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 $Fb 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 $th^ 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 $th^ 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.
Machine Learning