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 \emph{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 $ \widetilde{\mathcal{O}} \left( \left( \frac{1}{T} \right)^{ \frac{1}{d_s} } \right) $, where $ d_s \ge 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 $ \widetilde{\mathcal{O}} \left( \left( \frac{1}{T} \right)^{ \frac{1}{d_s + 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 emdowed with a Borel measure, and show that BLiN-MOS achieves a regret rate of order $ \widetilde{\mathcal{O}} \left( T^{ \frac{d_z}{d_z + 1} } \right) $, where $d_z$ is the zooming dimension of the problem instance. Our results show that in metric spaces with a Borel measure, the classic theory of Lipschitz bandits can be improved. This result suggests an intrinsic axiomatic gap between metric spaces and metric measure spaces from an algorithmic perspective, since the upper bound in a metric measure space breaks the known information-theoretical lower bounds for Lipschitz bandits in a metric space with no measure structure.