We study the noise-free Gaussian Process (GP) bandits problem, in which the learner seeks to minimize regret through noise-free observations of the black-box objective function lying on the known reproducing kernel Hilbert space (RKHS). Gaussian process upper confidence bound (GP-UCB) is the well-known GP-bandits algorithm whose query points are adaptively chosen based on the GP-based upper confidence bound score. Although several existing works have reported the practical success of GP-UCB, the current theoretical results indicate its suboptimal performance. However, GP-UCB tends to perform well empirically compared with other nearly optimal noise-free algorithms that rely on a non-adaptive sampling scheme of query points. This paper resolves this gap between theoretical and empirical performance by showing the nearly optimal regret upper bound of noise-free GP-UCB. Specifically, our analysis shows the first constant cumulative regret in the noise-free settings for the squared exponential kernel and Mat\'ern kernel with some degree of smoothness.

The optimization of black-box functions with noisy observations is a fundamental problem with widespread applications, and has been widely studied under the assumption that the function lies in a reproducing kernel Hilbert space (RKHS). This problem has been studied extensively in the stationary setting, and near-optimal regret bounds are known via developments in both upper and lower bounds. In this paper, we consider non-stationary scenarios, which are crucial for certain applications but are currently less well-understood. Specifically, we provide the first algorithm-independent lower bounds, where the time variations are subject satisfying a total variation budget according to some function norm. Under $\ell_{\infty}$-norm variations, our bounds are found to be close to the state-of-the-art upper bound (Hong \emph{et al.}, 2023). Under RKHS norm variations, the upper and lower bounds are still reasonably close but with more of a gap, raising the interesting open question of whether non-minor improvements in the upper bound are possible.

Kernelized bandit (KB) problem [Srinivas et al., 2010], also called Gaussian process bandit or Bayesian optimization, is one of the important sequential decision-making problems where one seeks to minimize the regret under an unknown reward function via sequentially acquiring function evaluations. As the name suggests, in the KB problem, the underlying reward function is assumed to be an element of reproducing kernel Hilbert space (RKHS) induced by a known fixed kernel function. KB has been applied in many applications, such as materials discovery [Ueno et al., 2016], drug discovery [Korovina et al., 2020], and robotics [Berkenkamp et al., 2023]. In addition, the near-optimal KB algorithms, whose regret upper bound matches the regret lower bound derived in Scarlett et al. [2017], have been shown [Camilleri et al., 2021, Salgia et al., 2021, Li and Scarlett, 2022, Salgia et al., 2024]. Non-stationary KB [Bogunovic et al., 2016] considers the optimization under a non-stationary environment; that is, the reward function may change over time within some RKHS. This modification is crucial in many practical applications where an objective function varies over time, such as financial markets [Heaton and Lucas, 1999] and recommender systems [Hariri et al., 2015]. For example, Zhou and Shroff [2021], Deng et al. [2022] have proposed upper confidence bound (UCB)-based algorithms for the non-stationary KB problem and derived the upper bound of the cumulative regret. Recently, Hong et al. [2023] have proposed an optimization-based KB

We consider the problem of sequentially optimizing a time-varying objective function using time-varying Bayesian optimization (TVBO). To cope with stale data arising from time variations, current approaches to TVBO require prior knowledge of a constant rate of change. However, in practice, the rate of change is usually unknown. We propose an event-triggered algorithm, ET-GP-UCB, that treats the optimization problem as static until it detects changes in the objective function online and then resets the dataset. This allows the algorithm to adapt to realized temporal changes without the need for prior knowledge. The event-trigger is based on probabilistic uniform error bounds used in Gaussian process regression. We show in numerical experiments that ET-GP-UCB outperforms state-of-the-art algorithms on synthetic and real-world data and provide regret bounds for the proposed algorithm. The results demonstrate that ET-GP-UCB is readily applicable without prior knowledge on the rate of change.

Finding out Taylor Swift was her 11th cousin twice-removed wasn't even the most shocking discovery Cher Scarlett made while exploring her family history. "There's a lot of stuff in my family that's weird and strange that we wouldn't know without Ancestry," says Scarlett, a software engineer and writer based in Kirkland, Washington. "I didn't even know who my mum's paternal grandparents were." In February 2022, the facial recognition search engine PimEyes surfaced non-consensual explicit photos of her at age 19, reigniting decades-old trauma. She attempted to get the pictures removed from the platform, which uses images scraped from the internet to create biometric "faceprints" of individuals.

Black box optimisation of an unknown function from expensive and noisy evaluations is a ubiquitous problem in machine learning, academic research and industrial production. An abstraction of the problem can be formulated as a kernel based bandit problem (also known as Bayesian optimisation), where a learner aims at optimising a kernelized function through sequential noisy observations. The existing work predominantly assumes feedback is immediately available; an assumption which fails in many real world situations, including recommendation systems, clinical trials and hyperparameter tuning. We consider a kernel bandit problem under stochastically delayed feedback, and propose an algorithm with $\tilde{\mathcal{O}}(\sqrt{\Gamma_k(T)T}+\mathbb{E}[\tau])$ regret, where $T$ is the number of time steps, $\Gamma_k(T)$ is the maximum information gain of the kernel with $T$ observations, and $\tau$ is the delay random variable. This represents a significant improvement over the state of the art regret bound of $\tilde{\mathcal{O}}(\Gamma_k(T)\sqrt{T}+\mathbb{E}[\tau]\Gamma_k(T))$ reported in Verma et al. (2022). In particular, for very non-smooth kernels, the information gain grows almost linearly in time, trivializing the existing results. We also validate our theoretical results with simulations.

In an interview with The Washington Post, Scarlett claimed she had gone through four rounds of interviews with Epic between November and December 2021. According to emails shared with The Post, on Dec. 8 a recruiting coordinator at Epic sent Scarlett an email with an attached "Request for Activity" form that asks for the disclosure of "any efforts you take outside of work that may overlap with your potential role at Epic." In the email, the recruiter wrote that the company "would like to get a head start on this process."

Confidence intervals are a crucial building block in the analysis of various online learning problems. The analysis of kernel based bandit and reinforcement learning problems utilize confidence intervals applicable to the elements of a reproducing kernel Hilbert space (RKHS). However, the existing confidence bounds do not appear to be tight, resulting in suboptimal regret bounds. In fact, the existing regret bounds for several kernelized bandit algorithms (e.g., GP-UCB, GP-TS, and their variants) may fail to even be sublinear. It is unclear whether the suboptimal regret bound is a fundamental shortcoming of these algorithms or an artifact of the proof, and the main challenge seems to stem from the online (sequential) nature of the observation points.

We consider the problem of Bayesian optimization of a one-dimensional Brownian motion in which the $T$ adaptively chosen observations are corrupted by Gaussian noise. We show that as the smallest possible expected simple regret and the smallest possible expected cumulative regret scale as $\Omega(1 / \sqrt{T \log (T)}) \cap \mathcal{O}(\log T / \sqrt{T})$ and $\Omega(\sqrt{T / \log (T)}) \cap \mathcal{O}(\sqrt{T} \cdot \log T)$ respectively. Thus, our upper and lower bounds are tight up to a factor of $\mathcal{O}( (\log T)^{1.5} )$. The upper bound uses an algorithm based on confidence bounds and the Markov property of Brownian motion, and the lower bound is based on a reduction to binary hypothesis testing.

An undercover investigation showed video of an Apple Genius Bar employee in Canada telling a customer that the simple repair to his Apple computer would cost as much as a new computer. The National went undercover with a hidden camera to an Apple Store in Toronto to see what the employees would say about a laptop with a darkened screen. After the employee went into the back of the store to run a diagnostic he came back with a shocking price quote. The employee said the most of the computer had been damaged with water, and therefore the customer would need to either spend $1,200 (USD $927) to fix it, or for around the same price, get a new one. However, when the same computer was brought to a third party repair shop in New York, it took Luis Rossman at Rossman Repair Group under two-minutes to fix the darkened LCD screen by bending back a pin, and he says he would never have even charged someone for fixing it.