In this paper, we consider the problem of Gaussian process (GP) optimization
with an added robustness requirement: The returned point may be perturbed by an
adversary, and we require the function value to remain as high as possible even
after this perturbation. This problem is motivated by settings in which the
underlying functions during optimization and implementation stages are
different, or when one is interested in finding an entire region of good inputs
rather than only a single point. We show that standard GP optimization
algorithms do not exhibit the desired robustness properties, and provide a
novel confidence-bound based algorithm StableOpt for this purpose. We
rigorously establish the required number of samples for StableOpt to find a
near-optimal point, and we complement this guarantee with an
algorithm-independent lower bound. We experimentally demonstrate several
potential applications of interest using real-world data sets, and we show that
StableOpt consistently succeeds in finding a stable maximizer where several
baseline methods fail.

考虑高斯过程优化问题，在最大化函数时增加健壮性，提出基于置信界限的 StableOpt 算法，该算法在真实数据集上的实验表明可以成功找到稳定的极大值点。

高斯过程在对抗鲁棒优化中的应用

Adversarially Robust Optimization with Gaussian Processes

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain satisfying a
Hayman-type asymmetry condition, and let $ D $ be an arbitrary bounded domain
referred to as "obstacle". We are interested in the behaviour of the first
Dirichlet eigenvalue $ \lambda_1(\Omega \setminus (x+D)) $. First, we prove an
upper bound on $ \lambda_1(\Omega \setminus (x+D)) $ in terms of the distance
of the set $ x+D $ to the set of maximum points $ x_0 $ of the first Dirichlet
ground state $ \phi_{\lambda_1} > 0 $ of $ \Omega $. In short, a direct
corollary is that if \begin{equation} \mu_\Omega := \max_{x}\lambda_1(\Omega
\setminus (x+D)) \end{equation} is large enough in terms of $ \lambda_1(\Omega)
$, then all maximizer sets $ x+D $ of $ \mu_\Omega $ are close to each maximum
point $ x_0 $ of $ \phi_{\lambda_1} $.
Second, we discuss the distribution of $ \phi_{\lambda_1(\Omega)} $ and the
possibility to inscribe wavelength balls at a given point in $ \Omega $.
Finally, we specify our observations to convex obstacles $ D $ and show that
if $ \mu_\Omega $ is sufficiently large with respect to $ \lambda_1(\Omega) $,
then all maximizers $ x+D $ of $ \mu_\Omega $ contain all maximum points $ x_0
$ of $ \phi_{\lambda_1(\Omega)} $.

研究了在满足 Hayman 型不对称条件的边界内限制在障碍物上的第一 Dirichlet 本征值的上界，并讨论了第一 Dirichlet 基态的分布及在凸障碍物上的情况.