Effective Dimension of Exp-concave Optimization

We investigate the role of the effective dimension $d_\lambda$ in determining both the statistical and the computational costs associated with exp-concave stochastic minimization. Our main statistical result is a nearly tight bound of order $d_\lambda/\epsilon$ on the sample complexity of any algorithm that approximately minimizes the empirical risk. Our main algorithmic contribution is a fast preconditioned method that solves the ERM problem in time $\tilde{O} \left(\min \left \{\frac{\lambda'}{\lambda} \left( \mathrm{nnz}(A)+\,d_{\lambda'}^{2}d\right) :\,\lambda' \ge \lambda \right \} \right)$, where $\mathrm{nnz}(A)$ is the number of nonzero entries in the data. Our results shed a light on two central sketching approaches named "sketch-and-solve" and "sketch-to-preconditioning". Our statistical result render the first approach redundant (in the context of bounded exp-concave minimization). On the contrary, our computation results highlight the efficacy of the latter approach. Our analysis emphasizes interesting connections between leverage scores, algorithmic stability and regularization, which might be of independent interest.