With the hardware development of mobile devices, it is possible to build the recommendation models on the mobile side to utilize the fine-grained features and the real-time feedbacks. Compared to the straightforward mobile-based modeling appended to the cloud-based modeling, we propose a Slow-Fast learning mechanism to make the Mobile-Cloud Collaborative recommendation (MC$^2$-SF) mutual benefit. Specially, in our MC$^2$-SF, the cloud-based model and the mobile-based model are respectively treated as the slow component and the fast component, according to their interaction frequency in real-world scenarios. During training and serving, they will communicate the prior/privileged knowledge to each other to help better capture the user interests about the candidates, resembling the role of System I and System II in the human cognition. We conduct the extensive experiments on three benchmark datasets and demonstrate the proposed MC$^2$-SF outperforms several state-of-the-art methods.

Continuous-time Bayesian networks (CTBNs) are graphical representations of multi-component continuous-time Markov processes as directed graphs. The edges in the network represent direct influences among components. The joint rate matrix of the multi-component process is specified by means of conditional rate matrices for each component separately. This paper addresses the situation where some of the components evolve on a time scale that is much shorter compared to the time scale of the other components. In this paper, we prove that in the limit where the separation of scales is infinite, the Markov process converges (in distribution, or weakly) to a reduced, or effective Markov process that only involves the slow components. We also demonstrate that for reasonable separation of scale (an order of magnitude) the reduced process is a good approximation of the marginal process over the slow components. We provide a simple procedure for building a reduced CTBN for this effective process, with conditional rate matrices that can be directly calculated from the original CTBN, and discuss the implications for approximate reasoning in large systems.