As a fundamental tool in the fields of data mining and computer vision, robust low rank subspace learning is to recover a low rank matrix under gross corruptions that are often modeled by another sparse matrix. Within this learning, we investigate the spectral k-support norm, a more appealing convex relaxation than the popular nuclear norm, as a low rank penalty in this paper. Despite the better recovering performance, the spectral k-support norm entails the model difficult to be optimized efficiently, which severely limits its scalability from the practical perspective. Therefore, this paper proposes a scalable and efficient algorithm which considers the dual objective of the original problem that can take advantage of the more computational efficient linear oracle of the spectral k-support norm to be evaluated. Further, by studying the sub-gradient of the loss of the dual objective, a line-search strategy is adopted in the algorithm to enable it to adapt to the Holder smoothness. Experiments on various tasks demonstrate the superior prediction performance and computation efficiency of the proposed algorithm.