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This research applies information geometry to Bayesian optimization by using the Fisher information metric to derive upper bounds on gradients of acquisition functions. The authors demonstrate that this geometric perspective explains why gradients vanish in high-dimensional optimization problems and provides theoretical justification for existing heuristics. They propose FITR, a new trust-region method that uses local Fisher information weights instead of lengthscale-based scaling, showing competitive performance across benchmarks while being more generalizable to different surrogate models.
Why it matters
This work addresses a fundamental challenge in high-dimensional Bayesian optimization, which is widely used in hyperparameter tuning, materials discovery, and engineering design. The theoretical framework and practical FITR method could improve optimization efficiency in applications where evaluating objective functions is expensive or time-consuming.
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⚠️ Preprint – Noch nicht peer-reviewed
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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.
Source: Bayesian Optimization with Fisher Information Geometry: Gradient Bounds and Trust-Region Methods