We present a new high-probability PAC-Bayes oracle bound for unbounded
losses. This result can be understood as a PAC-Bayes version of the Chernoff
bound. The proof technique relies on uniformly bounding the tail of certain
random variable based on the Cram\'er transform of the loss. We highlight two
applications of our main result. First, we show that our bound solves the open
problem of optimizing the free parameter on many PAC-Bayes bounds. Finally, we
show that our approach allows working with flexible assumptions on the loss
function, resulting in novel bounds that generalize previous ones and can be
minimized to obtain Gibbs-like posteriors.

我们提出了一个新的高概率 PAC-Bayes 预言界，用于无界损失。该结果可以理解为 Chernoff 界的 PAC-Bayes 版本。证明技巧依赖于基于损失的 Cramé 变换来统一地对某些随机变量的尾部进行界定。我们强调了我们主要结果的两个应用。首先，我们展示了我们的界解决了许多 PAC-Bayes 界上优化自由参数的开放问题。最后，我们展示了我们的方法允许在损失函数上进行灵活的假设，从而得到泛化之前界的新界，并且可以最小化以获取类似于 Gibbs 的后验概率。

对无界损失的 PAC-Bayes-Chernoff 界限

PAC-Bayes-Chernoff bounds for unbounded losses

This paper discusses the problem of adaptive estimation of a univariate
object like the value of a regression function at a given point or a linear
functional in a linear inverse problem. We consider an adaptive procedure
originated from Lepski [Theory Probab. Appl. 35 (1990) 454--466.] that selects
in a data-driven way one estimate out of a given class of estimates ordered by
their variability. A serious problem with using this and similar procedures is
the choice of some tuning parameters like thresholds. Numerical results show
that the theoretically recommended proposals appear to be too conservative and
lead to a strong oversmoothing effect. A careful choice of the parameters of
the procedure is extremely important for getting the reasonable quality of
estimation. The main contribution of this paper is the new approach for
choosing the parameters of the procedure by providing the prescribed behavior
of the resulting estimate in the simple parametric situation. We establish a
non-asymptotical "oracle" bound, which shows that the estimation risk is, up to
a logarithmic multiplier, equal to the risk of the "oracle" estimate that is
optimally selected from the given family. A numerical study demonstrates a good
performance of the resulting procedure in a number of simulated examples.

本文提出了使用新方法来选择 Lepski 程序的参数，以获得合理的估计质量，通过建立非渐近的 “Oracle” 界限，证明了估计风险与最优选择的 “Oracle” 估计风险相等。