Efron-Stein PAC-Bayesian不等式
Efron-Stein PAC-Bayesian Inequalities
摘要：
证明了随机变量为独立随机变量的可能非线性函数的半经验集中不等式。这些不等式描述的浓度的随机变量的数据/分布相关的Efron-Stein（ES）估计的方差，他们不需要任何额外的假设的时刻。特别是，这使我们能够陈述无界随机变量一般函数的半经验伯恩斯坦型不等式，这为相关方法（例如有界差）可能应用起来更具挑战性的情况提供了用户友好的浓度界限。我们将这些结果扩展到Efron-Stein PAC-贝叶斯不等式，该不等式适用于任意概率核，定义了一个随机的，数据相关的函数选择。最后，我们展示了一些应用程序，包括PAC贝叶斯推广界的无界损失函数，经验伯恩斯坦型推广界，新的截断自由界的政策外评估加权重要性采样（WIS），和政策外PAC贝叶斯学习与WIS。
英文摘要：
We prove semi-empirical concentration inequalities for random variables which are given as possibly nonlinear functions of independent random variables. These inequalities describe concentration of random variable in terms of the data/distribution-dependent Efron-Stein (ES) estimate of its variance and they do not require any additional assumptions on the moments. In particular, this allows us to state semi-empirical Bernstein type inequalities for general functions of unbounded random variables, which gives user-friendly concentration bounds for cases where related methods (e.g. bounded differences) might be more challenging to apply. We extend these results to Efron-Stein PAC-Bayesian inequalities which hold for arbitrary probability kernels that define a random, data-dependent choice of the function of interest. Finally, we demonstrate a number of applications, including PAC-Bayesian generalization bounds for unbounded loss functions, empirical Bernstein type generalization bounds, new truncation-free bounds for off-policy evaluation with Weighted Importance Sampling (WIS), and off-policy PAC-Bayesian learning with WIS.
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