Surrogate regret bounds for generalized classification performance metrics

We consider optimization of generalized performance metrics for binary classification by means of surrogate losses. We focus on a class of metrics, which are linear-fractional functions of the false positive and false negative rates (examples of which include $F_{\beta}$-measure, Jaccard similarity coefficient, AM measure, and many others). Our analysis concerns the following two-step procedure. First, a real-valued function $f$ is learned by minimizing a surrogate loss for binary classification on the training sample. It is assumed that the surrogate loss is a strongly proper composite loss function (examples of which include logistic loss, squared-error loss, exponential loss, etc.). Then, given $f$, a threshold $\widehat{\theta}$ is tuned on a separate validation sample, by direct optimization of the target performance metric. We show that the regret of the resulting classifier (obtained from thresholding $f$ on $\widehat{\theta}$) measured with respect to the target metric is upperbounded by the regret of $f$ measured with respect to the surrogate loss. We also extend our results to cover multilabel classification and provide regret bounds for micro- and macro-averaging measures. Our findings are further analyzed in a computational study on both synthetic and real data sets.