In this paper, we present new high-probability PAC-Bayes bounds for different types of losses. Firstly, for losses with a bounded range, we recover a strengthened version of Catoni's bound that holds uniformly for all parameter values. This leads to new fast-rate and mixed-rate bounds that are interpretable and tighter than previous bounds in the literature. In particular, the fast-rate bound is equivalent to the Seeger--Langford bound. Secondly, for losses with more general tail behaviors, we introduce two new parameter-free bounds: a PAC-Bayes Chernoff analogue when the loss' cumulative generating function is bounded, and a bound when the loss' second moment is bounded. These two bounds are obtained using a new technique based on a discretization of the space of possible events for the ``in probability'' parameter optimization problem. This technique is both simpler and more general than previous approaches optimizing over a grid on the parameters' space. Finally, using a simple technique that is applicable to any existing bound, we extend all previous results to anytime-valid bounds.

翻译：本文针对不同类型的损失函数提出了新的高概率PAC-Bayes界。首先，对于取值范围有界的损失函数，我们恢复了Catoni界的强化版本，该版本对所有参数值一致成立。这导出了新的快速率界和混合率界，这些界比文献中现有界更具可解释性且更紧致。特别地，快速率界等价于Seeger-Langford界。其次，针对具有更一般尾部行为的损失函数，我们引入了两个新的无参数界：当损失的累积生成函数有界时的PAC-Bayes切尔诺夫类比界，以及当损失的二阶矩有界时的界。这两个界是通过一种基于“概率内”参数优化问题可能事件空间离散化的新技术获得的。该技术比以往在参数空间网格上进行优化的方法更简单且更通用。最后，通过一种适用于任何现有界的简单技术，我们将所有先前结果扩展为任意时间有效界。