This paper studies the truncation method from Alquier [1] to derive
high-probability PAC-Bayes bounds for unbounded losses with heavy tails.
Assuming that the $p$-th moment is bounded, the resulting bounds interpolate
between a slow rate $1 / \sqrt{n}$ when $p=2$, and a fast rate $1 / n$ when $p
\to \infty$ and the loss is essentially bounded. Moreover, the paper derives a
high-probability PAC-Bayes bound for losses with a bounded variance. This bound
has an exponentially better dependence on the confidence parameter and the
dependency measure than previous bounds in the literature. Finally, the paper
extends all results to guarantees in expectation and single-draw PAC-Bayes. In
order to so, it obtains analogues of the PAC-Bayes fast rate bound for bounded
losses from [2] in these settings.

本文研究了来自 Alquier [1] 的截断方法，以推导具有重尾的无界损失的高概率 PAC-Bayes 边界。假设第 p 阶矩有界，则得到的边界在 p=2 时呈现慢速率 1/sqrt (n)，在 p→∞且损失基本有界时呈现快速率 1/n。此外，本文还推导了具有有界方差的损失的高概率 PAC-Bayes 边界，其对置信参数和依赖度量的依赖关系与文献中的之前边界有指数级的改进。最后，本文将所有结果推广到期望保证和单次抽取的 PAC-Bayes 中，并以此获得了 [2] 中有界损失的 PAC-Bayes 快速率边界在这些设置中的类比。

论具有有限矩的损失函数的泛化界注记

A note on generalization bounds for losses with finite moments

Many deep generative models are defined as a push-forward of a Gaussian
measure by a continuous generator, such as Generative Adversarial Networks
(GANs) or Variational Auto-Encoders (VAEs). This work explores the latent space
of such deep generative models. A key issue with these models is their tendency
to output samples outside of the support of the target distribution when
learning disconnected distributions. We investigate the relationship between
the performance of these models and the geometry of their latent space.
Building on recent developments in geometric measure theory, we prove a
sufficient condition for optimality in the case where the dimension of the
latent space is larger than the number of modes. Through experiments on GANs,
we demonstrate the validity of our theoretical results and gain new insights
into the latent space geometry of these models. Additionally, we propose a
truncation method that enforces a simplicial cluster structure in the latent
space and improves the performance of GANs.

研究深度生成模型的潜在空间几何结构及优化条件，通过实验提出限制方法，提高生成对抗网络的性能。