A General Approximation-Optimization Approach to Large Margin Estimation of HMMs

The most successful modeling approach to automatic speech recognition (ASR) is to use a set of hidden Markov models (HMMs) as the acoustic models for subword or whole-word speech units and to use the statistical N-gram model as language model for words and/or word classes in sentences. All the model parameters, including HMMs and N-gram models, are estimated from a large amount of training data according to certain criterion. It has been shown that success of this kind of data-driven modeling approach highly depends on the goodness of estimated models. As for HMM-based acoustic models, the dominant estimation method is the Baum-Welch algorithm which is based on the maximum likelihood (ML) criterion. As an alternative to the ML estimation, discriminative training (DT) has also been extensively studied for HMMs in ASR. It has been demonstrated that various DT techniques, such as maximum mutual information (MMI), minimum classification error (MCE) and minimum phone error (MPE), can significantly improve speech recognition performance over the conventional maximum likelihood (ML) estimation. More recently, we have proposed the large margin estimation (LME) of HMMs for speech recognition (Li et al., 2005; Liu et al., 2005a; Li & Jiang, 2005; Jiang et al., 2006), where Gaussian mixture HMMs are estimated based on the principle of maximizing the minimum margin. From the theoretical results in machine learning (Vapnik, 1998), a large margin classifier implies a good generalization power and generally yields much lower generalization errors in new test data, as shown in support vector machine and boosting method. As in Li et al., 2005 and Li & Jiang, 2005, estimation of large margin CDHMMs turns out to be a constrained minimax optimization problem. In the past few years, several optimization methods have been proposed to solve this problem, such as iterative localized optimization in Li et al., 2005, constrained joint optimization method in Li & Jiang, 2005 and Jiang et al., 2006, and semi-definite programming (SDP) method in Li & Jiang, 2006a and Li & Jiang 2006b. In this paper, we present a general Approximation-optiMization (AM) approach to solve the LME problem of Gaussian mixture HMMs in ASR. Similar to the EM algorithm, each iteration of the AM method consists of two distinct steps: namely A-step and M-step. In A-step, the original LME problem is approximated by a simple convex optimization problem in a close proximity of initial model parameters. In M-step, the approximate convex optimization problem is solved by using efficient convex optimization algorithms.