Learning Using Privileged Information: SVM+ and Weighted SVM

Prior knowledge can be used to improve predictive performance of learning algorithms or reduce the amount of data required for training. The same goal is pursued within the learning using privileged information paradigm which was recently introduced by Vapnik et al. and is aimed at utilizing additional information available only at training time -- a framework implemented by SVM+. We relate the privileged information to importance weighting and show that the prior knowledge expressible with privileged features can also be encoded by weights associated with every training example. We show that a weighted SVM can always replicate an SVM+ solution, while the converse is not true and we construct a counterexample highlighting the limitations of SVM+. Finally, we touch on the problem of choosing weights for weighted SVMs when privileged features are not available.Comment: 18 pages, 8 figures; integrated reviewer comments, improved typesettin

Block-Coordinate Frank-Wolfe Optimization for Structural SVMs

We propose a randomized block-coordinate variant of the classic Frank-Wolfe algorithm for convex optimization with block-separable constraints. Despite its lower iteration cost, we show that it achieves a similar convergence rate in duality gap as the full Frank-Wolfe algorithm. We also show that, when applied to the dual structural support vector machine (SVM) objective, this yields an online algorithm that has the same low iteration complexity as primal stochastic subgradient methods. However, unlike stochastic subgradient methods, the block-coordinate Frank-Wolfe algorithm allows us to compute the optimal step-size and yields a computable duality gap guarantee. Our experiments indicate that this simple algorithm outperforms competing structural SVM solvers.Comment: Appears in Proceedings of the 30th International Conference on Machine Learning (ICML 2013). 9 pages main text + 22 pages appendix. Changes from v3 to v4: 1) Re-organized appendix; improved & clarified duality gap proofs; re-drew all plots; 2) Changed convention for Cf definition; 3) Added weighted averaging experiments + convergence results; 4) Clarified main text and relationship with appendi