## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 6 (2012), 958-992.

### Calibrated asymmetric surrogate losses

#### Abstract

Surrogate losses underlie numerous state-of-the-art binary classification algorithms, such as support vector machines and boosting. The impact of a surrogate loss on the statistical performance of an algorithm is well-understood in symmetric classification settings, where the misclassification costs are equal and the loss is a margin loss. In particular, classification-calibrated losses are known to imply desirable properties such as consistency. While numerous efforts have been made to extend surrogate loss-based algorithms to asymmetric settings, to deal with unequal misclassification costs or training data imbalance, considerably less attention has been paid to whether the modified loss is still calibrated in some sense. This article extends the theory of classification-calibrated losses to asymmetric problems. As in the symmetric case, it is shown that calibrated asymmetric surrogate losses give rise to excess risk bounds, which control the expected misclassification cost in terms of the excess surrogate risk. This theory is illustrated on the class of uneven margin losses, and the uneven hinge, squared error, exponential, and sigmoid losses are treated in detail.

#### Article information

**Source**

Electron. J. Statist. Volume 6 (2012), 958-992.

**Dates**

First available in Project Euclid: 25 May 2012

**Permanent link to this document**

http://projecteuclid.org/euclid.ejs/1337951630

**Digital Object Identifier**

doi:10.1214/12-EJS699

**Mathematical Reviews number (MathSciNet)**

MR2988435

**Zentralblatt MATH identifier**

1335.62108

**Keywords**

Surrogate loss classification calibrated cost-sensitive classification imbalanced data uneven margin excess risk bound

#### Citation

Scott, Clayton. Calibrated asymmetric surrogate losses. Electron. J. Statist. 6 (2012), 958--992. doi:10.1214/12-EJS699. http://projecteuclid.org/euclid.ejs/1337951630.