Effective Dimension of Exp-concave Optimization

We investigate the role of the effective (a.k.a. statistical) dimension in determining both the statistical and the computational costs associated with exp-concave stochastic minimization. We derive sample complexity bounds that scale with $\frac{d_{\lambda}}{\epsilon}$, where $d_{\lambda}$ is the effective dimension associated with the regularization parameter $\lambda$. These are the first fast rates in this setting that do not exhibit any explicit dependence either on the intrinsic dimension or the $\ell_{2}$-norm of the optimal classifier. We also propose fast preconditioned methods that solve the ERM problem in time $\tilde{O} \left(nnz(X)+\min_{\lambda'\ge\lambda}\frac{\lambda'}{\lambda}~d_{\lambda'}^{2}d \right)$, where $nnz(X)$ is the number of nonzero entries in the data. Our analysis emphasizes interesting connections between leverage scores, algorithmic stability and regularization. In particular, our algorithm involves a novel technique for choosing a regularization parameter $\lambda'$ that minimizes the complexity bound $\frac{\lambda'}{\lambda}\,d_{\lambda'}^{2}d$, while avoiding the entire (approximate) computation of the effective dimension for each candidate $\lambda'$. All of our result extend to the kernel setting.