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 method that solves the ERM problem in time $\tilde{O} \left(\min \left \{\frac{\lambda'}{\lambda} \left( nnz(A)+d_{\lambda'}^{2}d\right) :~\lambda' \ge \lambda \right \} \right)$, where $nnz(A)$ 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 optimizing a tradeoff between oracle complexity and effective dimension. All of our results extend to the kernel setting.