# Probability Bounding: Post-Hoc Calibration via Box-Constrained Softmax

arXiv: https://arxiv.org/abs/2506.10572 · HTML: https://mlsanity.com/paper/2506.10572/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 62 references that are in the map.
- First posted on arXiv 12 Jun 2025.

## Builds on (most cited first, 30 of 39)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration](https://mlsanity.com/paper/1910.12656/) (2019)
- [Calibrating Deep Neural Networks using Focal Loss](https://mlsanity.com/paper/2002.09437/) (2020)
- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Mix-n-Match: Ensemble and Compositional Methods for Uncertainty Calibration in Deep Learning](https://mlsanity.com/paper/2003.07329/) (2020)
- [The Effect of Sampling Temperature on Problem Solving in Large Language Models](https://mlsanity.com/paper/2402.05201/) (2024)
- [The Road Less Scheduled](https://mlsanity.com/paper/2405.15682/) (2024)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Local Temperature Scaling for Probability Calibration](https://mlsanity.com/paper/2008.05105/) (2020)
- [Is Temperature the Creativity Parameter of Large Language Models?](https://mlsanity.com/paper/2405.00492/) (2024)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Intra Order-preserving Functions for Calibration of Multi-Class Neural Networks](https://mlsanity.com/paper/2003.06820/) (2020)
- [Smooth ECE: Principled Reliability Diagrams via Kernel Smoothing](https://mlsanity.com/paper/2309.12236/) (2023)
- [SparseMAP: Differentiable Sparse Structured Inference](https://mlsanity.com/paper/1802.04223/) (2018)
- [Non-Parametric Calibration for Classification](https://mlsanity.com/paper/1906.04933/) (2019)
- [A Unifying Theory of Distance from Calibration](https://mlsanity.com/paper/2211.16886/) (2022)
- [Parameterized Temperature Scaling for Boosting the Expressive Power in Post-Hoc Uncertainty Calibration](https://mlsanity.com/paper/2102.12182/) (2021)
- [When Does Optimizing a Proper Loss Yield Calibration?](https://mlsanity.com/paper/2305.18764/) (2023)
- [Meta-Cal: Well-controlled Post-hoc Calibration by Ranking](https://mlsanity.com/paper/2105.04290/) (2021)
- [Dual Focal Loss for Calibration](https://mlsanity.com/paper/2305.13665/) (2023)
- [AdaFocal: Calibration-aware Adaptive Focal Loss](https://mlsanity.com/paper/2211.11838/) (2022)
- [Overconfidence is a Dangerous Thing: Mitigating Membership Inference Attacks by Enforcing Less Confident Prediction](https://mlsanity.com/paper/2307.01610/) (2023)

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

- [Estimating Learners' Skill Acquisition Without Temporal Information](https://mlsanity.com/paper/2606.20611/) (2026)

## Abstract (arXiv)

Many studies have observed that modern neural networks achieve high accuracy while producing poorly calibrated probabilities, making calibration a critical practical issue. In this work, we propose probability bounding (PB), a novel post-hoc calibration method that mitigates both underconfidence and overconfidence by learning lower and upper bounds on the output probabilities. To implement PB, we introduce the box-constrained softmax (BCSoftmax) function, a generalization of Softmax that explicitly enforces lower and upper bounds on the output probabilities. While BCSoftmax is formulated as the solution to a box-constrained optimization problem, we develop an exact and efficient algorithm for computing BCSoftmax. We further provide theoretical guarantees for PB and introduce two variants of PB. We demonstrate the effectiveness of our methods experimentally on four real-world datasets, consistently reducing calibration errors. Our Python implementation is available at https://github.com/neonnnnn/torchbcsoftmax.