# Convex Risk Minimization and Conditional Probability Estimation

arXiv: https://arxiv.org/abs/1506.04513 · HTML: https://mlsanity.com/paper/1506.04513/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2015.

## Built on by (most cited first, 2 of 2)

- [Non-asymptotic Excess Risk Bounds for Classification with Deep Convolutional Neural Networks](https://mlsanity.com/paper/2105.00292/) (2021)
- [Proper losses regret at least 1/2-order](https://mlsanity.com/paper/2407.10417/) (2024)

## Abstract (arXiv)

This paper proves, in very general settings, that convex risk minimization is a procedure to select a unique conditional probability model determined by the classification problem. Unlike most previous work, we give results that are general enough to include cases in which no minimum exists, as occurs typically, for instance, with standard boosting algorithms. Concretely, we first show that any sequence of predictors minimizing convex risk over the source distribution will converge to this unique model when the class of predictors is linear (but potentially of infinite dimension). Secondly, we show the same result holds for empirical risk minimization whenever this class of predictors is finite dimensional, where the essential technical contribution is a norm-free generalization bound.