Online Learning for Structured Loss Spaces
This provides theoretical guarantees for online learning in structured loss settings, which is incremental but addresses specific practical constraints like noise and sparsity.
The paper tackles online learning with expert advice when loss vectors have structured constraints (low-rank, sparse, or both with noise), deriving a general regret bound for a modified online mirror descent algorithm and proving matching lower bounds based on rank and sparsity.
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.