PAC-Bayes generalisation bounds are derived via change-of-measure inequalities that transfer concentration properties from a reference measure to all posterior measures. The specific choice of change of measure determines the assumptions required on the empirical risk; in particular, the classical Donsker--Varadhan theorem leads to bounds relying on bounded exponential moments. We study change-of-measure inequalities based on \(f\)-divergences, obtained by combining the Legendre transform of \(f\) with the Fenchel--Young inequality. Beyond their intrinsic interest in probability theory, we show how these inequalities are helpful in learning theory and yield PAC-Bayes bounds under tailored assumptions on the empirical risk, thereby extending the range of conditions under which PAC-Bayesian guarantees can be established.

翻译：PAC-贝叶斯泛化界限是通过测度变换不等式推导得出的，这些不等式将参考测度的集中性质传递至所有后验测度。测度变换的具体选择决定了经验风险所需满足的假设；特别地，经典的Donsker-Varadhan定理导出了依赖于有界指数矩的界限。我们研究了基于\(f\)-散度的测度变换不等式，这些不等式通过将\(f\)的勒让德变换与Fenchel-Young不等式结合得到。除了在概率论中的内在意义外，我们还展示了这些不等式在学习理论中的有用性，并能在针对经验风险定制的假设下推导出PAC-贝叶斯界限，从而拓展了可建立PAC-贝叶斯保证的条件范围。