Bayesian Optimization with Fisher Information Geometry: Gradient Bounds and Trust-Region Methods
Saksham Kiroriwal, Julius Pfrommer, Jürgen Beyerer
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.
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