Amortized Acquisition Optimization for Bayesian Optimization with Variational Mutual Information

Bayesian optimization (BO) is a powerful paradigm for optimizing expensive black-box functions. Traditional BO methods typically rely on separate hand-crafted acquisition functions and surrogate models for the underlying function, and often operate in a myopic manner. In this paper, we propose a novel direct regret optimization approach that jointly learns the optimal model and non-myopic acquisition by distilling from a set of candidate models and acquisitions, and explicitly targets minimizing the multi-step regret. Our framework leverages an ensemble of Gaussian Processes (GPs) with varying hyperparameters to generate simulated BO trajectories, each guided by an acquisition function drawn from a pool of conventional choices and terminated by a Bayesian early stop criterion. These trajectories train an end-to-end Decision Transformer that selects the next query so as to improve the ultimate objective, following a dense training sparse learning paradigm: the transformer is trained on abundant simulated data, while a limited number of real evaluations refine the GPs online. On synthetic and real-world benchmarks, our method attains the best or near-best final simple regret against standard, lookahead, trust-region and amortized BO baselines, with the largest gains in high-dimensional settings. Ablations attribute the gains jointly to region-of-interest filtering and the learned policy, and matched-budget comparisons against explicit two-step lookahead acquisitions show that the advantage is not shared by lookahead alone.

Bayesian optimization is widely used for hyperparameter optimization when model evaluations are expensive; however, noisy acquisition estimates can lead to unstable decisions. We identify acquisition estimation noise as a failure mode that was previously overlooked: even when the surrogate model and acquisition target are correctly specified, finite-sample Monte Carlo error can perturb acquisition values. This can, in turn, flip candidate rankings and lead to suboptimal BO decisions. As a remedy, we aim at variance reduction and propose an orthogonal acquisition estimator that subtracts an optimally weighted score-function control variate, which yields an acquisition residual orthogonal to posterior score directions and which thus reduces Monte Carlo variance. We further introduce OrthoBO: a Bayesian optimization framework that combines our orthogonal acquisition estimator with ensemble surrogates and an outer log transformation. We show theoretically that our estimator preserves the target, leads to variance reduction, and improves pairwise ranking stability. We further verify the theoretical properties of OrthoBO through numerical experiments where our framework reduces acquisition estimation variance, stabilizes candidate rankings, and achieves strong performance. We also demonstrate the downstream utility of OrthoBO in hyperparameter optimization for neural network training and fine-tuning.

Bayesian Optimisation (BO) is a powerful framework for the optimisation of expensive black-box functions, but typically requires refitting a surrogate and maximising an acquisition function at every evaluation step. In-context approaches based on Prior-data Fitted Networks (PFNs) amortise part of this cost by pre-training transformers on functions drawn from synthetic priors. PFNs4BO amortises the surrogate but still relies on a numerically maximised acquisition function, while FIBO performs BO fully in-context by sampling optimiser locations from a learned density, which fixes the decision rule and admits no surrogate. Learned acquisition functions score a finite candidate set with a trained network, but, lacking a label for the query, learn the score by reinforcement learning on previously solved tasks. We propose IQS-BO, a PFN that learns the query decision by supervised learning on synthetic priors. In a single forward pass, IQS-BO predicts the probability that each candidate maximises the objective over the set, and we show that the minimiser of its objective is the posterior probability of this event. The model can be pre-trained without a surrogate for fully in-context BO, or take the predictions of a fixed probabilistic surrogate as additional input, amortising only the decision step. Our method proposes queries at a fraction of the cost of acquisition-based methods, while either matching or outperforming standard BO with Gaussian processes (GPs) and available in-context methods on synthetic and real-world benchmarks. Finally, we propose a mixture prior for pre-training PFNs which combines samples from GPs with functions exhibiting warped inputs, isolated narrow optima, or plateaus that are poorly modeled by stationary kernels common in GP surrogates. We show that pre-training on this prior can lead to improved optimisation performance.