In this paper we consider the problem of finding stable maxima of expensive (to evaluate) functions. We are motivated by the optimisation of physical and industrial processes where, for some input ranges, small and unavoidable variations in inputs lead to unacceptably large variation in outputs. Our approach uses multiple gradient Gaussian Process models to estimate the probability that worst-case output variation for specified input perturbation exceeded the desired maxima, and these probabilities are then used to (a) guide the optimisation process toward solutions satisfying our stability criteria and (b) post-filter results to find the best stable solution. We exhibit our algorithm on synthetic and real-world problems and demonstrate that it is able to effectively find stable maxima.
 Safe and robust grasping of unknown objects is a major challenge in robotics, which has no general solution yet. A promising approach relies on haptic exploration, where active optimization strategies can be employed to reduce the number of exploration trials. One critical problem is that certain optimal grasps discoverd by the optimization procedure may be very sensitive to small deviations of the parameters from their nominal values: we call these unsafe grasps because small errors during motor execution may turn optimal grasps into bad grasps. To reduce the risk of grasp failure, safe grasps should be favoured. Therefore, we propose a new algorithm, unscented Bayesian optimization, that performs efficient optimization while considering uncertainty in the input space, leading to the discovery of safe optima. The results highlight how our method outperforms the classical Bayesian optimization both in synthetic problems and in realistic robot grasp simulations, finding robust and safe grasps after a few exploration trials. Tuning hyperparameters of machine learning models is important for their performance. Bayesian optimization has recently emerged as a de-facto method for this task. The hyperparameter tuning is usually performed by looking at model performance on a validation set. Bayesian optimization is used to find the hyperparameter set corresponding to the best model performance. However, in many cases, the function representing the model performance on the validation set contains several spurious sharp peaks due to limited datapoints. The Bayesian optimization, in such cases, has a tendency to converge to sharp peaks instead of other more stable peaks. When a model trained using these hyperparameters is deployed in the real world, its performance suffers dramatically. We address this problem through a novel stable Bayesian optimization framework. We construct two new acquisition functions that help Bayesian optimization to avoid the convergence to the sharp peaks. We conduct a theoretical analysis and guarantee that Bayesian optimization using the proposed acquisition functions prefers stable peaks over unstable ones. Experiments with synthetic function optimization and hyperparameter tuning for support vector machines show the effectiveness of our proposed framework.