Spectral Smoothing via Random Matrix Perturbations

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.