We introduce a new PAC-Bayes oracle bound for unbounded losses. This result can be understood as a PAC-Bayesian version of the Cram\'er-Chernoff bound. The proof technique relies on controlling the tails of certain random variables involving the Cram\'er transform of the loss. We highlight several applications of the main theorem. First, we show that our result naturally allows exact optimization of the free parameter on many PAC-Bayes bounds. Second, we recover and generalize previous results. Finally, we show that our approach allows working with richer assumptions that result in more informative and potentially tighter bounds. In this direction, we provide a general bound under a new ``model-dependent bounded CGF" assumption from which we obtain bounds based on parameter norms and log-Sobolev inequalities. All these bounds can be minimized to obtain novel posteriors.

翻译：我们提出了一个新的针对无界损失的 PAC-Bayes 预报界。该结果可视为 Cramér-Chernoff 界的 PAC-Bayesian 版本。证明技巧依赖于控制涉及损失 Cramér 变换的某些随机变量的尾部行为。我们强调了主要定理的若干应用。首先，我们证明该结果能自然地实现对许多 PAC-Bayes 界中自由参数的精确优化。其次，我们恢复并推广了先前的结果。最后，我们证明该方法允许在更丰富的假设下工作，从而得到信息量更大且可能更紧的界。在这一方向上，我们基于一个新的“模型依赖有界 CGF”假设给出了一个通用界，并由此得到了基于参数范数和对数 Sobolev 不等式的界。所有这些界均可通过最小化来获得新的后验分布。