From Random Search to Bandit Learning in Metric Measure Spaces

TL;DR

This paper provides a theoretical framework for understanding random search in hyperparameter optimization, introduces the scattering dimension to quantify landscape complexity, and proposes a new algorithm for Lipschitz bandits with proven regret bounds.

Contribution

It introduces the scattering dimension to analyze random search performance and develops BLiN-MOS, a new algorithm with regret guarantees for Lipschitz bandits in metric spaces.

Findings

Random search converges at rate depending on scattering dimension.

In noisy settings, convergence rate is affected by noise and scattering dimension.

BLiN-MOS achieves near-optimal regret in Lipschitz bandit problems.

Abstract

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 endowed with a probability measure, and show that under mild conditions, 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.