While discriminative training (e.g., CRF, structural SVM) holds much promise for machine translation, image segmentation, and clustering, the complex inference these applications require make exact training intractable. This leads to a need for approximate training methods. Unfortunately, knowledge about how to perform efficient and effective approximate training is limited. Focusing on structural SVMs, we provide and explore algorithms for two different classes of approximate training algorithms, which we call undergenerating (e.g., greedy) and overgenerating (e.g., relaxations) algorithms. We provide a theoretical and empirical analysis of both types of approximate trained structural SVMs, focusing on fully connected pairwise Markov random fields. We find that models trained with overgenerating methods have theoretic advantages over undergenerating methods, are empirically robust relative to their undergenerating brethren, and relaxed trained models favor non-fractional predictions from relaxed predictor