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. 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. main algorithmic contribution is a fast preconditioned method that solves the ERM problem in time $tilde{O} left(min left {frac{lambdau0027}{lambda} left( mathrm{nnz}(A)+,d_{lambdau0027}^{2}dright) :,lambdau0027 ge lambda right } right)$, 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. 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. analysis emphasizes interesting connections between leverage scores, algorithmic stability and regularization, which might be of independent interest.