We show that there is a largely unexplored class of functions (positive polyma-troids) that can define proper discrete metrics over pairs of binary vectors and that are fairly tractable to optimize over.

Neural Information Processing Systems http://nips.cc/

We first generalize existing algebraic characterizations of acyclicity to a class of matrix polynomials.

This estimator has been the subject of extensive empirical inquiry [e.g.

We consider Empirical Risk Minimization (ERM) in the context of stochastic optimization with exp-concave and smooth losses--a general optimization framework that captures several important learning problems including linear and logistic regression, learning SVMs with the squared hinge-loss, portfolio selection and more. In this setting, we establish the first evidence that ERM is able to attain fast generalization rates, and show that the expected loss of the ERM solution in d dimensions converges to the optimal expected loss in a rate of d/n. This rate matches existing lower bounds up to constants and improves by a log n factor upon the state-of-the-art, which is only known to be attained by an online-to-batch conversion of computationally expensive online algorithms.

We study the problem of black-box optimization of a function f of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one of its global optima, but this smoothness is unknown. Our contribution is an adaptive optimization algorithm, POO or parallel optimistic optimization, that is able to deal with this setting. POO performs almost as well as the best known algorithms requiring the knowledge of the smoothness. Furthermore, POO works for a larger class of functions than what was previously considered, especially for functions that are difficult to optimize, in a very precise sense. We provide a finite-time analysis of POO's performance, which shows that its error after n evaluations is at most a factor of ln n away from the error of the best known optimization algorithms using the knowledge of the smoothness.