ON THE MODIFIED LOGISTIC EQUATION
A simple slow-fast system based on the logistic equation for which the exact solution is known, is considered here. We are especially interested in the lift-off point i.e. the point where the solution suddenly leaves the unstable slow manifold after being exponentially close to it for quite some time. Sharp estimates of this point are given. Also a slightly modified system is considered which has a periodic solution of canard type. The explicit expression of this periodic solution is given as well as estimates of its lift-off point. This type of equation can be valuable in modeling population dynamics. Lift-off points can then be used to predict outbreaks of epidemics, which nowadays is an important item in our society.
Diversified retrieval and online learning are two core research areas in the design of modern information retrieval systems. In this paper, we propose the linear sub-modular bandits problem, which is an online learning setting for optimizing a general class of feature-rich submodular utility models for diversified retrieval. We present an algorithm, called LSBGREEDY, and prove that it efficiently converges to a near-optimal model. As a case study, we applied our approach to the setting of personalized news recommendation, where the system must recommend small sets of news articles selected from tens of thousands of available articles each day. In a live user study, we found that LSBGREEDY significantly outperforms existing online learning approaches.