  Online learning to rank is a core problem in machine learning. In Lattimore
et al. (2018), a novel online learning algorithm was proposed based on
topological sorting. In the paper they provided a set of self-normalized
inequalities (a) in the algorithm as a criterion in iterations and (b) to
provide an upper bound for cumulative regret, which is a measure of algorithm
performance. In this work, we utilized method of mixtures and asymptotic
expansions of certain implicit function to provide a tighter, iterated-log-like
boundary for the inequalities, and as a consequence improve both the algorithm
itself as well as its performance estimation.
