Embedded Bandits for Large-Scale Black-Box Optimization

TL;DR

The paper introduces EmbeddedHunter, a hierarchical bandit algorithm leveraging random embedding for efficient large-scale black-box optimization, with theoretical regret bounds and superior empirical performance in low-dimensional problems.

Contribution

It presents the EmbeddedHunter algorithm that integrates random embedding into a hierarchical bandit framework, providing finite-time regret bounds and improved performance over existing methods.

Findings

EmbeddedHunter outperforms recent random embedding methods.

Theoretical regret bounds are established for the algorithm.

Numerical experiments confirm the efficiency in low-dimensional settings.

Abstract

Random embedding has been applied with empirical success to large-scale black-box optimization problems with low effective dimensions. This paper proposes the EmbeddedHunter algorithm, which incorporates the technique in a hierarchical stochastic bandit setting, following the optimism in the face of uncertainty principle and breaking away from the multiple-run framework in which random embedding has been conventionally applied similar to stochastic black-box optimization solvers. Our proposition is motivated by the bounded mean variation in the objective value for a low-dimensional point projected randomly into the decision space of Lipschitz-continuous problems. In essence, the EmbeddedHunter algorithm expands optimistically a partitioning tree over a low-dimensional---equal to the effective dimension of the problem---search space based on a bounded number of random embeddings of sampled points from the low-dimensional space. In contrast to the probabilistic theoretical guarantees of multiple-run random-embedding algorithms, the finite-time analysis of the proposed algorithm presents a theoretical upper bound on the regret as a function of the algorithm's number of iterations. Furthermore, numerical experiments were conducted to validate its performance. The results show a clear performance gain over recently proposed random embedding methods for large-scale problems, provided the intrinsic dimensionality is low.