We study Gaussian Process Thompson Sampling (GP-TS) for sequential decision-making over compact, continuous action spaces and provide a frequentist regret analysis based on fractional Gaussian process posteriors, without relying on domain discretization as in prior work. We show that the variance inflation commonly assumed in existing analyses of GP-TS can be interpreted as Thompson Sampling with respect to a fractional posterior with tempering parameter $α\in (0,1)$. We derive a kernel-agnostic regret bound expressed in terms of the information gain parameter $γ_t$ and the posterior contraction rate $ε_t$, and identify conditions on the Gaussian process prior under which $ε_t$ can be controlled. As special cases of our general bound, we recover regret of order $\tilde{\mathcal{O}}(T^{\frac{1}{2}})$ for the squared exponential kernel, $\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}} )$ for the Matérn-$ν$ kernel, and a bound of order $\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}})$ for the rational quadratic kernel. Overall, our analysis provides a unified and discretization-free regret framework for GP-TS that applies broadly across kernel classes.

We design and evaluate a Bayesian optimization framework for resource efficient pre-training of Transformer-based language models (TLMs). TLM pre-training requires high computational resources and introduces many unresolved design choices, such as selecting its pre-training hyperparameters. We propose a multi-armed bandit framework for the sequential selection of TLM pre-training hyperparameters, aimed at optimizing language model performance, in a resource efficient manner. We design a Thompson sampling algorithm, with a surrogate Gaussian process reward model of the Masked Language Model (MLM) pre-training objective, for its sequential minimization. Instead of MLM pre-training with fixed masking probabilities, the proposed Gaussian process-based Thompson sampling (GP-TS) accelerates pre-training by sequentially selecting masking hyperparameters that improve performance. We empirically demonstrate how GP-TS pre-trains language models efficiently, i.e., it achieves lower MLM loss in fewer epochs, across a variety of settings. In addition, GP-TS pre-trained TLMs attain competitive downstream performance, while avoiding expensive hyperparameter grid search. GP-TS provides an interactive framework for efficient and optimized TLM pre-training that, by circumventing costly hyperparameter selection, enables substantial computational savings.

Bayesian optimisation is a powerful method for non-convex black-box optimization in low data regimes. However, the question of establishing tight upper bounds for common algorithms in the noiseless setting remains a largely open question. In this paper, we establish new and tightest bounds for two algorithms, namely GP-UCB and Thompson sampling, under the assumption that the objective function is smooth in terms of having a bounded norm in a Mat\'ern RKHS. Importantly, unlike several related works, we do not consider perfect knowledge of the kernel of the Gaussian process emulator used within the Bayesian optimization loop. This allows us to provide results for practical algorithms that sequentially estimate the Gaussian process kernel parameters from the available data.