Bayesian optimization (BO) is a widely used iterative algorithm for optimizing black-box functions. Each iteration requires maximizing an acquisition function, such as the upper confidence bound (UCB) or a sample path from the Gaussian process (GP) posterior, as in Thompson sampling (TS). However, finding an exact solution to these maximization problems is often intractable and computationally expensive. Reflecting such realistic situations, in this paper, we delve into the effect of inexact maximizers of the acquisition functions. Defining a measure of inaccuracy in acquisition solutions, we establish cumulative regret bounds for both GP-UCB and GP-TS without requiring exact solutions of acquisition function maximization. Our results show that under appropriate conditions on accumulated inaccuracy, inexact BO algorithms can still achieve sublinear cumulative regret. Motivated from such findings, we provide both theoretical justification and numerical validation for random grid search as an effective and computationally efficient acquisition function solver.

The presence of distribution shifts poses a significant challenge for deploying modern machine learning models in real-world applications. This work focuses on the target shift problem in a regression setting (Zhang et al., 2013; Nguyen et al., 2016). More specifically, the target variable $y$ (also known as the response variable), which is continuous, has different marginal distributions in the training source and testing domain, while the conditional distribution of features $\boldsymbol{x}$ given $y$ remains the same. While most literature focuses on classification tasks with finite target space, the regression problem has an *infinite dimensional* target space, which makes many of the existing methods inapplicable. In this work, we show that the continuous target shift problem can be addressed by estimating the importance weight function from an ill-posed integral equation. We propose a nonparametric regularized approach named *ReTaSA* to solve the ill-posed integral equation and provide theoretical justification for the estimated importance weight function. The effectiveness of the proposed method has been demonstrated with extensive numerical studies on synthetic and real-world datasets.

In this paper, we develop a statistical inference procedure using stochastic gradient descent (SGD)-based confidence intervals. These intervals are of the simplest possible form: $\theta_{N,j} \pm 2\sqrt{}(\gamma/N)$ , where $\theta_N$ is the SGD estimate of model parameters $\theta$ over N data points, and $\gamma$ is the learning rate. This construction relies only on a proper selection of the learning rate to ensure the standard SGD conditions for O(1/n) convergence. The procedure performs well in our empirical evaluations, achieving near-nominal coverage intervals scaling up to 20$\times$ as many parameters as other SGD-based inference methods. We also demonstrate our method’s practical significance on modeling adverse events in emergency general surgery patients using a novel dataset from the Hospital of the University of Pennsylvania.