Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure anduniform error bounds havebeen derived,which allowsafe control based on thesemodels.

Neural Information Processing Systems http://nips.cc/

This is achieved e.g. if a constant fraction of all samples lies on the point Theorem 3.3 by reformulating lines 190-191 as follows: "Furthermore, consider an infinite data stream of observations ". Making Theorem 3.3 quantitative as suggested by Reviewer #2 Although unbounded, they grow slow enough to allow the proof of Theorem 3.3 such that the main We will add a brief discussion on this in the updated paper. Reviewer #1 pointed out, that Assumption 3.1. Therefore, Assumption 3.1 is valid for our experimental setup. We will include the given reasoning in the updated paper.

--This paper addresses the integration of additional information sources into a Bayesian optimization framework while ensuring that safety constraints are satisfied. The interdependencies between these information sources are modeled using an unknown correlation matrix. We explore how uniform error bounds must be adjusted to maintain constraint satisfaction throughout the optimization process, considering both Bayesian and frequentist statistical perspectives. This is achieved by appropriately scaling the error bounds based on a confidence interval that can be estimated from the data. Furthermore, the efficacy of the proposed approach is demonstrated through experiments on two benchmark functions and a controller parameter optimization problem. Our results highlight a significant improvement in sample efficiency, demonstrating the method's suitability for optimizing expensive-to-evaluate functions. Many practical optimization problems can be formulated as the optimization of a black-box function, e. g., because of their complex underlying physics or the requirement of impractical identification processes. Black-box optimization algorithms bypass the need of models for optimizations. In essence, these algorithms sequentially evaluate the black-box function for some input while reducing the cost. In the last decade, Bayesian optimization (BO) has emerged as a promising method for solving exactly this set of problems. This method involves constructing a probabilistic surrogate model of an arbitrary objective function with minimal assumptions. The utilization of Gaussian processes (GPs) enables the incorporation of prior knowledge about the objective function, making BO particularly well-suited for scenarios where function evaluations are costly and observations may be noisy. As a simple example of BO, consider the optimization of a PID controller for unit step reference tracking, where the plant dynamics are unknown. A potential cost function that measures tracking accuracy could be the mean-squared error of the plant output and the step reference for a designated time window. The black-box function is now the function that maps the PID parameters to the image of the cost function. An evaluation corresponds to running the step response of the system with the specified PID parameters.

We propose a novel family of decision-aware surrogate losses, called Perturbation Gradient (PG) losses, for the predict-then-optimize framework. These losses directly approximate the downstream decision loss and can be optimized using off-the-shelf gradient-based methods. Importantly, unlike existing surrogate losses, the approximation error of our PG losses vanishes as the number of samples grows. This implies that optimizing our surrogate loss yields a best-in-class policy asymptotically, even in misspecified settings. This is the first such result in misspecified settings and we provide numerical evidence confirming our PG losses substantively outperform existing proposals when the underlying model is misspecified and the noise is not centrally symmetric. Insofar as misspecification is commonplace in practice -- especially when we might prefer a simpler, more interpretable model -- PG losses offer a novel, theoretically justified, method for computationally tractable decision-aware learning.

Gaussian Process Upper Confidence Bound (GP-UCB) is one of the most popular methods for optimizing black-box functions with noisy observations, due to its simple structure and superior performance. Its empirical successes lead to a natural, yet unresolved question: Is GP-UCB regret optimal? In this paper, we offer the first generally affirmative answer to this important open question in the Bayesian optimization literature. We establish new upper bounds on both the simple and cumulative regret of GP-UCB when the objective function to optimize admits certain smoothness property. These upper bounds match the known minimax lower bounds (up to logarithmic factors independent of the feasible region's dimensionality) for optimizing functions with the same smoothness. Intriguingly, our findings indicate that, with the same level of exploration, GP-UCB can simultaneously achieve optimality in both simple and cumulative regret. The crux of our analysis hinges on a refined uniform error bound for online estimation of functions in reproducing kernel Hilbert spaces. This error bound, which we derive from empirical process theory, is of independent interest, and its potential applications may reach beyond the scope of this study.

In application areas where data generation is expensive, Gaussian processes are a preferred supervised learning model due to their high data-efficiency. Particularly in model-based control, Gaussian processes allow the derivation of performance guarantees using probabilistic model error bounds. To make these approaches applicable in practice, two open challenges must be solved i) Existing error bounds rely on prior knowledge, which might not be available for many real-world tasks. (ii) The relationship between training data and the posterior variance, which mainly drives the error bound, is not well understood and prevents the asymptotic analysis. This article addresses these issues by presenting a novel uniform error bound using Lipschitz continuity and an analysis of the posterior variance function for a large class of kernels. Additionally, we show how these results can be used to guarantee safe control of an unknown dynamical system and provide numerical illustration examples.

Classification is a fundamental problem in statistics and machine learning that arises in many scientific and engineering problems. Scientific applications include identifying plant and animal species from body measurements, determining cancer types based on gene expression, and satellite image processing (Fisher, 1936, 1938; Khan et al., 2001; Lee et al., 2004); in modern engineering contexts, credit card fraud detection, handwritten digit recognition, word sense disambiguation, and object detection in images are all examples of classification tasks. These applications have brought two new challenges: multiclass classification with a potentially large number of classes and imbalanced data. For example, in online retailing, websites have hundreds of thousands or millions of products, and they may like to categorize these products within a preexisting taxonomy based on product descriptions (Lin et al., 2018). While the number of classes alone makes the problem difficult, an added difficulty with text data is that it is usually highly imbalanced, meaning that a few classes may constitute a large fraction of the data while many classes have only a few examples. In fact, Feldman (2019) notes that if the data follows the classical Zipf distribution for text data (Zipf, 1936), i.e., the class probabilities satisfy a power-law distribution, then up to 35% of seen examples may appear only once in the training data. Additionally, natural image data also seems to have the problems of many classes and imbalanced data (Salakhutdinov et al., 2011; Zhu et al., 2014). Focusing on the problem of imbalanced data, researchers have found that a few heuristics help "do better," and the most principled and studied of these is weighting. There are a number of forms of weighting; we consider the most basic in which we incur a loss of weight for misclassifying an example of class and refer to this method as class-weighting.

Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on these models. However, existing error bounds require restrictive assumptions. In this paper, we employ the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions. Furthermore, we demonstrate how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of our bound. Finally, we derive safety conditions for the control of unknown dynamical systems based on Gaussian process models and evaluate them in simulations of a robotic manipulator.