# Loss Minimization Yields Multicalibration for Large Neural Networks

arXiv: https://arxiv.org/abs/2304.09424 · HTML: https://mlsanity.com/paper/2304.09424/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 53 references that are in the map.
- First posted on arXiv 19 Apr 2023.

## Builds on (most cited first, 24 of 24)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Inherent Trade-Offs in the Fair Determination of Risk Scores](https://mlsanity.com/paper/1609.05807/) (2016)
- [Revisiting the Calibration of Modern Neural Networks](https://mlsanity.com/paper/2106.07998/) (2021)
- [Preventing Fairness Gerrymandering: Auditing and Learning for Subgroup Fairness](https://mlsanity.com/paper/1711.05144/) (2017)
- [Calibration of Pre-trained Transformers](https://mlsanity.com/paper/2003.07892/) (2020)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (2019)
- [Multiaccuracy: Black-Box Post-Processing for Fairness in Classification](https://mlsanity.com/paper/1805.12317/) (2018)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Soft Calibration Objectives for Neural Networks](https://mlsanity.com/paper/2108.00106/) (2021)
- [Outcome Indistinguishability](https://mlsanity.com/paper/2011.13426/) (2020)
- [Omnipredictors](https://mlsanity.com/paper/2109.05389/) (2021)
- [Moment Multicalibration for Uncertainty Estimation](https://mlsanity.com/paper/2008.08037/) (2020)
- [Loss Minimization through the Lens of Outcome Indistinguishability](https://mlsanity.com/paper/2210.08649/) (2022)
- [A Unifying Theory of Distance from Calibration](https://mlsanity.com/paper/2211.16886/) (2022)
- [Low-Degree Multicalibration](https://mlsanity.com/paper/2203.01255/) (2022)
- [Making Decisions under Outcome Performativity](https://mlsanity.com/paper/2210.01745/) (2022)
- [Multicalibration as Boosting for Regression](https://mlsanity.com/paper/2301.13767/) (2023)
- [The Calibration Generalization Gap](https://mlsanity.com/paper/2210.01964/) (2022)
- [HappyMap: A Generalized Multi-calibration Method](https://mlsanity.com/paper/2303.04379/) (2023)
- [Omnipredictors for Constrained Optimization](https://mlsanity.com/paper/2209.07463/) (2022)
- [Multicalibrated Partitions for Importance Weights](https://mlsanity.com/paper/2103.05853/) (2021)

## Built on by (most cited first, 19 of 19)

- [Calibrated Language Models Must Hallucinate](https://mlsanity.com/paper/2311.14648/) (2023)
- [When Does Optimizing a Proper Loss Yield Calibration?](https://mlsanity.com/paper/2305.18764/) (2023)
- [When is Multicalibration Post-Processing Necessary?](https://mlsanity.com/paper/2406.06487/) (2024)
- [Omnipredictors for Regression and the Approximate Rank of Convex Functions](https://mlsanity.com/paper/2401.14645/) (2024)
- [Calibration Error for Decision Making](https://mlsanity.com/paper/2404.13503/) (2024)
- [Optimal Multiclass U-Calibration Error and Beyond](https://mlsanity.com/paper/2405.19374/) (2024)
- [ElicitationGPT: Text Elicitation Mechanisms via Language Models](https://mlsanity.com/paper/2406.09363/) (2024)
- [Discretization-free Multicalibration through Loss Minimization over Tree Ensembles](https://mlsanity.com/paper/2505.17435/) (2025)
- [MCGrad: Multicalibration at Web Scale](https://mlsanity.com/paper/2509.19884/) (2025)
- [Comparing Uncertainty Measurement and Mitigation Methods for Large Language Models: A Systematic Review](https://mlsanity.com/paper/2504.18346/) (2025)
- [Supersimulators](https://mlsanity.com/paper/2509.17994/) (2025)
- [Multigroup Robustness](https://mlsanity.com/paper/2405.00614/) (2024)
- [Confidential Guardian: Cryptographically Prohibiting the Abuse of Model Abstention](https://mlsanity.com/paper/2505.23968/) (2025)
- [Multicalibration Yields Better Matchings](https://mlsanity.com/paper/2511.11413/) (2025)
- [Model Agreement via Anchoring](https://mlsanity.com/paper/2602.23360/) (2026)
- [Uniform convergence of the smooth calibration error and its relationship with functional gradient](https://mlsanity.com/paper/2505.19396/) (2025)
- [L2-Regularized Empirical Risk Minimization Guarantees Small Smooth Calibration Error](https://mlsanity.com/paper/2510.13450/) (2025)
- [Provable Long-Range Benefits of Next-Token Prediction](https://mlsanity.com/paper/2512.07818/) (2025)
- [Augmented Feature Boosting for Multicalibration](https://mlsanity.com/paper/2609.33891/) (2026)

## Abstract (arXiv)

Multicalibration is a notion of fairness for predictors that requires them to provide calibrated predictions across a large set of protected groups. Multicalibration is known to be a distinct goal than loss minimization, even for simple predictors such as linear functions. In this work, we consider the setting where the protected groups can be represented by neural networks of size k, and the predictors are neural networks of size n > k. We show that minimizing the squared loss over all neural nets of size n implies multicalibration for all but a bounded number of unlucky values of n. We also give evidence that our bound on the number of unlucky values is tight, given our proof technique. Previously, results of the flavor that loss minimization yields multicalibration were known only for predictors that were near the ground truth, hence were rather limited in applicability. Unlike these, our results rely on the expressivity of neural nets and utilize the representation of the predictor.