This paper proposes an efficient online method that trains a classifier with many conjunctive features. We employ kernel computation called kernel slicing, which explicitly considers conjunctions among frequent features in computing the polynomial kernel, to combine the merits of linear and kernel-based training. To improve the scalability of this training, we reuse the temporal margins of partial feature vectors and terminate unnecessary margin computations. Experiments on dependency parsing and hyponymy-relation extraction demonstrated that our method could train a classifier orders of magnitude faster than kernel-based online learning, while retaining its space efficiency.