  A fundamental challenge in machine learning is the choice of a loss as it
characterizes our learning task, is minimized in the training phase, and serves
as an evaluation criterion for estimators. Proper losses are commonly chosen,
ensuring minimizers of the full risk match the true probability vector.
Estimators induced from a proper loss are widely used to construct forecasters
for downstream tasks such as classification and ranking. In this procedure, how
does the forecaster based on the obtained estimator perform well under a given
downstream task? This question is substantially relevant to the behavior of the
$p$-norm between the estimated and true probability vectors when the estimator
is updated. In the proper loss framework, the suboptimality of the estimated
probability vector from the true probability vector is measured by a surrogate
regret. First, we analyze a surrogate regret and show that the strict
properness of a loss is necessary and sufficient to establish a non-vacuous
surrogate regret bound. Second, we solve an important open question that the
order of convergence in p-norm cannot be faster than the $1/2$-order of
surrogate regrets for a broad class of strictly proper losses. This implies
that strongly proper losses entail the optimal convergence rate.
