Optimistic optimization of a Brownian
This provides a significant improvement in sample complexity for stochastic optimization in continuous domains, moving from polynomial to logarithmic rates.
The paper tackles the problem of optimizing a Brownian motion to find an ε-approximation of its maximum with minimal function evaluations, achieving a sample complexity of order log²(1/ε).
We address the problem of optimizing a Brownian motion. We consider a (random) realization $W$ of a Brownian motion with input space in $[0,1]$. Given $W$, our goal is to return an $ε$-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order $\log^2(1/ε)$. This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive---each query depends on previous values---and is an instance of the optimism-in-the-face-of-uncertainty principle.