The convex hull of $n$-symbol Huffman trees is known to have exponentially many facets/constraints. This makes the standard on-line learning techniques for learning Huffman trees impractical, since they use multiplicative updates followed by projections to satisfy all of the constraints. However, there are general extended formulation techniques that encode the convex hull of Huffman trees as a polytope in a higher dimensional space with only polynomially many facets. This extended formulation methodology can also be used to encode the $n$-element permutahedron in $O(n \log n)$ dimensions with only a polynomial number of facets. We develop a general technique for converting these extended formulations into efficient on-line algorithms with good relative loss bounds. The resulting algorithms have nearly the same regret bounds as state of the art algorithms for permutations, and are the first efficient algorithms for the on-line learning of Huffman trees.
['Holakou Rahmanian', 'S. V. N. Vishwanathan', 'David P. Helmbold']
Extended Formulation for Online Learning of Combinatorial Objects