Parallel Gaussian Process Optimization with Upper Confidence Bound and Pure Exploration

In this paper, we consider the challenge of maximizing an unknown function f for which evaluations are noisy and are acquired with high cost. An iterative procedure uses the previous measures to actively select the next estimation of f which is predicted to be the most useful. We focus on the case where the function can be evaluated in parallel with batches of fixed size and analyze the benefit compared to the purely sequential procedure in terms of cumulative regret. We introduce the Gaussian Process Upper Confidence Bound and Pure Exploration algorithm (GP-UCB-PE) which combines the UCB strategy and Pure Exploration in the same batch of evaluations along the parallel iterations. We prove theoretical upper bounds on the regret with batches of size K for this procedure which show the improvement of the order of sqrt{K} for fixed iteration cost over purely sequential versions. Moreover, the multiplicative constants involved have the property of being dimension-free. We also confirm empirically the efficiency of GP-UCB-PE on real and synthetic problems compared to state-of-the-art competitors

Novel Exploration Techniques (NETs) for Malaria Policy Interventions

The task of decision-making under uncertainty is daunting, especially for problems which have significant complexity. Healthcare policy makers across the globe are facing problems under challenging constraints, with limited tools to help them make data driven decisions. In this work we frame the process of finding an optimal malaria policy as a stochastic multi-armed bandit problem, and implement three agent based strategies to explore the policy space. We apply a Gaussian Process regression to the findings of each agent, both for comparison and to account for stochastic results from simulating the spread of malaria in a fixed population. The generated policy spaces are compared with published results to give a direct reference with human expert decisions for the same simulated population. Our novel approach provides a powerful resource for policy makers, and a platform which can be readily extended to capture future more nuanced policy spaces.Comment: Under-revie

Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization

In this paper, we consider the problem of sequentially optimizing a black-box function $f$ based on noisy samples and bandit feedback. We assume that $f$ is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process bandit optimization. We provide algorithm-independent lower bounds on the simple regret, measuring the suboptimality of a single point reported after $T$ rounds, and on the cumulative regret, measuring the sum of regrets over the $T$ chosen points. For the isotropic squared-exponential kernel in $d$ dimensions, we find that an average simple regret of $\epsilon$ requires $T = \Omega\big(\frac{1}{\epsilon^2} (\log\frac{1}{\epsilon})^{d/2}\big)$, and the average cumulative regret is at least $\Omega\big( \sqrt{T(\log T)^{d/2}} \big)$, thus matching existing upper bounds up to the replacement of $d/2$ by $2d+O(1)$ in both cases. For the Mat\'ern-$\nu$ kernel, we give analogous bounds of the form $\Omega\big( (\frac{1}{\epsilon})^{2+d/\nu}\big)$ and $\Omega\big( T^{\frac{\nu + d}{2\nu + d}} \big)$, and discuss the resulting gaps to the existing upper bounds.Comment: Appearing in COLT 2017. This version corrects a few minor mistakes in Table I, which summarizes the new and existing regret bound