Multiclass classification problems such as image annotation can involve a
large number of classes. In this context, confusion between classes can occur,
and single label classification may be misleading. We provide in the present
paper a general device that, given an unlabeled dataset and a score function
defined as the minimizer of some empirical and convex risk, outputs a set of
class labels, instead of a single one. Interestingly, this procedure does not
require that the unlabeled dataset explores the whole classes. Even more, the
method is calibrated to control the expected size of the output set while
minimizing the classification risk. We show the statistical optimality of the
procedure and establish rates of convergence under the Tsybakov margin
condition. It turns out that these rates are linear on the number of labels. We
apply our methodology to convex aggregation of confidence sets based on the
V-fold cross validation principle also known as the superlearning principle. We
illustrate the numerical performance of the procedure on real data and
demonstrate in particular that with moderate expected size, w.r.t. the number
of labels, the procedure provides significant improvement of the classification
risk.

本文提出了一种在多类分类问题中，通过最小化经验凸风险函数定义的评分函数提供输出类标签集合而不是单一标签的方法，该方法利用凸聚合与 V 折交叉验证原则，具有预期大小的控制和分类风险的最小化性质。

多类别分类的预期尺寸置信区间

Confidence sets with expected sizes for Multiclass Classification

Let $F$ be a finite model of cardinality $M$ and denote by $\operatorname
{conv}(F)$ its convex hull. The problem of convex aggregation is to construct a
procedure having a risk as close as possible to the minimal risk over
$\operatorname {conv}(F)$. Consider the bounded regression model with respect
to the squared risk denoted by $R(\cdot)$. If
${\widehat{f}}_n^{\mathit{ERM-C}}$ denotes the empirical risk minimization
procedure over $\operatorname {conv}(F)$, then we prove that for any $x>0$,
with probability greater than $1-4\exp(-x)$,
\[R({\widehat{f}}_n^{\mathit{ERM-C}})\leq\min_{f\in \operatorname
{conv}(F)}R(f)+c_0\max \biggl(\psi_n^{(C)}(M),\frac{x}{n}\biggr),\] where
$c_0>0$ is an absolute constant and $\psi_n^{(C)}(M)$ is the optimal rate of
convex aggregation defined in (In Computational Learning Theory and Kernel
Machines (COLT-2003) (2003) 303-313 Springer) by $\psi_n^{(C)}(M)=M/n$ when
$M\leq \sqrt{n}$ and $\psi_n^{(C)}(M)=\sqrt{\log (\mathrm{e}M/\sqrt{n})/n}$
when $M>\sqrt{n}$.

本文研究了凸聚合问题，通过考虑经验风险最小化过程，证明了一种风险近似最小风险程序，在概率上具有误差较小的优点，同时给出了最优凸聚合率的详细定义。

凸聚合问题中的经验风险最小化是最优的

Empirical risk minimization is optimal for the convex aggregation  problem

We show that empirical risk minimization procedures and regularized empirical
risk minimization procedures satisfy nonexact oracle inequalities in an
unbounded framework, under the assumption that the class has a subexponential
envelope function. The main novelty, in addition to the boundedness assumption
free setup, is that those inequalities can yield fast rates even in situations
in which exact oracle inequalities only hold with slower rates. We apply these
results to show that procedures based on $\ell_1$ and nuclear norms
regularization functions satisfy oracle inequalities with a residual term that
decreases like $1/n$ for every $L_q$-loss functions ($q\geq2$), while only
assuming that the tail behavior of the input and output variables are well
behaved. In particular, no RIP type of assumption or "incoherence condition"
are needed to obtain fast residual terms in those setups. We also apply these
results to the problems of convex aggregation and model selection.

研究表明，在类具有次指数包络函数的假设下，经验风险最小化程序和正则经验风险最小化程序满足非精确 Oracle 不等式。 该研究将这些结果应用于凸聚合和模型选择问题。

具有次指数包络的类别的普适非精确预言机不等式

General nonexact oracle inequalities for classes with a subexponential  envelope

We study the problem of linear and convex aggregation of $M$ estimators of a
density with respect to the mean squared risk. We provide procedures for linear
and convex aggregation and we prove oracle inequalities for their risks. We
also obtain lower bounds showing that these procedures are rate optimal in a
minimax sense. As an example, we apply general results to aggregation of
multivariate kernel density estimators with different bandwidths. We show that
linear and convex aggregates mimic the kernel oracles in asymptotically exact
sense for a large class of kernels including Gaussian, Silverman's and
Pinsker's ones. We prove that, for Pinsker's kernel, the proposed aggregates
are sharp asymptotically minimax simultaneously over a large scale of Sobolev
classes of densities. Finally, we provide simulations demonstrating performance
of the convex aggregation procedure.

本文研究了密度估计器线性和凸聚合的问题，并提供相应的程序及其风险的奥卡尔不等式，以及相似核函数的线性和凸聚合方案可在 Sobolev 密度族中进行一致最优的最小化。