Amortized Acquisition Optimization for Bayesian Optimization with Variational Mutual Information

Farhad Mirkarimi

cs.LGJan 13, 2026

Amortized Acquisition Optimization for Bayesian Optimization with Variational Mutual Information

Authors: Farhad Mirkarimi

Abstract

Bayesian optimization (BO) of expensive black-box functions is traditionally addressed with Gaussian processes (GPs), which scale cubically with observations, or Bayesian neural networks (BNNs), which incur costly posterior sampling and inner-loop acquisition optimization. We propose VBO-MI (Variational Bayesian Optimization with Mutual Information), a fully gradient-based BO framework that requires no explicit GP prior or fixed parametric posterior family over objective function and treats it as a strict black box. An actor-critic architecture pairs an action-net with a variational critic that estimates information gain, eliminating the acquisition optimization bottleneck and achieving up to $10^{2}\times$ fewer FLOPs than BNN-BO baselines. A lightweight surrogate network further reduces real function queries to one batch per iteration. We establish consistency guarantees and evaluate VBO-MI on synthetic benchmarks (Ackley, Levy, Griewank) and real-world tasks (Rover Trajectory, Lunar Lander, Pest Control), demonstrating competitive or superior performance over the baselines.

Figures & tables

Panel of Figure 1 — Figure 1 : VBO-MI framework. An action-net $E_{\phi}$ maps random seeds $\mathbf{s}_{t}$ to candidate inputs $\mathbf{x}_{t}^{(i)}$ . A surrogate $\hat{f}_{\psi}$ predicts rewards $\hat{y}_{t}^{(i)}$ , enabling oracle-free updates of the helper $D_{\theta}$ ( $K_{a}$ steps) and action-net ( $K_{b}$ steps) via the acquisition loss $L_{E_{\phi}}$ , which balances exploitation (surrogate mean) and exploration (variational MI estimate $I_{\theta}$ ). One real oracle query per iteration yields $y_{t}^{(i)}$ , which replenishes the replay buffer $\mathcal{H}$ used to retrain $\hat{f}_{\psi}$ via MSE ( $K_{s}$ steps). Dashed arrows indicate gradient flows.

Panel of Figure 2 — Figure 2 : Synthetic benchmarks. Global optimum is 0 (dashed line). Mean $\pm$ 1 std, 5 trials.

Panel of Figure 3 — Figure 3 : Real-world benchmarks. Mean $\pm$ 1 std, 5 trials.

Method	Surrogate Upd.	Acq. Opt.	$f$ /iter	Scaling
GP	$\mathcal{O}(T^{3})$	$N\mathcal{O}(T^{2})$	$B$	Cubic
HMC-BNN	$\mathcal{O}(SLTW)$	$N\mathcal{O}(SW)$	$B$	Hi. Lin.
DKL	$\mathcal{O}(TW\!+\!T^{3})$	$N\mathcal{O}(T^{2})$	$B$	Cubic
LLA	$\mathcal{O}(TW\!+\!W^{3})$	$N\mathcal{O}(W^{2})$	$B$	Linear
VBO-MI	$\mathcal{O}(K_{s}BW_{s}\!+\!K_{a}BH)$	$\mathcal{O}(K_{b}W_{e})$	$\mathbf{B}$	Const.

Table 1 : FLOP and oracle complexity per BO iteration.

T

: observations;

W

: BNN weights;

W_{s}

W_{e}

: surrogate/action-net weights;

H

: LSTM dim;

S

: HMC samples;

L

: leapfrog steps.

Panel of Figure 4 — Figure 4 : Ablation I: surrogate vs. direct queries. Mean $\pm$ 1 std, 3 trials.

Panel of Figure 5 — Figure 5 : Ablation II: acquisition term contribution. Full VBO-MI vs. GP-exploration and GP-exploitation hybrids. Mean $\pm$ 1 std, 3 trials.

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Jul 9, 2025, cs.LGJ/K move · Enter open · S save

Direct Regret Optimization in Bayesian Optimization

Fengxue Zhang, Yuxin Chen

Department of Computer Science University of Chicago Chicago, IL 60637

Gaussian Process Surrogate Gradient

Open paper AskPDF arXiv

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.

Abstract

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