On the duality of strong convexity and strong smoothness: Learning applications and matrix regularization

We show 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.