The fact that gLDA+SVM performs better than the standard gSLDA is due to the same reason, since the SVM part of gL-DA+SVM can well capture the supervision information to learn a classifier for good prediction, while standard sLDA can't well-balance the influence of supervision.
in training, we run the Markov chain for M iterations (i.e., the burn-in stage), as outlined in Algorithm 1.