  The standard techniques for online learning of combinatorial objects perform
multiplicative updates followed by projections into the convex hull of all the
objects. However, this methodology can be expensive if the convex hull contains
many facets. For example, the convex hull of $n$-symbol Huffman trees is known
to have exponentially many facets (Maurras et al., 2010). We get around this
difficulty by exploiting extended formulations (Kaibel, 2011), which encode the
polytope of combinatorial objects in a higher dimensional "extended" space with
only polynomially many facets. We develop a general framework for converting
extended formulations into efficient online algorithms with good relative loss
bounds. We present applications of our framework to online learning of Huffman
trees and permutations. The regret bounds of the resulting algorithms are
within a factor of $O(\sqrt{\log(n)})$ of the state-of-the-art specialized
algorithms for permutations, and depending on the loss regimes, improve on or
match the state-of-the-art for Huffman trees. Our method is general and can be
applied to other combinatorial objects.
