# Regularization Techniques for Learning with Matrices

## TL;DR

This paper introduces a systematic framework for designing matrix-based regularization methods in machine learning, leveraging duality between strong convexity and smoothness to improve generalization and regret bounds.

## Contribution

It presents a novel methodology connecting matrix regularization functions with their conjugates, enabling tailored regularization based on problem properties.

## Key findings

- Derived new generalization bounds for multi-task learning
- Established regret bounds for multi-class learning
- Applied framework to kernel learning scenarios

## Abstract

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.