Online Learning of Eigenvectors

Computing the leading eigenvector of a symmetric real matrix is a fundamental primitive of numerical linear algebra with numerous applications. We consider a natural online extension of the leading eigenvector problem: a sequence of matrices is presented and the goal is to predict for each matrix a unit vector, with the overall goal of competing with the leading eigenvector of the cumulative matrix. Existing regret-minimization algorithms for this problem either require to compute an eigen decompostion every iteration, or suffer from a large dependency of the regret bound on the dimension. In both cases the algorithms are not practical for large scale applications. In this paper we present new algorithms that avoid both issues. On one hand they do not require any expensive matrix decompositions and on the other, they guarantee regret rates with a mild dependence on the dimension at most. In contrast to previous algorithms, our algorithms also admit implementations that enable to leverage sparsity in the data to further reduce computation. We extend our results to also handle non-symmetric matrices.