We derive generic information-theoretic and PAC-Bayesian generalization
bounds involving an arbitrary convex comparator function, which measures the
discrepancy between the training and population loss. The bounds hold under the
assumption that the cumulant-generating function (CGF) of the comparator is
upper-bounded by the corresponding CGF within a family of bounding
distributions. We show that the tightest possible bound is obtained with the
comparator being the convex conjugate of the CGF of the bounding distribution,
also known as the Cram\'er function. This conclusion applies more broadly to
generalization bounds with a similar structure. This confirms the
near-optimality of known bounds for bounded and sub-Gaussian losses and leads
to novel bounds under other bounding distributions.

通过对任意凸比较函数进行信息论和 PAC-Bayesian 广义泛化界推导，我们得到泛化界，这些界利用该函数衡量训练和总体损失之间的差异。当比较函数是边界分布的 CGF 凸共轭时，最紧密的界得以实现，这也适用于结构类似的广义泛化界。这证实了对于有界和亚高斯损失的已知界的近最优性，并得出了关于其他边界分布的新界。

比较泛化界中的比较器

Comparing Comparators in Generalization Bounds

This paper describes a new form of unsupervised learning, whose input is a
set of unlabeled points that are assumed to be local maxima of an unknown value
function v in an unknown subset of the vector space. Two functions are learned:
(i) a set indicator c, which is a binary classifier, and (ii) a comparator
function h that given two nearby samples, predicts which sample has the higher
value of the unknown function v. Loss terms are used to ensure that all
training samples x are a local maxima of v, according to h and satisfy c(x)=1.
Therefore, c and h provide training signals to each other: a point x' in the
vicinity of x satisfies c(x)=-1 or is deemed by h to be lower in value than x.
We present an algorithm, show an example where it is more efficient to use
local maxima as an indicator function than to employ conventional
classification, and derive a suitable generalization bound. Our experiments
show that the method is able to outperform one-class classification algorithms
in the task of anomaly detection and also provide an additional signal that is
extracted in a completely unsupervised way.

本文介绍了一种新的无监督学习方法，用于处理无标记点集，并通过比较函数和判别器函数来提供互补的训练信号，使训练样本 x 成为未知函数 v 在局部最大值，实现异常检测任务并以完全无监督的方式提取了其他信号。