A sequential Monte Carlo approach to Thompson sampling for Bayesian optimization
This work addresses a specific bottleneck in Bayesian optimization for expensive function evaluations, offering an incremental improvement by enabling Thompson sampling in continuous domains.
The paper tackled the problem of approximating the distribution of the maximum in Gaussian process-based Bayesian optimization, which is analytically intractable, by developing a sequential Monte Carlo algorithm that enables Thompson sampling for continuous input spaces without optimizing acquisition functions, resulting in competitive performance in limiting cumulative regret.
Bayesian optimization through Gaussian process regression is an effective method of optimizing an unknown function for which every measurement is expensive. It approximates the objective function and then recommends a new measurement point to try out. This recommendation is usually selected by optimizing a given acquisition function. After a sufficient number of measurements, a recommendation about the maximum is made. However, a key realization is that the maximum of a Gaussian process is not a deterministic point, but a random variable with a distribution of its own. This distribution cannot be calculated analytically. Our main contribution is an algorithm, inspired by sequential Monte Carlo samplers, that approximates this maximum distribution. Subsequently, by taking samples from this distribution, we enable Thompson sampling to be applied to (armed-bandit) optimization problems with a continuous input space. All this is done without requiring the optimization of a nonlinear acquisition function. Experiments have shown that the resulting optimization method has a competitive performance at keeping the cumulative regret limited.