This paper tackles Bayesian optimization using Gaussian processes, focusing on minimizing regret when the objective function is drawn from a known GP. For a Matérn kernel with sufficient smoothness, the authors prove that the GP-UCB algorithm achieves a cumulative regret of “~O(√T) with high probability. They also establish an “O(√T ln²T)" regret bound for the squared exponential kernel, bridging the gap between previous upper bounds and Scarlett’s 2018 results. The analysis hinges on capturing the concentration of the input sequence generated by GP-UCB to refine the GP’s information‑gain estimates.
By Shogo Iwazaki
The paper analyzes Bayesian linear bandits with isotropic Gaussian parameters, independent Gaussian arms, and Gaussian reward noise when the time horizon scales with the dimension. It derives explicit limits for the normalized posterior uncertainty and parameter overlaps, yielding exact regret curves for several policies—including Thompson sampling, posterior‑mean greedy selection, and scaled‑covariance variants. The results show that posterior‑mean greedy selection achieves the optimal Bayes regret, while Thompson sampling incurs a strictly larger leading regret whose ratio to greedy lies between one and two, approaching two for long horizons.
By Prakhar Singhvi (Abstract Math Institute), Yi Zou (Abstract Math Institute), Abhishek Bhattacharjee (Abstract Math Institute)
arXiv:2512. 00517v3 Announce Type: replace-cross Abstract: Sequential optimization of black-box functions from noisy evaluations has been widely studied, with Gaussian Process bandit algorithms such as GP-UCB guaranteeing no-regret in stationary settings.
By Eliabelle Mauduit, Elo\"ise Berthier, Andrea Simonetto
arXiv:2603. 09276v2 Announce Type: replace-cross Abstract: We study a widely used Bayesian optimization method, Gaussian process Thompson sampling (GP-TS), under the assumption that the objective function is a sample path from a GP.
By Shion Takeno, Shogo Iwazaki
The paper presents the first high‑probability regret bounds for both cumulative and current‑step variants of Feel‑Good Thompson Sampling (FGTS) in contextual bandits with general realizable function approximation. The bounds rely on decoupling coefficients and prior mass near the true reward function, and in the linear setting they achieve ≠d√T log(2T)+√T log(1/δ) regret with probability at least 1‑δ. This matches the dimension and horizon dependence of upper‑confidence‑bound algorithms up to logarithmic factors, while the high‑probability cost adds only linear dependence on log(1/δ).
By Xuheng Li, Qingyue Zhao, Quanquan Gu
The paper studies the trade‑off between maximizing rewards and accurately estimating current treatment effects in nonstationary multi‑armed bandits. For $K$‑armed bandits with H"older‑smooth mean rewards, the authors derive minimax bounds on expected dynamic regret under a high‑probability simultaneous coverage requirement for all current effects. They propose the TRACE algorithm, which achieves a regret–precision trade‑off that separates reward‑learning regret from the unavoidable cost of refreshing current‑effect estimates, and extend the method with an empirical‑Bernstein e‑process to adapt exploration while preserving worst‑case guarantees.
By Sharan Sahu, Martin T. Wells