Gaussian processes (GPs) are widely used for regression and optimization tasks such as Bayesian optimization (BO) due to their expressiveness and principled uncertainty estimates. However, in settings with large datasets corrupted by outliers, standard GPs and their sparse approximations struggle with computational tractability and robustness. We introduce Robust Computation-aware Gaussian Process (RCaGP), a novel GP model that jointly addresses these challenges by combining a principled treatment of approximation-induced uncertainty with robust generalized Bayesian updating. The key insight is that robustness and approximation-awareness are not orthogonal but intertwined: approximations can exacerbate the impact of outliers, and mitigating one without the other is insufficient. Unlike previous work that focuses narrowly on either robustness or approximation quality, RCaGP combines both in a principled and scalable framework, thus effectively managing both outliers and computational uncertainties introduced by approximations such as low-rank matrix multiplications. Our model ensures more conservative and reliable uncertainty estimates, a property we rigorously demonstrate. Additionally, we establish a robustness property and show that the mean function is key to preserving it, motivating a tailored model selection scheme for robust mean functions. Empirical results confirm that solving these challenges jointly leads to superior performance across both clean and outlier-contaminated settings, both on regression and high-throughput Bayesian optimization benchmarks.

This work presents a framework for control theory based on constructive analysis to account for discrepancy between mathematical results and their implementation in a computer, also referred to as computational uncertainty. In control engineering, the latter is usually either neglected or considered submerged into some other type of uncertainty, such as system noise, and addressed within robust control. However, even robust control methods may be compromised when the mathematical objects involved in the respective algorithms fail to exist in exact form and subsequently fail to satisfy the required properties. For instance, in general stabilization using a control Lyapunov function, computational uncertainty may distort stability certificates or even destabilize the system despite robustness of the stabilization routine with regards to system, actuator and measurement noise. In fact, battling numerical problems in practical implementation of controllers is common among control engineers. Such observations indicate that computational uncertainty should indeed be addressed explicitly in controller synthesis and system analysis. The major contribution here is a fairly general framework for proof techniques in analysis and synthesis of control systems based on constructive analysis which explicitly states that every computation be doable only up to a finite precision thus accounting for computational uncertainty. A series of previous works is overviewed, including constructive system stability and stabilization, approximate optimal controls, eigenvalue problems, Caratheodory trajectories, measurable selectors. Additionally, a new constructive version of the Danskin's theorem, which is crucial in adversarial defense, is presented.

Simulation-based inference methods have been shown to be inaccurate in the data-poor regime, when training simulations are limited or expensive. Under these circumstances, the inference network is particularly prone to overfitting, and using it without accounting for the computational uncertainty arising from the lack of identifiability of the network weights can lead to unreliable results. To address this issue, we propose using Bayesian neural networks in low-budget simulation-based inference, thereby explicitly accounting for the computational uncertainty of the posterior approximation. We design a family of Bayesian neural network priors that are tailored for inference and show that they lead to well-calibrated posteriors on tested benchmarks, even when as few as $O(10)$ simulations are available. This opens up the possibility of performing reliable simulation-based inference using very expensive simulators, as we demonstrate on a problem from the field of cosmology where single simulations are computationally expensive. We show that Bayesian neural networks produce informative and well-calibrated posterior estimates with only a few hundred simulations.

Gaussian processes scale prohibitively with the size of the dataset. In response, many approximation methods have been developed, which inevitably introduce approximation error. This additional source of uncertainty, due to limited computation, is entirely ignored when using the approximate posterior. Therefore in practice, GP models are often as much about the approximation method as they are about the data. Here, we develop a new class of methods that provides consistent estimation of the combined uncertainty arising from both the finite number of data observed and the finite amount of computation expended. The most common GP approximations map to an instance in this class, such as methods based on the Cholesky factorization, conjugate gradients, and inducing points. For any method in this class, we prove (i) convergence of its posterior mean in the associated RKHS, (ii) decomposability of its combined posterior covariance into mathematical and computational covariances, and (iii) that the combined variance is a tight worst-case bound for the squared error between the method's posterior mean and the latent function. Finally, we empirically demonstrate the consequences of ignoring computational uncertainty and show how implicitly modeling it improves generalization performance on benchmark datasets.