In this paper, we provide three applications for $f$-divergences: (i) we introduce Sanov's upper bound on the tail probability of the sum of independent random variables based on super-modular $f$-divergence and show that our generalized Sanov's bound strictly improves over ordinary one, (ii) we consider the lossy compression problem which studies the set of achievable rates for a given distortion and code length. We extend the rate-distortion function using mutual $f$-information and provide new and strictly better bounds on achievable rates in the finite blocklength regime using super-modular $f$-divergences, and (iii) we provide a connection between the generalization error of algorithms with bounded input/output mutual $f$-information and a generalized rate-distortion problem. This connection allows us to bound the generalization error of learning algorithms using lower bounds on the $f$-rate-distortion function. Our bound is based on a new lower bound on the rate-distortion function that (for some examples) strictly improves over previously best-known bounds.

翻译：本文给出了$f$-散度的三个应用：（i）基于超模$f$-散度，我们引入了独立随机变量和的尾部概率的Sanov上界，并证明我们的广义Sanov界严格优于经典Sanov界；（ii）我们研究了给定失真和码长条件下可达速率集合的失真压缩问题。利用互$f$-信息扩展了率失真函数，并基于超模$f$-散度给出了有限块长体制下可达速率新的且严格更优的边界；（iii）我们建立了具有有界输入/输出互$f$-信息的算法泛化误差与广义率失真问题之间的联系。这一联系使我们能够利用$f$-率失真函数的下界来限制学习算法的泛化误差。我们的边界基于一个率失真函数的新下界，该下界（在某些示例中）严格优于先前已知的最佳边界。