. It leverages the use of classical multivariate Taylor expansion and degenerate U-statistic and yields an explicit formula. In the univariate case, the formula expresses the error of the proposed estimator as the sum of a Taylor-Hoeffding series and an explicit remainder term in the form of the Riemann-Liouville integral as in the Taylor expansion around the true mean vector. The Taylor-Hoeffding series replaces the power of the average noise in the classical Taylor series by its degenerate version to give a Hoeffding decomposition as a weighted sum of degenerate U-products of the noises. A similar formula holds in general dimension. This makes the proposed method a natural statistical version of the classical Taylor expansion. The proposed estimator can be viewed as a jackknife estimator of the Taylor-Hoeffding series and can be approximated by bootstrap. Thus, the jackknife, bootstrap and Taylor expansion approaches all converge to the proposed estimator. We develop risk bounds for the proposed estimator under proper moment conditions and a central limit theorem under a second moment condition even in expansions of higher than the second order. We apply this new method to several smooth and non-smooth problems under minimum moment constraints. This is joint work with Fan Zhou and Ping Li.)

13:00 Lunch

17:00 session by junior speakers (3 speakers 25 minutes each)

Learning dynamical systems – a transfer-operator approach, Pietro Novelli (Non-linear dynamical systems can be handily described by the associated transfer operator, whose action returns the expected value of every observable of the system forward in time. These operators are instrumental to forecasting and interpreting the system dynamics, and have broad applications in science and engineering. After a gentle introduction to this topic, I’ll present recent work on learning transfer operators. In particular, I’ll describe how we used a statistical learning approach to develop reduced rank estimators, efficiently learn a representation for the dynamics, and justify the use of Nyström sketching techniques in the context of transfer operator learning.)

Contextual continuum bandits: Static versus dynamic regret, Arya Akhavan (In this work, we study the problem of contextual continuum bandits. Under the minimal condition that the objective functions are continuous with respect to the received contexts, we demonstrate that any algorithm achieving a sub-linear static regret can be extended to one achieving a sub-linear dynamic (contextual) regret. Moreover, in the case of strongly convex and smooth functions, inspired by the interior point method and employing self-concordant barriers, we propose an algorithm that achieves a sub-linear dynamic regret. Furthermore, we conclude by providing a minmax lower bound indicating that no algorithm achieves sub-linear dynamic regret over functions lacking continuity with respect to the context variable. Additionally, our lower bound outlines that our proposed approach achieves minimax optimal regret, up to a logarithmic factor, concerning the horizon parameter.)

Robust estimation of a regression function in exponential families, Juntong Chen (In this talk, our aim is to offer a unified treatment of estimating a regression function in one-parameter exponential families. Furthermore, we want to go beyond the common assumption that the true distribution of the data exactly belongs to the statistical model we consider. Our estimation methodology is based on Rho-estimation. Strategies based on a single model and model selection will be discussed. We present non-asymptotic risk bounds for the resulting estimators and explain their robustness with respect to data contamination, the presence of outliers, and model misspecification. To remedy the curse of dimensionality, we handle estimation under structural assumptions on the regression functions. We consider specific models in these cases and derive VC dimension bounds for them. By combining the existing approximation results, we show that, under a suitable parametrization of the exponential family, the rates of convergence we obtain coincide with those derived in the Gaussian regression setting under the same structural assumption. At the end of the talk, we conduct a simulation study to compare the performance of Rho-estimators with the maximum likelihood estimator and median-based ones.)

19:00 Dinner

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FRIDAY 7 JUNE

Departure in the morning