Peter Bartlett, Michael Collins, Ben Taskar, David McAllester
We consider the problem of structured classiﬁcation, where the task is to predict a label y from an input x, and y has meaningful internal struc- ture. Our framework includes supervised training of Markov random ﬁelds and weighted context-free grammars as special cases. We describe an algorithm that solves the large-margin optimization problem deﬁned in , using an exponential-family (Gibbs distribution) representation of structured objects. The algorithm is efﬁcient—even in cases where the number of labels y is exponential in size—provided that certain expecta- tions under Gibbs distributions can be calculated efﬁciently. The method for structured labels relies on a more general result, speciﬁcally the ap- plication of exponentiated gradient updates [7, 8] to quadratic programs.