多任务学习是指在一次学习过程中同时完成多个学习任务的方法。相比于传统的单任务学习方法，多任务学习更关注于各个任务之间的相关性，并利用此相关性以更低的运算成本得到更具泛化性的结果。尽管多任务学习方法已经被广泛应用到许多领域，但是其理论分析工作还处于起步阶段并仍有许多问题有待进一步的解决。在本项目中，我们将研究多任务学习过程的泛化界、一致性、收敛率和学习模型的可学习性等。考虑到多任务学习的特性，我们拟将经典的统计学习理论结果推广到向量值函数情形。我们将得到对应于向量值函数的偏差不等式、对称不等式进而求得多任务学习过程的泛化界。我们还将研究向量值函数类复杂度和任务相关性度量，并讨论任务相关性对多任务学习过程的一致性和收敛率的影响。以得到的理论结果为基础，我们将对已有的多任务学习算法模型进行改进。

Multi-task learning is referring to a learning process where multiple learning tasks are achieved simultaneously. Compared to the traditional single-task learning, the multi-task learning performs better at a lower cost by exploiting the relatedness among multiple tasks, and the learning results have a stronger generalization ability. At present, the multi-task learning has been widely used for many practical applications, e.g., biological image processing and pattern recognition. However, the theoretical research of multi-task learning is still in the beginning stages with many unsolved theoretical issues. In this project, we will study the theoretical properties of the multi-task learning process including the generalization bounds, the consistence and the rate of convergence. In view of the inherent characteristic of the multi-task learning, we will extend the classical results of the statistical learning theory to the setting of vector-valued functions. In particular, we will develop the deviation inequalities and the symmetrization inequalities for the vector-valued functions as well as the generalization bounds for multi-task learning. We will also study the complexity measures of the vector-valued function classes. Next, we will find some quantities to describe the relatedness between two tasks and then analyze how the task relatedness affects the consistence and the rate of convergence of the multi-task learning processes. Based on the theoretical findings, we will further improve the existing multi-task learning models.