# Online Learning for Structured Loss Spaces

## TL;DR

This paper develops a unified online learning framework for structured loss spaces using atomic norms, providing new regret bounds for low-rank and sparse structured losses and establishing fundamental lower bounds.

## Contribution

It introduces a general regret bound for an adaptive online mirror descent algorithm tailored to structured loss spaces, covering new low-rank and sparse settings.

## Key findings

- Regret bounds for structured loss spaces including low-rank and sparse cases.
- Lower bounds on regret based on rank and sparsity of loss vectors.
- Unified framework recovering standard regret bounds and extending to new structured settings.

## Abstract

We consider prediction with expert advice when the loss vectors are assumed to lie in a set described by the sum of atomic norm balls. We derive a regret bound for a general version of the online mirror descent (OMD) algorithm that uses a combination of regularizers, each adapted to the constituent atomic norms. The general result recovers standard OMD regret bounds, and yields regret bounds for new structured settings where the loss vectors are (i) noisy versions of points from a low-rank subspace, (ii) sparse vectors corrupted with noise, and (iii) sparse perturbations of low-rank vectors. For the problem of online learning with structured losses, we also show lower bounds on regret in terms of rank and sparsity of the source set of the loss vectors, which implies lower bounds for the above additive loss settings as well.