Online regression with partial information: Generalization and linear projection

We investigate an online regression problem in which the learner makes predictions sequentially while only the limited information on features is observable. In this paper, we propose a general setting for the limitation of the available information, where the observed information is determined by a function chosen from a given set of observation functions. Our problem setting is a generalization of the online sparse linear regression problem, which has been actively studied. For our general problem, we present an algorithm by combining multi-armed bandit algorithms and online learning methods. This algorithm admits a sublinear regret bound when the number of observation functions is constant. We also show that the dependency on the number of observation functions is inevitable unless additional assumptions are adopted. To mitigate this inefficiency, we focus on a special case of practical importance, in which the observed information is expressed through linear combinations of the original features. We propose efficient algorithms for this special case. Finally, we also demonstrate the efficiency of the proposed algorithms by simulation studies using both artificial and real data.