Mathematical Sciences: Sequential Experimentation and Adaptive Control

本研究将研究三个问题。第一类问题称为序列设计与调度中的强盗问题。多臂盗贼问题在每个阶段n的k个独立随机序列中根据特定的规则进行选择。“索引规则”选择具有最大“Gittins index”的序列。这些指数并不容易计算。研究人员将继续开发最优停止问题的简单近似。第二组问题使用样本均值的边界跨越理论来寻找对某些贝叶斯检验有用的近似值。研究者已经为一般指数族找到了这样的近似。现在，他建议将结果从指数族扩展到更高的维度。第三组问题涉及随机回归和自适应控制。经典的随机自适应控制是将输出设计为先前的p个输出和q个输入的线性组合时对输出的最优调节。根据各种标准来调节输出，例如最小化平方和等。这位调查员已经在这一领域取得了几项成果，他计划进一步研究这些问题。

This research will study three problems. The first set is called bandit problems in sequential design and scheduling. A multi-armed bandit problem selects between k independent stochastic sequences at every stage n according to a specified rule. The "index rule" selects a sequence with maximum "Gittins index". These indices are not easily computable. The investigator will continue to develop simple approximations to the optimal stopping problems. The second set of problems uses the boundary crossing theory for sample means to find useful approximations to certain Bayes tests. The investigator has already found such approximations for general exponential families. Now he proposes to extend the results beyond the exponential families and to higher dimensions. The third set of problems deals with stochastic regression and adaptive control. The classical stochastic adaptive control is the optimal regulation of the output when it is designed as a linear combination of previous p outputs and q inputs. The output is regulated according to various criteria such as mimimizing the sum of squares etc.. The investigator already has several results in this area and he plans to pursue these problems further.