From Random Search to Bandit Learning in Metric Measure Spaces

Random Search is one of the most widely-used method for Hyperparameter Optimization, and is critical to the success of deep learning models. Despite its astonishing performance, little non-heuristic theory has been developed to describe the underlying working mechanism. This paper gives a theoretical accounting of Random Search. We introduce the concept of scattering dimension that describes the landscape of the underlying function, and quantifies the performance of random search. We show that, when the environment is noise-free, the output of random search converges to the optimal value in probability at rate widetildeO left( left( frac1T right)^ frac1ds right) , where ds ≥ 0 is the scattering dimension of the underlying function. When the observed function values are corrupted by bounded iid noise, the output of random search converges to the optimal value in probability at rate widetildeO left( left( frac1T right)^ frac1ds + 1 right) . In addition, based on the principles of random search, we introduce an algorithm, called BLiN-MOS, for Lipschitz bandits in doubling metric spaces that are also endowed with a probability measure, and show that under mild conditions, BLiN-MOS achieves a regret rate of order widetildeO left( T^ fracdzdz + 1 right) , where dz is the zooming dimension of the problem instance.