In some challenging BO tasks, input uncertainty arises due to the inevitable randomness in the optimization process, such as machining errors, execution noise, or contextual variability.

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Bayesian Optimization (BO) is a sample-efficient optimization algorithm widely employed across various applications. In some challenging BO tasks, input uncertainty arises due to the inevitable randomness in the optimization process, such as machining errors, execution noise, or contextual variability. This uncertainty deviates the input from the intended value before evaluation, resulting in significant performance fluctuations in the final result. In this paper, we introduce a novel robust Bayesian Optimization algorithm, AIRBO, which can effectively identify a robust optimum that performs consistently well under arbitrary input uncertainty. Our method directly models the uncertain inputs of arbitrary distributions by empowering the Gaussian Process with the Maximum Mean Discrepancy (MMD) and further accelerates the posterior inference via Nystrom approximation. Rigorous theoretical regret bound is established under MMD estimation error and extensive experiments on synthetic functions and real problems demonstrate that our approach can handle various input uncertainties and achieve state-of-the-art performance.

Conventional Gaussian process regression exclusively assumes the existence of noise in the output data of model observations. In many scientific and engineering applications, however, the input locations of observational data may also be compromised with uncertainties owing to modeling assumptions, measurement errors, etc. In this work, we propose a Bayesian method that integrates the variability of input data into Gaussian process regression. Considering two types of observables -- noise-corrupted outputs with fixed inputs and those with prior-distribution-defined uncertain inputs, a posterior distribution is estimated via a Bayesian framework to infer the uncertain data locations. Thereafter, such quantified uncertainties of inputs are incorporated into Gaussian process predictions by means of marginalization. The effectiveness of this new regression technique is demonstrated through several numerical examples, in which a consistently good performance of generalization is observed, while a substantial reduction in the predictive uncertainties is achieved by the Bayesian inference of uncertain inputs.

To interpret uncertainty estimates from differentiable probabilistic models, recent work has proposed generating a single Counterfactual Latent Uncertainty Explanation (CLUE) for a given data point where the model is uncertain, identifying a single, on-manifold change to the input such that the model becomes more certain in its prediction. We broaden the exploration to examine $\delta$-CLUE, the set of potential CLUEs within a $\delta$ ball of the original input in latent space. We study the diversity of such sets and find that many CLUEs are redundant; as such, we propose DIVerse CLUE ($\nabla$-CLUE), a set of CLUEs which each propose a distinct explanation as to how one can decrease the uncertainty associated with an input. We then further propose GLobal AMortised CLUE (GLAM-CLUE), a distinct and novel method which learns amortised mappings on specific groups of uncertain inputs, taking them and efficiently transforming them in a single function call into inputs for which a model will be certain. Our experiments show that $\delta$-CLUE, $\nabla$-CLUE, and GLAM-CLUE all address shortcomings of CLUE and provide beneficial explanations of uncertainty estimates to practitioners.

To interpret uncertainty estimates from differentiable probabilistic models, recent work has proposed generating Counterfactual Latent Uncertainty Explanations (CLUEs). However, for a single input, such approaches could output a variety of explanations due to the lack of constraints placed on the explanation. Here we augment the original CLUE approach, to provide what we call $\delta$-CLUE. CLUE indicates $\it{one}$ way to change an input, while remaining on the data manifold, such that the model becomes more confident about its prediction. We instead return a $\it{set}$ of plausible CLUEs: multiple, diverse inputs that are within a $\delta$ ball of the original input in latent space, all yielding confident predictions.

We propose a method for automatically answering questions about images by bringing together recent advances from natural language processing and computer vision. We combine discrete reasoning with uncertain predictions by a multi-world approach that represents uncertainty about the perceived world in a bayesian framework. Our approach can handle human questions of high complexity about realistic scenes and replies with range of answer like counts, object classes, instances and lists of them. The system is directly trained from question-answer pairs. We establish a first benchmark for this task that can be seen as a modern attempt at a visual turing test.

The Gaussian Process (GP) framework flexibility has enabled its use in several data modeling scenarios. The setting where we have unavailable or uncertain inputs that generate possibly noisy observations is usually tackled by the well known Gaussian Process Latent Variable Model (GPLVM). However, the standard variational approach to perform inference with the GPLVM presents some expressions that are tractable for only a few kernel functions, which may hinder its general application. While other quadrature or sampling approaches could be used in that case, they usually are very slow and/or non-deterministic. In the present paper, we propose the use of the unscented transformation to enable the use of any kernel function within the Bayesian GPLVM. Our approach maintains the fully deterministic feature of tractable kernels and presents a simple implementation with only moderate computational cost. Experiments on dimensionality reduction and multistep-ahead prediction with uncertainty propagation indicate the feasibility of our proposal.

Bayesian optimisation (BO) has been a successful approach to optimise functions which are expensive to evaluate and whose observations are noisy. Classical BO algorithms, however, do not account for errors about the location where observations are taken, which is a common issue in problems with physical components. In these cases, the estimation of the actual query location is also subject to uncertainty. In this context, we propose an upper confidence bound (UCB) algorithm for BO problems where both the outcome of a query and the true query location are uncertain. The algorithm employs a Gaussian process model that takes probability distributions as inputs. Theoretical results are provided for both the proposed algorithm and a conventional UCB approach within the uncertain-inputs setting. Finally, we evaluate each method's performance experimentally, comparing them to other input noise aware BO approaches on simulated scenarios involving synthetic and real data.