The Behavior and Convergence of Local Bayesian Optimization
This work addresses a theoretical gap for researchers and practitioners in optimization and machine learning, offering foundational insights into local strategies, though it is incremental as it builds on prior algorithms.
The paper tackles the lack of theoretical understanding of local Bayesian optimization methods, which are empirically effective in high dimensions, by analyzing their behavior and providing the first rigorous convergence rates for a specific algorithm, showing strong statistical properties and derived rates in noisy and noiseless settings.
A recent development in Bayesian optimization is the use of local optimization strategies, which can deliver strong empirical performance on high-dimensional problems compared to traditional global strategies. The "folk wisdom" in the literature is that the focus on local optimization sidesteps the curse of dimensionality; however, little is known concretely about the expected behavior or convergence of Bayesian local optimization routines. We first study the behavior of the local approach, and find that the statistics of individual local solutions of Gaussian process sample paths are surprisingly good compared to what we would expect to recover from global methods. We then present the first rigorous analysis of such a Bayesian local optimization algorithm recently proposed by Müller et al. (2021), and derive convergence rates in both the noisy and noiseless settings.