10/01/1995 09:00 AM

Computer Science

We investigate the problem of estimating the proportion vector which maximizes the likelihood of a given sample for a mixture of given densities. We adapt a framework developed for supervised learning and give simple derivations for many of the standard iterative algorithms like gradient projection and EM. In this framework, the distance between the new and old proportion vectors is used as a penalty term. The square distance leads to the gradient projection update, and the relative entropy to a new update which we call the exponentiated gradient update ($EG_\\eta$). Curiously, when a second order Taylor expansion of the relative entropy is used, we arrive at an update $\\EMeta$ which, for $\\eta=1$, gives the usual EM update. Experimentally, both the $EM_\\eta$-update and the $EG_\\eta$-update for $\\eta>1$ outperform the EM algorithm and its variants. We also prove a polynomial bound on the worst-case global rate of convergence of the $EG_\\eta$ algorithm.