The generalization error (risk) of a supervised statistical learning algorithm quantifies its prediction ability on previously unseen data. Inspired by exponential tilting, Li et al. (2021) proposed the tilted empirical risk as a non-linear risk metric for machine learning applications such as classification and regression problems. In this work, we examine the generalization error of the tilted empirical risk. In particular, we provide uniform and information-theoretic bounds on the tilted generalization error, defined as the difference between the population risk and the tilted empirical risk, with a convergence rate of $O(1/\sqrt{n})$ where $n$ is the number of training samples. Furthermore, we study the solution to the KL-regularized expected tilted empirical risk minimization problem and derive an upper bound on the expected tilted generalization error with a convergence rate of $O(1/n)$.

翻译：监督统计学习算法的泛化误差（风险）量化了其在未见数据上的预测能力。受指数倾斜的启发，Li等人（2021）提出了倾斜经验风险，作为分类和回归等机器学习应用中的一种非线性风险度量。在本工作中，我们研究了倾斜经验风险的泛化误差。具体而言，我们为倾斜泛化误差（定义为总体风险与倾斜经验风险之差）提供了均匀界和信息论界，其收敛速度为$O(1/\sqrt{n})$，其中$n$为训练样本数。此外，我们研究了KL正则化期望倾斜经验风险最小化问题的解，并推导了期望倾斜泛化误差的上界，其收敛速度为$O(1/n)$。