Regret Bounds for Non-decomposable Metrics with Missing Labels
This addresses the challenge of accurate label recommendation under realistic conditions of missing labels for applications like information retrieval or content tagging, though it is incremental in extending existing methods to handle missing data explicitly.
The paper tackles the problem of recommending relevant labels with missing training data and non-decomposable evaluation metrics like F1, proposing a framework that derives regret bounds for collaborative filtering, multilabel classification, and PU learning, achieving significantly better performance in experiments.
We consider the problem of recommending relevant labels (items) for a given data point (user). In particular, we are interested in the practically important setting where the evaluation is with respect to non-decomposable (over labels) performance metrics like the $F_1$ measure, and the training data has missing labels. To this end, we propose a generic framework that given a performance metric $Ψ$, can devise a regularized objective function and a threshold such that all the values in the predicted score vector above and only above the threshold are selected to be positive. We show that the regret or generalization error in the given metric $Ψ$ is bounded ultimately by estimation error of certain underlying parameters. In particular, we derive regret bounds under three popular settings: a) collaborative filtering, b) multilabel classification, and c) PU (positive-unlabeled) learning. For each of the above problems, we can obtain precise non-asymptotic regret bound which is small even when a large fraction of labels is missing. Our empirical results on synthetic and benchmark datasets demonstrate that by explicitly modeling for missing labels and optimizing the desired performance metric, our algorithm indeed achieves significantly better performance (like $F_1$ score) when compared to methods that do not model missing label information carefully.