Adaptive Parameter Optimization in Gaussian Processes: A Comprehensive Study of Uncertainty Quantification and Dimensional Scaling

Gaussian Process (GP) models have also become extremely useful for optimization under uncertainty algorithms, especially where the objective functions are costly to compute. Yet, the more classical methods usually adopt strategies that, in certain circumstances, might be effective but not flexible to be applied to a wide range of problem terrains. This study aims to adapt parameter optimization in GP models and especially how uncertainty quantification can assist in the learning process. We investigate the effect of adaptive kappa parameters that govern the exploration-exploitation trade-off and the interplay between dimensionality, penalty on uncertainty, and noise levels to influence optimization results. Uncertainty quantification is built directly into our comprehensive theoretical framework and gives us new algorithms to dynamically tune exploration-exploitation trade-offs according to the uncertainty trend observed in nature. We rigorously empirically test various strategies, parametrizing our tests by dimensionality, noise, penalty terms, and evaluate the performance of any given strategery in a wide variety of test settings, and show conclusively that adaptive strategies always outperform fixed ones, but in difficult settings, where the dimensions are large and the noise is severe, the advantage is enormous. We build theoretical assurances of convergence under different settings as well as furnish a sensible direction on the application of adaptive GP-based optimization even in very complicated conditions. The results of our work will help in the development of more efficient and robust methods of optimization of realistic problems in which there are only a few functions available for evaluation, and when quantifying the uncertainty, there is a need to know more about the uncertainty.