Proper losses regret at least 1/2-order

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.

💡 Research Summary

This paper investigates the fundamental relationship between proper loss functions and the convergence rate of estimated probability vectors in multiclass classification. Proper losses—those whose expected risk is minimized by the true probability vector—are widely used to train probabilistic models, and their strict version guarantees a unique minimizer. The authors focus on the surrogate regret
(R(p,q)=L(p,q)-L(q,q))
where (L(p,q)) is the conditional risk of an estimate (p) when the true distribution is (q). They ask two central questions: (1) When does a non‑trivial bound linking the surrogate regret to the (p)-norm distance (|q-p|_p) exist? (2) How fast can this bound shrink as the regret tends to zero?

To answer (1), the paper introduces a “modulus of convexity” for the convex generator associated with a proper loss. By extending the classical Bregman divergence framework to multivariate functions on the probability simplex, they define a rate function (\psi) such that
\