Title: Online Learning for Supervised Dimension Reduction. Abstract:  Online learning has attracted great attention due to the increasing demand for systems that have the ability of learning and evolving. When the data to be processed is also high dimensional and dimension reduction is necessary for visualization or prediction enhancement, online dimension reduction will play an essential role. The purpose of this paper is to propose new online learning approaches for supervised dimension reduction. Our first algorithm is motivated by adapting the sliced inverse regression (SIR), a pioneer and effective algorithm for supervised dimension reduction, and making it implementable in an incremental manner. The new algorithm, called incremental sliced inverse regression (ISIR), is able to update the subspace of significant factors with intrinsic lower dimensionality fast and efficiently when new observations come in. We also refine the algorithm by using an overlapping technique  and develop an incremental overlapping sliced inverse regression (IOSIR) algorithm. We verify the effectiveness and efficiency of both algorithms by simulations and real data applications.