标题： 广义信息散度测度的超额最小风险界 显示英文标题

标题： Bounds on the Excess Minimum Risk via Generalized Information Divergence Measures

摘要： 已知有限维随机向量$Y$、$X$和$Z$按顺序构成马尔可夫链（即$Y \to X \to Z$），我们利用广义信息散度测度推导出超额最小风险的上界。 这里， $Y$是目标向量，需从观测特征向量$X$或其随机退化版本$Z$中进行估计。 超额最小风险被定义为从$X$和$Z$估计$Y$时的最小期望损失之间的差值。 我们提出了一组广义的界，这些界推广了 Györfi 等人 (2023) 基于互信息的界，使用了 Rényi 和$\alpha$-Jensen-Shannon 散度，以及 Sibson 的互信息。 我们的界与 Modak 等人 (2021) 和 Aminian 等人 (2024) 为学习算法的泛化误差开发的界类似。 然而，与这些工作不同的是，我们的界不需要子高斯参数为常数，因此适用于$Y$、$X$和$Z$上更广泛的联合分布类。 我们还在常数和非常数子高斯假设下提供了数值例子，说明了对于参数$\alpha$的某些取值范围，基于广义散度的界可以比基于互信息的界更紧。 显示英文摘要

摘要： Given finite-dimensional random vectors $Y$, $X$, and $Z$ that form a Markov chain in that order (i.e., $Y \to X \to Z$), we derive upper bounds on the excess minimum risk using generalized information divergence measures. Here, $Y$ is a target vector to be estimated from an observed feature vector $X$ or its stochastically degraded version $Z$. The excess minimum risk is defined as the difference between the minimum expected loss in estimating $Y$ from $X$ and from $Z$. We present a family of bounds that generalize the mutual information based bound of Gy\"orfi et al. (2023), using the R\'enyi and $\alpha$-Jensen-Shannon divergences, as well as Sibson's mutual information. Our bounds are similar to those developed by Modak et al. (2021) and Aminian et al. (2024) for the generalization error of learning algorithms. However, unlike these works, our bounds do not require the sub-Gaussian parameter to be constant and therefore apply to a broader class of joint distributions over $Y$, $X$, and $Z$. We also provide numerical examples under both constant and non-constant sub-Gaussianity assumptions, illustrating that our generalized divergence based bounds can be tighter than the one based on mutual information for certain regimes of the parameter $\alpha$.