295 research outputs found
Exponential Regret Bounds for Gaussian Process Bandits with Deterministic Observations
This paper analyzes the problem of Gaussian process (GP) bandits with
deterministic observations. The analysis uses a branch and bound algorithm that
is related to the UCB algorithm of (Srinivas et al, 2010). For GPs with
Gaussian observation noise, with variance strictly greater than zero, Srinivas
et al proved that the regret vanishes at the approximate rate of
, where t is the number of observations. To complement their
result, we attack the deterministic case and attain a much faster exponential
convergence rate. Under some regularity assumptions, we show that the regret
decreases asymptotically according to
with high probability. Here, d is the dimension of the search space and tau is
a constant that depends on the behaviour of the objective function near its
global maximum.Comment: Appears in Proceedings of the 29th International Conference on
Machine Learning (ICML 2012). arXiv admin note: substantial text overlap with
arXiv:1203.217
Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization
In this paper, we consider the problem of sequentially optimizing a black-box
function based on noisy samples and bandit feedback. We assume that is
smooth in the sense of having a bounded norm in some reproducing kernel Hilbert
space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian
process bandit optimization. We provide algorithm-independent lower bounds on
the simple regret, measuring the suboptimality of a single point reported after
rounds, and on the cumulative regret, measuring the sum of regrets over the
chosen points. For the isotropic squared-exponential kernel in
dimensions, we find that an average simple regret of requires , and the
average cumulative regret is at least , thus matching existing upper bounds up to the replacement of by
in both cases. For the Mat\'ern- kernel, we give analogous
bounds of the form and
, and discuss the resulting
gaps to the existing upper bounds.Comment: Appearing in COLT 2017. This version corrects a few minor mistakes in
Table I, which summarizes the new and existing regret bound
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