Online Learning of Combinatorial Objects
This thesis develops algorithms for learning combinatorial objects. A combinatorial object is a structured concept composed of components. Examples are permutations, Huffman trees, binary search trees and paths in a directed graph. Learning combinatorial objects is a challenging problem: First, the number of combinatorial objects is typically exponential in terms of number of components. Second, the convex hull of these objects is a polytope whose characterization in the original space may have exponentially many facets or a description of the polytope in terms of facets/inequalities may not be even known. Finally, the loss of each object could be a complicated function of its component and may not be simply additive as a function of the components. In this thesis, we explore a wide variety of combinatorial objects and address the challenges above. For each combinatorial object, we go beyond the original space of the problem and introduce auxiliary spaces and representations. The representation of the objects in these auxiliary spaces admits additive losses and polytopes with polynomially many facets. This allows us to extend well-known algorithms like Expanded Hedge and Component Hedge to these combinatorial objects for the first time.