Tight Regret Bounds for Noisy Optimization of a Brownian Motion
IEEE Transactions on Signal Processing(2022)
摘要
We consider the problem of Bayesian optimization of a one-dimensional Brownian motion in which the
$T$
adaptively chosen observations are corrupted by Gaussian noise. We show that the smallest possible expected cumulative regret and the smallest possible expected simple regret scale as
$\Omega (\sigma \sqrt {T / \log (T)}) \cap \mathcal {O}(\sigma \sqrt {T} \cdot \log T)$
and
$\Omega (\sigma / \sqrt {T \log (T)}) \cap \mathcal {O}(\sigma \log T / \sqrt {T})$
respectively, where
$\sigma ^2$
is the noise variance. Thus, our upper and lower bounds are tight up to a factor of
$\mathcal {O} ((\log T)^{1.5})$
. The upper bound uses an algorithm based on confidence bounds and the Markov property of Brownian motion (among other useful properties), and the lower bound is based on a reduction to binary hypothesis testing.
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关键词
Bayesian optimization,Brownian motion,non-smooth optimization,continuum-armed bandits,regret bounds,information-theoretic limits
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