The expected improvement (EI) is one of the most popular acquisition functions for Bayesian optimization (BO) and has demonstrated good empirical performances in many applications for the minimization of simple regret. However, under the evaluation metric of cumulative regret, the performance of EI may not be competitive, and its existing theoretical regret upper bound still has room for improvement. To adapt the EI for better performance under cumulative regret, we introduce a novel quantity called the evaluation cost which is compared against the acquisition function, and with this, develop the expected improvement-cost (EIC) algorithm. In each iteration of EIC, a new point with the largest acquisition function value is sampled, only if that value exceeds its evaluation cost. If none meets this criteria, the current best point is resampled. This evaluation cost quantifies the potential downside of sampling a point, which is important under the cumulative regret metric as the objective function value in every iteration affects the performance measure. We establish in theory a high-probability regret upper bound of EIC based on the maximum information gain, which is tighter than the bound of existing EI-based algorithms. It is also comparable to the regret bound of other popular BO algorithms such as Thompson sampling (GP-TS) and upper confidence bound (GP-UCB). We further perform experiments to illustrate the improvement of EIC over several popular BO algorithms.

We consider here the identification of change-points on large-scale data streams. The objective is to find the most efficient way of combining information across data stream so that detection is possible under the smallest detectable change magnitude. The challenge comes from the sparsity of change-points when only a small fraction of data streams undergo change at any point in time. The most successful approach to the sparsity issue so far has been the application of hard thresholding such that only local scores from data streams exhibiting significant changes are considered and added. However the identification of an optimal threshold is a difficult one. In particular it is unlikely that the same threshold is optimal for different levels of sparsity. We propose here a sparse likelihood score for identifying a sparse signal. The score is a likelihood ratio for testing between the null hypothesis of no change against an alternative hypothesis in which the change-points or signals are barely detectable. By the Neyman-Pearson Lemma this score has maximum detection power at the given alternative. The outcome is that we have a scoring of data streams that is successful in detecting at the boundary of the detectable region of signals and change-points. The likelihood score can be seen as a soft thresholding approach to sparse signal and change-point detection in which local scores that indicate small changes are down-weighted much more than local scores indicating large changes. We are able to show sharp optimality of the sparsity likelihood score in the sense of achieving successful detection at the minimum detectable order of change magnitude as well as the best constant with respect this order of change.