  The online problem of computing the top eigenvector is fundamental to machine
learning. In both adversarial and stochastic settings, previous results (such
as matrix multiplicative weight update, follow the regularized leader, follow
the compressed leader, block power method) either achieve optimal regret but
run slow, or run fast at the expense of loosing a $\sqrt{d}$ factor in total
regret where $d$ is the matrix dimension.
  We propose a $\textit{follow-the-compressed-leader (FTCL)}$ framework which
achieves optimal regret without sacrificing the running time. Our idea is to
"compress" the matrix strategy to dimension 3 in the adversarial setting, or
dimension 1 in the stochastic setting. These respectively resolve two open
questions regarding the design of optimal and efficient algorithms for the
online eigenvector problem.
