Bounding Neyman-Pearson Region with $f$-Divergences

The Neyman-Pearson region of a simple binary hypothesis testing is the set of points whose coordinates represent the false positive rate and false negative rate of some test. The lower boundary of this region is given by the Neyman-Pearson lemma, and is up to a coordinate change, equivalent to the optimal ROC curve. We establish a novel lower bound for the boundary in terms of any $f$-divergence. Since the bound generated by hockey-stick $f$-divergences characterizes the Neyman-Pearson boundary, this bound is best possible. In the case of KL divergence, this bound improves Pinsker's inequality. Furthermore, we obtain a closed-form refined upper bound for the Neyman-Pearson boundary in terms of the Chernoff $\alpha$-coefficient. Finally, we present methods for constructing pairs of distributions that can approximately or exactly realize any given Neyman-Pearson boundary.

HELLINGER-UCB: A novel algorithm for stochastic multi-armed bandit problem and cold start problem in recommender system

In this paper, we study the stochastic multi-armed bandit problem, where the reward is driven by an unknown random variable. We propose a new variant of the Upper Confidence Bound (UCB) algorithm called Hellinger-UCB, which leverages the squared Hellinger distance to build the upper confidence bound. We prove that the Hellinger-UCB reaches the theoretical lower bound. We also show that the Hellinger-UCB has a solid statistical interpretation. We show that Hellinger-UCB is effective in finite time horizons with numerical experiments between Hellinger-UCB and other variants of the UCB algorithm. As a real-world example, we apply the Hellinger-UCB algorithm to solve the cold-start problem for a content recommender system of a financial app. With reasonable assumption, the Hellinger-UCB algorithm has a convenient but important lower latency feature. The online experiment also illustrates that the Hellinger-UCB outperforms both KL-UCB and UCB1 in the sense of a higher click-through rate (CTR).