  In this paper we refine the process of computing calibration functions for a
number of multiclass classification surrogate losses. Calibration functions are
a powerful tool for easily converting bounds for the surrogate risk (which can
be computed through well-known methods) into bounds for the true risk, the
probability of making a mistake. They are particularly suitable in
non-parametric settings, where the approximation error can be controlled, and
provide tighter bounds than the common technique of upper-bounding the 0-1 loss
by the surrogate loss.
  The abstract nature of the more sophisticated existing calibration function
results requires calibration functions to be explicitly derived on a
case-by-case basis, requiring repeated efforts whenever bounds for a new
surrogate loss are required. We devise a streamlined analysis that simplifies
the process of deriving calibration functions for a large number of surrogate
losses that have been proposed in the literature. The effort of deriving
calibration functions is then surmised in verifying, for a chosen surrogate
loss, a small number of conditions that we introduce.
  As case studies, we recover existing calibration functions for the well-known
loss of Lee et al. (2004), and also provide novel calibration functions for
well-known losses, including the one-versus-all loss and the logistic
regression loss, plus a number of other losses that have been shown to be
classification-calibrated in the past, but for which no calibration function
had been derived.
