We develop minimax optimal risk bounds for the general learning task consisting in predicting as well as the best function in a reference set $\mathcal{G}$ up to the smallest possible additive term, called the convergence rate. When the reference set is finite and when $n$ denotes the size of the training data, we provide minimax convergence rates of the form $C(\frac{\log|\mathcal{G}|}{n})^v$ with tight evaluation of the positive constant $C$ and with exact $0<v\le1$, the latter value depending on the convexity of the loss function and on the level of noise in the output distribution. The risk upper bounds are based on a sequential randomized algorithm, which at each step concentrates on functions having both low risk and low variance with respect to the previous step prediction function. Our analysis puts forward the links between the probabilistic and worst-case viewpoints, and allows to obtain risk bounds unachievable with the standard statistical learning approach. One of the key ideas of this work is to use probabilistic inequalities with respect to appropriate (Gibbs) distributions on the prediction function space instead of using them with respect to the distribution generating the data. The risk lower bounds are based on refinements of the Assouad lemma taking particularly into account the properties of the loss function. Our key example to illustrate the upper and lower bounds is to consider the $L_q$-regression setting for which an exhaustive analysis of the convergence rates is given while $q$ ranges in $[1;+\infty[$.
A parsimonious tail compliant multiscale statistical model for aggregated rainfall
Univ Brest（布雷斯特大学）；CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique（法国国家科学研究中心 UMR 6205，布列塔尼大西洋数学实验室）；Dipartimento di Scienze Ambientali, Informatica e Statistica, Universita Ca’ Foscari di Venezia（威尼斯卡福斯卡里大学环境、信息学与统计学系）；Laboratoire des Sciences du Climat et l’Environnement（气候与环境科学实验室）
Towards a Statistical Understanding of Mixture-of-Experts
Qiuzhen College, Tsinghua University（清华大学求真书院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；East China Normal University（华东师范大学）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学科学与应用研究院）
