Paper:
Predtron: A Family of Online Algorithms for General Prediction Problems
Modern prediction problems arising in multilabel learning and learning to rank pose unique challenges to the classical theory of supervised learning. These problems have large prediction and label spaces of a combinatorial nature and involve sophisticated loss functions. We offer a general framework to derive mistake driven online algorithms and associated loss bounds. The key ingredients in our framework are a general loss function, a general vector space representation of predictions, and a notion of margin with respect to a general norm. Our general algorithm, Predtron, yields the perceptron algorithm and its variants when instantiated on classic problems such as binary classification, multiclass classification, ordinal regression, and multilabel classification. For multilabel ranking and subset ranking, we derive novel algorithms, notions of margins, and loss bounds. A simulation study confirms the behavior predicted by our bounds and demonstrates the flexibility of the design choices in our framework.

Review:
The paper proposes a new general perceptron-like algorithm, Predtron, for learning problems with large prediction and label spaces possibly of combinatorial nature, where predictions and labels are not necessary the same objects. First, a general mistake bound is given for the case in which the data is separable with a margin. Then, the algorithm is instantiated for several specific learning problems, such as binary classification (with or without a reject option), multiclass classification, ordinal regression, multilabel classification, subset ranking and multilabel ranking. The paper ends with a computational experiment.The paper is very nicely written, clear and easy to follow. The results look technically sound (although I did not check all the proof in the appendix). It is hard for me to assess the novelty of the proposed algorithm. The algorithm is a version of perceptron based on a generalization of hinge loss for binary classification. Such a generalization was already considered before (mainly in the context of structural SVMs), and is usually referred to as structural hinge loss. I think the contribution here is to define separate prediction and label spaces and explicitly use representation of predictions (which is usually hidden in the feature vector for SVMs). As far as I can tell, the mistake bound for the algorithm is a novel, original result. The proof is actually quite simple and more or less follow standard techniques from the analysis of online gradient descent (and its generalization, online mirror descent, for Predtron with a link function) and from the mistake-bound analysis of the original perceptron algorithm.What is a bit dissatisfying is that both the theoretical analysis and the experiment only concern a very special case of linear separability with margin. This is a very strong assumption on the data, and rarely appears in practice. For instance, the labels (relevance judgments) generated in the experiment are deterministic functions of the input X, so there is no label noise involved. It would be nice to see a counterpart of Theorem 1 in the nonseparable case, where the regret would be expressed as the total loss of the algorithm minus the total phi-loss (structural hinge loss) of the best W. A bound on such defined regret can be obtained for the standard perceptron algorithm (see e.g. Cesa-Bianchi, Lugosi: "Prediction Learning and Games", Chapter 12).The results on subset ranking are clearly interesting, especially the analysis of representation function and the value of the margin (Lemma 4). However, from what I know, solving max within a surrogate for NDCG through a linear assignment problem is known; see, e.g.: Chapelle, Le, Smola: Large margin optimization of ranking measures Le, Smola, Chapelle, Teo: Optimization of Ranking Measures