Combining Bayesian Optimization and Lipschitz Optimization
This work addresses global optimization challenges for researchers and practitioners, but it is incremental as it builds on existing techniques.
The paper tackles the problem of optimizing black-box functions by combining Bayesian optimization and Lipschitz optimization, resulting in a method called Lipschitz Bayesian optimization (LBO) that maintains asymptotic runtime and can drastically improve performance, with experiments on 15 datasets showing substantial gains, such as Thompson sampling often outperforming other acquisition functions.
Bayesian optimization and Lipschitz optimization have developed alternative techniques for optimizing black-box functions. They each exploit a different form of prior about the function. In this work, we explore strategies to combine these techniques for better global optimization. In particular, we propose ways to use the Lipschitz continuity assumption within traditional BO algorithms, which we call Lipschitz Bayesian optimization (LBO). This approach does not increase the asymptotic runtime and in some cases drastically improves the performance (while in the worst-case the performance is similar). Indeed, in a particular setting, we prove that using the Lipschitz information yields the same or a better bound on the regret compared to using Bayesian optimization on its own. Moreover, we propose a simple heuristics to estimate the Lipschitz constant, and prove that a growing estimate of the Lipschitz constant is in some sense ``harmless''. Our experiments on 15 datasets with 4 acquisition functions show that in the worst case LBO performs similar to the underlying BO method while in some cases it performs substantially better. Thompson sampling in particular typically saw drastic improvements (as the Lipschitz information corrected for its well-known ``over-exploration'' phenomenon) and its LBO variant often outperformed other acquisition functions.