Bayesian Optimization with Fisher Information Geometry: Gradient Bounds and Trust-Region Methods
Abstract
We study Bayesian optimization (BO) through the lens of information geometry. Pulling back the Fisher information metric through the surrogate posterior map yields a local sensitivity tensor on the input space, which leads to an upper bound on the gradient of reparameterizable acquisition functions (AFs). This view explains vanishing-gradient behavior in high-dimensional BO and provides a common interpretation of heuristics such as RAASP and dimension-scaled lengthscales. The upper bound can also serve as a generic recipe to understand the behavior of new acquisition functions or surrogates in high input dimensions as well. Using the bounds, we also identify necessary conditions for an input location to be an optimizer of the AF for a univariate Gaussian predictive posterior. Building on this analysis, we propose FITR, a trust-region-based BO method that replaces lengthscale-based scaling with local pullback-Fisher weights. FITR is not restricted to GP kernels with explicit lengthscales and easily generalizes to non-isotropic surrogates. FITR shows competitive performance on several benchmarks.
Related
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BibTeX
@misc{kiroriwal2026bayesian,
title = {{Bayesian Optimization with Fisher Information Geometry: Gradient Bounds and Trust-Region Methods}},
author = {Saksham Kiroriwal and Julius Pfrommer and Jürgen Beyerer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.31107},
eprint = {2609.31107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31107},
}
RIS
TY - UNPB
TI - Bayesian Optimization with Fisher Information Geometry: Gradient Bounds and Trust-Region Methods
AU - Saksham Kiroriwal
AU - Julius Pfrommer
AU - Jürgen Beyerer
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.31107
UR - https://arxiv.org/abs/2609.31107
AB - We study Bayesian optimization (BO) through the lens of information geometry. Pulling back the Fisher information metric through the surrogate posterior map yields a local sensitivity tensor on the input space, which leads to an upper bound on the gradient of reparameterizable acquisition functions (AFs). This view explains vanishing-gradient behavior in high-dimensional BO and provides a common interpretation of heuristics such as RAASP and dimension-scaled lengthscales. The upper bound can also serve as a generic recipe to understand the behavior of new acquisition functions or surrogates in high input dimensions as well. Using the bounds, we also identify necessary conditions for an input location to be an optimizer of the AF for a univariate Gaussian predictive posterior. Building on this analysis, we propose FITR, a trust-region-based BO method that replaces lengthscale-based scaling with local pullback-Fisher weights. FITR is not restricted to GP kernels with explicit lengthscales and easily generalizes to non-isotropic surrogates. FITR shows competitive performance on several benchmarks.
ER -