We consider active learning (AL) in an uncertain environment in which
trade-off between multiple risk measures need to be considered. As an AL
problem in such an uncertain environment, we study Mean-Variance Analysis in
Bayesian Optimization (MVA-BO) setting. Mean-variance analysis was developed in
the field of financial engineering and has been used to make decisions that
take into account the trade-off between the average and variance of investment
uncertainty. In this paper, we specifically focus on BO setting with an
uncertain component and consider multi-task, multi-objective, and constrained
optimization scenarios for the mean-variance trade-off of the uncertain
component. When the target blackbox function is modeled by Gaussian Process
(GP), we derive the bounds of the two risk measures and propose AL algorithm
for each of the above three problems based on the risk measure bounds. We show
the effectiveness of the proposed AL algorithms through theoretical analysis
and numerical experiments.

本文研究了在不确定环境下的主动学习问题，着重于基于贝叶斯优化的均值 - 方差分析问题，针对高斯过程模型的多任务、多目标和约束优化场景，提出了考虑不确定因素的均值 - 方差权衡的风险度量上下界的主动学习算法，并通过理论和数值实验证明了其有效性。

贝叶斯优化中的均值方差分析

Mean-Variance Analysis in Bayesian Optimization under Uncertainty

We consider optimal decision-making problems in an uncertain environment. In
particular, we consider the case in which the distribution of the input is
unknown, yet there is abundant historical data drawn from the distribution. In
this paper, we propose a new type of distributionally robust optimization model
called the likelihood robust optimization (LRO) model for this class of
problems. In contrast to previous work on distributionally robust optimization
that focuses on certain parameters (e.g., mean, variance, etc.) of the input
distribution, we exploit the historical data and define the accessible
distribution set to contain only those distributions that make the observed
data achieve a certain level of likelihood. Then we formulate the targeting
problem as one of optimizing the expected value of the objective function under
the worst-case distribution in that set. Our model avoids the
over-conservativeness of some prior robust approaches by ruling out unrealistic
distributions while maintaining robustness of the solution for any
statistically likely outcomes. We present statistical analyses of our model
using Bayesian statistics and empirical likelihood theory. Specifically, we
prove the asymptotic behavior of our distribution set and establish the
relationship between our model and other distributionally robust models. To
test the performance of our model, we apply it to the newsvendor problem and
the portfolio selection problem. The test results show that the solutions of
our model indeed have desirable performance.

本文提出了一种新型的分布鲁棒优化模型 —— 似然鲁棒优化模型，以历史数据为依据，用置信区间代替概率分布，解决了当环境不确定且输入分布未知的决策问题，避免了以往过于谨慎的方法对真实分布的偏离和对输出的限制。

面向数据驱动问题的似然鲁棒优化

Likelihood Robust Optimization for Data-driven Problems

Many tasks in human environments require performing a sequence of navigation
and manipulation steps involving objects. In unstructured human environments,
the location and configuration of the objects involved often change in
unpredictable ways. This requires a high-level planning strategy that is robust
and flexible in an uncertain environment. We propose a novel dynamic planning
strategy, which can be trained from a set of example sequences. High level
tasks are expressed as a sequence of primitive actions or controllers (with
appropriate parameters). Our score function, based on Markov Random Field
(MRF), captures the relations between environment, controllers, and their
arguments. By expressing the environment using sets of attributes, the approach
generalizes well to unseen scenarios. We train the parameters of our MRF using
a maximum margin learning method. We provide a detailed empirical validation of
our overall framework demonstrating successful plan strategies for a variety of
tasks.

研究提出了一种基于 Markov 随机场（MRF）的动态规划策略，通过训练参数可以更好地适应不确定的人工环境，并且可以通过一系列原始动作或控制器来表达高水平任务，经过详细的实验验证，该方法可以成功地规划各种任务。