Invited 03

## Potential Theory for Non-local Operators and Jump Processes (Organizer: Panki Kim)

Conference
10:30 PM — 11:00 PM KST
Local
Jul 21 Wed, 9:30 AM — 10:00 AM EDT

### SDEs driven by multiplicative stable-like Levy processes

Zhen-Qing Chen (University of Washington)

6
In this talk, I will present results on weak as well as strong well-poshness results for solutions to time-inhomogeneous SDEs driven by stable-like Levy processes with Holder continuous coefficients. The Levy measure of the Levy process can be anisotropic and singular with respect to the Lebesgue measure on R^d and its support can be a proper subset of R^d.
Based on joint work with Xicheng Zhang and Guohuan Zhao.

### Periodic homogenization of non-symmetric Lévy-type processes

Takashi Kumagai (Kyoto University)

6

### Optimal Hardy identities and inequalites for the fractional Laplacian on $L^p$

Krzysztof Bogdan (Wrocław University of Science and Technology)

6
We will present a route from symmetric Markovian semigroups to Hardy inequalities, to nonexplosion and contractivity results for Feynman-Kac semigroups on $L^p$. We will focus on the fractional Laplacian on $\mathbb{R}^d$, in which case the constants, estimates of the Feynman-Kac semigroups and tresholds for contractivity and explosion are sharp. Namely we will discuss selected results from joint work with Bartłomiej Dyda, Tomasz Grzywny, Tomasz Jakubowski, Panki Kim, Julia Lenczewska, Katarzyna Pietruska-Pałuba and Dominika Pilarczyk (see arXiv).

### Q&A for Invited Session 03

0
This talk does not have an abstract.

###### Session Chair

Panki Kim (Seoul National University)

Invited 10

## Change-point Problems for Complex Data (Organizer: Claudia Kirch)

Conference
10:30 PM — 11:00 PM KST
Local
Jul 21 Wed, 9:30 AM — 10:00 AM EDT

### Two-sample tests for relevant differences in the eigenfunctions of covariance operators

Alexander Aue (University of California at Davis)

4
This talk deals with two-sample tests for functional time series data, which have become widely available in conjunction with the advent of modern complex observation systems. Here, particular interest is in evaluating whether two sets of functional time series observations share the shape of their primary modes of variation as encoded by the eigenfunctions of the respective covariance operators. To this end, a novel testing approach is introduced that connects with, and extends, existing literature in two main ways. First, tests are set up in the relevant testing framework, where interest is not in testing an exact null hypothesis but rather in detecting deviations deemed sufficiently relevant, with relevance determined by the practitioner and perhaps guided by domain experts. Second, the proposed test statistics rely on a self-normalization principle that helps to avoid the notoriously difficult task of estimating the long-run covariance structure of the underlying functional time series. The main theoretical result of this paper is the derivation of the large-sample behavior of the proposed test statistics. Empirical evidence, indicating that the proposed procedures work well in finite samples and compare favorably with competing methods, is provided through a simulation study, and an application to annual temperature data.

### Multiple change point detection under serial dependence

Haeran Cho (University of Bristol)

5
We propose a methodology for detecting multiple change points in the mean of an otherwise stationary, autocorrelated, linear time series. It combines solution path generation based on the wild energy maximisation principle, and an information criterion-based model selection strategy termed gappy Schwarz criterion. The former is well-suited to separating shifts in the mean from fluctuations due to serial correlations, while the latter simultaneously estimates the dependence structure and the number of change points without performing the difficult task of estimating the level of the noise as quantified e.g. by the long-run variance. We provide modular investigation into their theoretical properties and show that the combined methodology, named WEM.gSC, achieves consistency in estimating both the total number and the locations of the change points. The good performance of WEM.gSC is demonstrated via extensive simulation studies, and we further illustrate its usefulness by applying the methodology to London air quality data.

### An asymptotic test for constancy of the variance in a time series

Herold Dehling (Ruhr-University Bochum)

6
We present a novel approach to test for heteroscedasticity of a non-stationary time series that is based on Gini's mean difference of logarithmic local sample variances. In order to analyse the large sample behaviour of our test statistic, we establish new limit theorems for U-statistics of dependent triangular arrays. We derive the asymptotic distribution of the test statistic under the the null hypothesis of a constant variance and show that the test is consistent against a large class of alternatives, including multiple structural breaks in the variance. Our test is applicable even in the case of non-stationary processes, assuming a locally varying mean function. The performance of the test and its comparatively low computation time are illustrated in an extensive simulation study.

### Q&A for Invited Session 10

0
This talk does not have an abstract.

###### Session Chair

Claudia Kirch (Otto von Guericke University Magdeburg)

Invited 12

## Statistics for Data with Geometric Structure (Organizer: Sungkyu Jung)

Conference
10:30 PM — 11:00 PM KST
Local
Jul 21 Wed, 9:30 AM — 10:00 AM EDT

### Wasserstein regression

Hans-Georg Müller (University of California, Davis)

4
The analysis of samples of random objects that do not lie in a vector space has found increasing attention in statistics in recent years. An important class of such object data are univariate probability measures defined on the real line. Adopting the Wasserstein metric, we develop a class of regression models for data that include random distributions as predictors and distributions or scalars as responses. To study these regression models, we utilize the geometry of tangent bundles of the metric space of random measures with the Wasserstein metric and derive asymptotic rates of convergence for estimators of the regression coefficient function and for predicted distributions. We also study an extension to autoregressive models for distribution-valued time series. The proposed methods are illustrated with data that include distributional components in various regression settings.

### Finite sample smeariness for Fréchet means

Stephan Huckemann (Georg-August-Universitaet Goettingen)

4
It is well known for the Euclidean setting that a variety of statistical asymptotic tests, e.g. T-tests or MANOVA, are robust under nonnormality. It is much less known, that this cannot be taken for granted, for similar tests based on manifolds data, in particular for data on compact spaces. The reason lies in a recently discovered phenomenon: Smeariness lowers the classical square-root-of-n-rate for Fréchet means. While true smeariness is only present for a nullset of most parametric families, it surfaces in a finite sample regime for a large class of distributions: For instance, all nontrivial distributions on spheres are affected and all distributions on circles whose support extends beyond a half circle, like, e.g. all Fisher-von-Mises distributions. We give finite sample smeariness a precise definition and illustrate some effects in theory and practice. In particular, the presence of finite sample smeariness renders tests based on quantiles of asymptotic distributions ineffective up to considerably high sample sizes. Suitably designed bootstrap tests remain valid, however.

### Score matching for microbiome compositional data

Janice Scealy (Australian National University)

2
Compositional data and multivariate count data with known totals are challenging to analyse due to the non-negativity and sum constraint on the sample space. It is often the case with microbiome compositional data that many of the components are highly right-skewed, with large numbers of zeros. A major limitation of currently available estimators for compositional models is that they either cannot handle many zeros in the data or are not computationally feasible in moderate to high dimensions. We derive a new set of novel score matching estimators applicable to distributions on a Riemannian manifold with boundary, of which the standard simplex is a special case. The score matching method is applied to estimate the parameters in a new flexible model for compositional data and we show that the estimators are scalable and available in closed form. We apply the new model and estimators to real microbiome compositional data and show that the model provides a good fit to the data.

### Q&A for Invited Session 12

0
This talk does not have an abstract.

###### Session Chair

Sungkyu Jung (Seoul National University)

Invited 25

## Random Graphs (Organizer: Christina Goldschmidt)

Conference
10:30 PM — 11:00 PM KST
Local
Jul 21 Wed, 9:30 AM — 10:00 AM EDT

### An unexpected phase transition for percolation on scale-free networks

Souvik Dhara (Massachusetts Institute of Technology)

5
The talk concerns the critical behavior for percolation on finite, inhomogeneous random networks, where the weights of the vertices follow a power-law distribution with exponent $\tau \in (2,3)$. Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero, as the network-size becomes large. We identify the critical window for percolation phase transition. Rather surprisingly, the critical window turns out to be of finite length, which is in sharp contrast with the previously studied critical behaviors for$\tau \in (3,4)$ and $\tau >4$ regimes. The rescaled vector of maximum component sizes are shown to converge in distribution to an infinite vector of non-degenerate random variables that can be described in terms of components of a one-dimensional inhomogeneous percolation model studied in a seminal work by Durrett and Kesten (1990).

Based on joint work with Shankar Bhamidi, Remco van der Hofstad.

### Recent results for the graph alignment problem

Marc Lelarge (INRIA)

3
Random graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This can be viewed as an average-case and noisy version of the well-known NP-hard graph isomorphism problem. For the correlated Erdös-Rényi model, we give an impossibility result for partial recovery in the sparse regime. We also propose a machine learning approach to solve the problem and design a new graph neural network architecture showing great performances.

### Local law and Tracy-Widom limit for sparse stochastic block models

Ji Oon Lee (Korea Advanced Institute of Science and Technology (KAIST))

5
We consider the spectral properties of sparse stochastic block models, where N vertices are partitioned into K balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar order, we prove a local semicircle law up to the spectral edges, with an explicit formula on the deterministic shift of the spectral edge. We also prove that the fluctuation of the extremal eigenvalues is given by the GOE Tracy-Widom law after rescaling and centering the entries of sparse stochastic block models. Applying the result to sparse stochastic block models, we rigorously prove that there is a large gap between the outliers and the spectral edge without centering.

### Q&A for Invited Session 25

0
This talk does not have an abstract.

###### Session Chair

Christina Goldschmidt (University of Oxford)

Invited 36

## Problems and Approaches in Multi-Armed Bandits (Organizer: Vianney Perchet)

Conference
10:30 PM — 11:00 PM KST
Local
Jul 21 Wed, 9:30 AM — 10:00 AM EDT

### Dynamic pricing and learning under the Bass model

Shipra Agrawal (Columbia University)

1
We consider a novel formulation of the dynamic pricing and demand learning problem, where the evolution of demand in response to posted prices is governed by a stochastic variant of the popular Bass model with parameters (α, β) that are linked to the so-called "innovation" and "imitation" effects. Unlike the more commonly used i.i.d. demand models, in this model the price posted not only affects the demand and the revenue in the current round but also the evolution of demand, and hence the fraction of market potential that can be captured, in future rounds. Finding a revenue-maximizing dynamic pricing policy in this model is non-trivial even when model parameters are known, and requires solving for the optimal non-stationary policy of a continuous-time, continuous-state MDP. In this paper, we consider the problem of dynamic pricing is used in conjunction with learning the model parameters, with the objective of optimizing the cumulative revenues over a given selling horizon. Our main contribution is an algorithm with a regret guarantee of O (m^2/3), where m is mnemonic for the (known) market size. Moreover, we show that no algorithm can incur smaller order of loss by deriving a matching lower bound. We observe that in this problem the market size m, and not the time horizon T, is the fundamental driver of the complexity; our lower bound in fact indicates that for any fixed α,β, most non-trivial instances of the problem have constant T and large m. This insight sets the problem setting considered here uniquely apart from the MAB type formulations typically considered in the learning to price literature. Keywords: Dynamic Pricing, Multi-armed bandits, Bass model

### TensorPlan: A new, flexible, scalable and provably efficient local planner for huge MDPs

Csaba Szepesvari (Deepmind & University of Alberta)

1
In this talk I will consider provably efficient planning in huge MDPs when the planner is helped with a hint about the form of the optimal value function. In particular, a thoughtful oracle provides the planner with basis functions the linear combination of which give the optimal value function either exactly, or with small errors. The problem is to design a local planner, which, similarly to model-predictive control, is called to find a good action after every state transition, while it is given access to a simulator. We propose a new planner which when used continuously is guaranteed to induce a near-optimal policy. When the number of action is kept as a constant, the planner is shown to require only polynomially many simulator queries as a function of the horizon and the number of basis functions. The planner does not use dynamic programming as we know it, but is based on optimism and the "tensorization" of the Bellman optimality equation.

### On the importance of (linear) structure in contextual multi-armed bandit

Alessandro Lazaric (Facebook AI Research)

1
In this talk I will discuss how structural assumptions on the reward function impacts the regret performance of bandit algorithms. Notably, I will focus on linear contextual bandits and first review recent results showing how the structure of the arm set and reward function can be leveraged to achieve improved regret guarantees. Then, I will describe a novel incremental algorithm able to achieve asymptotic optimality, while ensuring finite-time worst-case optimality in the context-free case. Finally, I will discuss how stronger assumptions on context distribution and linear representation may be leveraged to achieve constant regret. This eventually leads to a representation-selection algorithm matching the regret of the best linear representation in a given set, up to a logarithmic factor in the number of representations.

Most relevant references:
T. Lattimore, Cs. Szepesvari. "The End of Optimism? An Asymptotic Analysis of Finite-Armed Linear Bandits", 2016.
B. Hao, T. Lattimore, Cs. Szepesvari, "Adaptive Exploration in Linear Contextual Bandit", 2019.
A. Tirinzoni, M. Pirotta, M. Restelli, A. Lazaric, "An Asymptotically Optimal Primal-Dual Incremental Algorithm for Contextual Linear Bandits", 2020.
M. Papini, A. Tirinzoni, M. Restelli, A. Lazaric, M. Pirotta, "Leveraging Good Representations in Linear Contextual Bandits", 2021.

### Q&A for Invited Session 36

0
This talk does not have an abstract.

###### Session Chair

Vianney Perchet (École nationale de la statistique et de l'administration économique Paris)

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