It is known 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. Utilizing this duality, we isolate a single inequality which seamlessly implies both generalization bounds and online regret bounds; and we show how to construct strongly convex functions over matrices based on strongly convex functions over vectors. The newly constructed functions (over matrices) inherit the strong convexity properties of the underlying vector functions. We demonstrate the potential of this framework by analyzing several learning algorithms including group Lasso, kernel learning, and online control with adversarial quadratic costs.

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。

学习矩阵的正则化技术