# Building a stable classifier with the inflated argmax

arXiv: https://arxiv.org/abs/2405.14064 · HTML: https://mlsanity.com/paper/2405.14064/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 35 references that are in the map.
- First posted on arXiv 22 May 2024.

## Builds on (most cited first, 14 of 14)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [Interpretable machine learning: definitions, methods, and applications](https://mlsanity.com/paper/1901.04592/) (2019)
- [Least Ambiguous Set-Valued Classifiers with Bounded Error Levels](https://mlsanity.com/paper/1609.00451/) (2016)
- [A comparative study of fairness-enhancing interventions in machine learning](https://mlsanity.com/paper/1802.04422/) (2018)
- [Conformal Language Modeling](https://mlsanity.com/paper/2306.10193/) (2023)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Veridical Data Science](https://mlsanity.com/paper/1901.08152/) (2019)
- [Cautious Deep Learning](https://mlsanity.com/paper/1805.09460/) (2018)
- [An Exploration of Softmax Alternatives Belonging to the Spherical Loss Family](https://mlsanity.com/paper/1511.05042/) (2015)
- [Set-valued classification -- overview via a unified framework](https://mlsanity.com/paper/2102.12318/) (2021)
- [Stable and Fair Classification](https://mlsanity.com/paper/1902.07823/) (2019)
- [Bagging Provides Assumption-free Stability](https://mlsanity.com/paper/2301.12600/) (2023)

## Built on by (most cited first, 8 of 8)

- [Stabilizing black-box model selection with the inflated argmax](https://mlsanity.com/paper/2410.18268/) (2024)
- [Assumption-free stability for ranking problems](https://mlsanity.com/paper/2506.02257/) (2025)
- [Multiclass threshold-based classification](https://mlsanity.com/paper/2505.11276/) (2025)
- [Aerial Image Classification in Scarce and Unconstrained Environments via Conformal Prediction](https://mlsanity.com/paper/2504.17655/) (2025)
- [Multiclass threshold-based classification and model evaluation](https://mlsanity.com/paper/2511.21794/) (2025)
- [The Multiclass Score-Oriented Loss (MultiSOL) on the Simplex](https://mlsanity.com/paper/2511.22587/) (2025)
- [The Price of Robustness: Stable Classifiers Need Overparameterization](https://mlsanity.com/paper/2603.02806/) (2026)
- [Stabilizing black-box algorithms through task-oriented randomization](https://mlsanity.com/paper/2606.25269/) (2026)

## Abstract (arXiv)

We propose a new framework for algorithmic stability in the context of multiclass classification. In practice, classification algorithms often operate by first assigning a continuous score (for instance, an estimated probability) to each possible label, then taking the maximizer -- i.e., selecting the class that has the highest score. A drawback of this type of approach is that it is inherently unstable, meaning that it is very sensitive to slight perturbations of the training data, since taking the maximizer is discontinuous. Motivated by this challenge, we propose a pipeline for constructing stable classifiers from data, using bagging (i.e., resampling and averaging) to produce stable continuous scores, and then using a stable relaxation of argmax, which we call the "inflated argmax," to convert these scores to a set of candidate labels. The resulting stability guarantee places no distributional assumptions on the data, does not depend on the number of classes or dimensionality of the covariates, and holds for any base classifier. Using a common benchmark data set, we demonstrate that the inflated argmax provides necessary protection against unstable classifiers, without loss of accuracy.