Optimistic optimization of a Brownian

Abstract

We address the problem of optimizing a Brownian motion. We consider a (random) realization W of a Brownian motion with input space in [0, 1]. Given W , our goal is to return an ε-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order log 2 (1/ε). This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive-each query depends on previous values-and is an instance of the optimism-in-the-face-of-uncertainty principle.

Power of human-algorithm collaboration in solving combinatorial optimization problems

“…[28] showed exponential convergence rate of Bayesian optimization to expected global optimum (that is, the expected highest or lowest value of the function depending on the context) if the function near to global optimum has a certain shape. [17] showed similar results to more While it is tempting to praise Bayesian optimization for its ability to find optimal values of the structures efficiently, one of the downsides of the approach is that the locally optimal values found and globally optimal values are based on how well the prior of the Gaussian process is defined. If the black box function is a realization of the prior, then BO works surprisingly well.…”

“…The expert knowledge can be supported with previously calculated values of the objective function and, for instance, Maximum likehood estimation [17]. [17] gave bounds on sample sizes of random variables (in our case, the function values) to derive a certain probability in optimization of a likehood function for covariance hyperparameters.…”

“…The expert knowledge can be supported with previously calculated values of the objective function and, for instance, Maximum likehood estimation [17]. [17] gave bounds on sample sizes of random variables (in our case, the function values) to derive a certain probability in optimization of a likehood function for covariance hyperparameters. Because the bounds do not depend on the domain size of the function, fixing a sample size and a covariance function for covariance matrix Σ yields a constant time estimation of a prior.…”

“…We then branch to the optimal regions of the domain where the optimal function values are likely to reside. Hence, our algorithm follows quite popular geometric branch and bound framework used in multiple global optimization algorithms [17] and [19].…”

“…In [17], the search space is shrunk after function evaluations and the intervals where the global optimizer is not likely to reside are discarded. [17] and [19] cannot be used as such to solve problems with exponential sized domains because constants in their algorithm depend on the domain size and the shape of the function near to optimizer(s).…”