泛不确定测度空间上的统计学习理论基础研究

In recent years, uncertain statistical learning theory has been widespread concern in uncertainty mathematic and machine learning. It has got good results in dealing with learning problems of small samples on probability measure space and non- probability measure space. However, statistical learning theory is merely referred to statistical learning theory on probability measure space and non-probability measure space, it has yet to establish a unified mathematical axiom system. In order to combine probability measure and three representative non-probability measure (possibility measure, credibility measure and uncertain measure), the project intends to define a pan uncertain measure and give statistical learning theory on pan uncertain measure space. First, define pan uncertain measure and pan uncertain variables by the use of pan operator, discuss pan uncertain variables’ numerical characteristics, and prove important inequalities of pan uncertain variables on pan uncertain measure space. Then, give empirical risk minimization principle (ERM) and key theorem of learning theory on pan uncertain measure space. In the end, under ERM consistency condition on pan uncertain measure space, give series of conclusions the rate of uniform convergence of learning process and establish the structural risk minimization principles on pan uncertain measure space.

近年来，不确定统计学习理论在不确定性数学和机器学习领域得到了广泛的关注。它能有效地处理概率测度空间或非概率测度空间上的小样本学习问题。然而，不确定统计学习理论仅是概率测度空间和非概率测度空间上统计学习理论的统称，目前尚未建立一个统一的数学化公理体系。为了将概率测度与三种有代表性的非概率测度——可能性测度、可信性测度和不确定测度相结合，本项目拟定义一类泛不确定测度，并在泛不确定测度空间上建立统计学习理论的理论基础。首先，利用泛算子定义泛不确定测度和相应的泛不确定变量，在泛不确定测度空间上给出并证明泛不确定变量的数字特征和重要不等式。在此基础上，给出泛不确定测度空间上经验风险最小化原则（ERM）和学习理论的关键定理。在泛不确定测度空间上ERM一致性条件下，研究泛不确定测度空间上学习过程一致收敛速度界的系列结论，并在这些界的基础上建立泛不确定测度空间上的结构风险最小原则。

在很多学习问题中，随机性和不确定性会存在于同一复杂系统中。因此，我们既要考虑用统计方法来估计概率分布，也要同时邀请专家来估计信度。然而，基于随机变量的统计学习理论还难以处理这类机器学习问题。基于此，本项目将不确定随机变量引入到统计学习理论中，进一步提高了统计学习理论的应用范围。以不确定随机理论为基础，构建了基于不确定随机变量的损失函数、经验风险泛函、期望风险泛函以及经验风险最小化原则严格一致性条件。在不确定随机变量的大数定律的基础上，给出了经验风险最小化方法的严格一致性的充要条件。在此基础上，还讨论了几种情形下基于不确定随机变量的学习过程一致收敛速度的界。