Information diffusion in social networks facilitates rapid and large-scale propagation of content. However, spontaneous diffusion behavior could also lead to the cascading of sensitive information, which is neglected in prior arts. In this paper, we present the first look into adaptive diffusion of sensitive information, which we aim to prevent from widely spreading without incurring much information loss. We undertake the investigation in networks with partially known topology, meaning that some users' ability of forwarding information is unknown. Formulating the problem into a bandit model, we propose BLAG (Bandit on Large Action set Graph), which adaptively diffuses sensitive information towards users with weak forwarding ability that is learnt from tentative transmissions and corresponding feedbacks. BLAG enjoys a low complexity of O(n), and is provably more efficient in the sense of half regret bound compared with prior learning method. Experiments on synthetic and three real datasets further demonstrate the superiority of BLAG in terms of adaptive diffusion of sensitive information over several baselines, with at least 40 percent less information loss, at least 10 times of learning efficiency given limited learning rounds and significantly postponed cascading of sensitive information.