# Density-Ratio Losses for Post-Hoc Learning to Defer

arXiv: https://arxiv.org/abs/2605.19557 · HTML: https://mlsanity.com/paper/2605.19557/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 87 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 30 of 40)

- [Generalized Cross Entropy Loss for Training Deep Neural Networks with Noisy Labels](https://mlsanity.com/paper/1805.07836/) (2018)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [SelectiveNet: A Deep Neural Network with an Integrated Reject Option](https://mlsanity.com/paper/1901.09192/) (2019)
- [Consistent Estimators for Learning to Defer to an Expert](https://mlsanity.com/paper/2006.01862/) (2020)
- [Predict Responsibly: Improving Fairness and Accuracy by Learning to Defer](https://mlsanity.com/paper/1711.06664/) (2017)
- [Large Language Models in Finance: A Survey](https://mlsanity.com/paper/2311.10723/) (2023)
- [MedMNIST Classification Decathlon: A Lightweight AutoML Benchmark for Medical Image Analysis](https://mlsanity.com/paper/2010.14925/) (2020)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Language Model Cascades: Token-level uncertainty and beyond](https://mlsanity.com/paper/2404.10136/) (2024)
- [Language Model Cascades](https://mlsanity.com/paper/2207.10342/) (2022)
- [Differentiable Learning Under Triage](https://mlsanity.com/paper/2103.08902/) (2021)
- [Coping with Label Shift via Distributionally Robust Optimisation](https://mlsanity.com/paper/2010.12230/) (2020)
- [Who Should Predict? Exact Algorithms For Learning to Defer to Humans](https://mlsanity.com/paper/2301.06197/) (2023)
- [Calibrated Learning to Defer with One-vs-All Classifiers](https://mlsanity.com/paper/2202.03673/) (2022)
- [Optimal strategies for reject option classifiers](https://mlsanity.com/paper/2101.12523/) (2021)
- [Learning to Defer to Multiple Experts: Consistent Surrogate Losses, Confidence Calibration, and Conformal Ensembles](https://mlsanity.com/paper/2210.16955/) (2022)
- [When Does Confidence-Based Cascade Deferral Suffice?](https://mlsanity.com/paper/2307.02764/) (2023)
- [Classification with Rejection Based on Cost-sensitive Classification](https://mlsanity.com/paper/2010.11748/) (2020)
- [Model Cascading: Towards Jointly Improving Efficiency and Accuracy of NLP Systems](https://mlsanity.com/paper/2210.05528/) (2022)
- [On the Calibration of Multiclass Classification with Rejection](https://mlsanity.com/paper/1901.10655/) (2019)
- [Wisdom of Committees: An Overlooked Approach To Faster and More Accurate Models](https://mlsanity.com/paper/2012.01988/) (2020)
- [Principled Approaches for Learning to Defer with Multiple Experts](https://mlsanity.com/paper/2310.14774/) (2023)
- [Faster Cascades via Speculative Decoding](https://mlsanity.com/paper/2405.19261/) (2024)
- [Predictor-Rejector Multi-Class Abstention: Theoretical Analysis and Algorithms](https://mlsanity.com/paper/2310.14772/) (2023)
- [Mastering Multiple-Expert Routing: Realizable H-Consistency and Strong Guarantees for Learning to Defer](https://mlsanity.com/paper/2506.20650/) (2025)
- [TangoBERT: Reducing Inference Cost by using Cascaded Architecture](https://mlsanity.com/paper/2204.06271/) (2022)
- [AUC-based Selective Classification](https://mlsanity.com/paper/2210.10703/) (2022)
- [Budgeted Multiple-Expert Deferral](https://mlsanity.com/paper/2510.26706/) (2025)

## Built on by (most cited first, 1 of 1)

- [Cost-Aware Post-Hoc Deferral Under Calibration and Shift: An Environmental AI Case Study](https://mlsanity.com/paper/2609.09235/) (2026)

## Abstract (arXiv)

We study post-hoc Learning to Defer (L2D) through the lens of ideal distributions: divergence-regularized reweightings of the data distribution under which a model attains low loss. We define deferral via the density-ratio between a model's and an expert's ideals. Using the reduction from density-ratio estimation to class-probability estimation, we derive the DR CPE losses for post-hoc L2D scorers. Deferral decisions are then made by thresholding the scorer, allowing deferral rates to be adjusted without retraining. For KL-based ideal distributions, our deferral rules recovers Chow's rule under the original distribution and a connection to an expert-tilted Bayes posterior -- which incorporates the expert's performance -- depending on if the ideal distributions are joint or marginal distributions. Experimentally, our approach is competitive compared to common baselines and more robust across dataset settings. More broadly, our results cast post-hoc L2D as density-ratio learning between ideal distributions, bridging Chow-style rules, expert comparison, and elucidating connections to related learning settings including anomaly detection.