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.

翻译：本文研究了Alquier [1]中的截断方法，用于推导重尾无界损失函数的高概率PAC-Bayes界。假设$p$阶矩有界，所得界在$p=2$时的慢速$1 / \sqrt{n}$与$p \to \infty$且损失本质有界时的快速$1 / n$之间插值。此外，本文推导了具有有界方差的损失函数的高概率PAC-Bayes界。该界对置信参数和依赖度量的依赖程度比文献中先前的界指数级更优。最后，本文将所有结果推广至期望保证和单次抽样PAC-Bayes。为此，本文获得了文献[2]中有界损失函数的PAC-Bayes快速率界在这些设定下的对应形式。