There is growing body of learning problems for which it is natural to
organize the parameters into matrix, so as to appropriately regularize the
parameters under some matrix norm (in order to impose some more sophisticated
prior knowledge). This work describes and analyzes a systematic method for
constructing such matrix-based, regularization methods. In particular, we focus
on how the underlying statistical properties of a given problem can help us
decide which regularization function is appropriate.
Our methodology is based on the known duality fact: that a function is
strongly convex with respect to some norm if and only if its conjugate function
is strongly smooth with respect to the dual norm. This result has already been
found to be a key component in deriving and analyzing several learning
algorithms. We demonstrate the potential of this framework by deriving novel
generalization and regret bounds for multi-task learning, multi-class learning,
and kernel learning.

本文研究了一种将参数组织成矩阵的学习方式，采用矩阵范数来适当地规范参数以实现更复杂的先验知识。基于已知的对偶事实，即函数相对于某些范数是强凸的，当且仅当其共轭函数相对于对偶范数是强平滑的，我们的方法是基于统计学性质来确定合适的正则化函数。通过将这个框架应用于多任务学习、多类学习和核学习中，并推导出新的广义化和失误上限，我们展示了这个方法的潜力。