A note on a confidence bound of Kuzborskij and Szepesv'ari

In an interesting recent work, Kuzborskij and Szepesv'ari derived a confidence bound for functions of independent random variables, which is based on an inequality that relates concentration to squared perturbations of the chosen function. Kuzborskij and Szepesv'ari also established the PAC-Bayes-ification of their confidence bound. Two important aspects of their work are that the random variables could be of unbounded range, and not necessarily of an identical distribution. The purpose of this note is to advertise/discuss these interesting results, with streamlined proofs. This expository note is written for persons who, metaphorically speaking, enjoy the "featured movie" but prefer to skip the preview sequence.