Conditional random fields(CRFs) are a class of undirected graphical models which have been widely used for classifying and labeling sequence data. The training of CRFs is typically formulated as an unconstrained optimization problem that maximizes the conditional likelihood. However, maximum likelihood training is prone to overfitting. To address this issue, we propose a novel constrained nonlinear optimization formulation in which the prediction accuracy of cross-validation sets are included as constraints. Instead of requiring multiple passes of training, the constrained formulation allows the cross-validation be handled in one pass of constrained optimization.

The new formulation is discontinuous, and classical Lagrangian based constraint handling methods are not applicable. A new constrained optimization algorithm based on the recently proposed extended saddle point theory is developed to learn the constrained CRF model. Experimental results on gene and stock-price prediction tasks show that the constrained formulation is able to significantly improve the generalization ability of CRF training.