  We consider stochastic smoothing of spectral functions of matrices using
perturbations commonly studied in random matrix theory. We show that a spectral
function remains spectral when smoothed using a unitarily invariant
perturbation distribution. We then derive state-of-the-art smoothing bounds for
the maximum eigenvalue function using the Gaussian Orthogonal Ensemble (GOE).
Smoothing the maximum eigenvalue function is important for applications in
semidefinite optimization and online learning. As a direct consequence of our
GOE smoothing results, we obtain an $O((N \log N)^{1/4} \sqrt{T})$ expected
regret bound for the online variance minimization problem using an algorithm
that performs only a single maximum eigenvector computation per time step. Here
$T$ is the number of rounds and $N$ is the matrix dimension. Our algorithm and
its analysis also extend to the more general online PCA problem where the
learner has to output a rank $k$ subspace. The algorithm just requires
computing $k$ maximum eigenvectors per step and enjoys an $O(k (N \log N)^{1/4}
\sqrt{T})$ expected regret bound.
