# ESD: Expected Squared Difference as a Tuning-Free Trainable Calibration Measure

arXiv: https://arxiv.org/abs/2303.02472 · HTML: https://mlsanity.com/paper/2303.02472/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 36 references that are in the map.
- First posted on arXiv 4 Mar 2023.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [A large annotated corpus for learning natural language inference](https://mlsanity.com/paper/1508.05326/) (2015)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [End to End Learning for Self-Driving Cars](https://mlsanity.com/paper/1604.07316/) (2016)
- [Adversarial NLI: A New Benchmark for Natural Language Understanding](https://mlsanity.com/paper/1910.14599/) (2019)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Calibrating Deep Neural Networks using Focal Loss](https://mlsanity.com/paper/2002.09437/) (2020)
- [Improving model calibration with accuracy versus uncertainty optimization](https://mlsanity.com/paper/2012.07923/) (2020)
- [Calibration of Neural Networks using Splines](https://mlsanity.com/paper/2006.12800/) (2020)
- [Soft Calibration Objectives for Neural Networks](https://mlsanity.com/paper/2108.00106/) (2021)
- [LAD: A Hybrid Deep Learning System for Benign Paroxysmal Positional Vertigo Disorders Diagnostic](https://mlsanity.com/paper/2210.08282/) (2022)
- [Local Interpretability of Calibrated Prediction Models: A Case of Type 2 Diabetes Mellitus Screening Test](https://mlsanity.com/paper/2006.13815/) (2020)

## Built on by (most cited first, 13 of 13)

- [C-TPT: Calibrated Test-Time Prompt Tuning for Vision-Language Models via Text Feature Dispersion](https://mlsanity.com/paper/2403.14119/) (2024)
- [A Large-Scale Study of Probabilistic Calibration in Neural Network Regression](https://mlsanity.com/paper/2306.02738/) (2023)
- [Uncertainty-Calibrated Test-Time Model Adaptation without Forgetting](https://mlsanity.com/paper/2403.11491/) (2024)
- [FRAG: Frequency Adapting Group for Diffusion Video Editing](https://mlsanity.com/paper/2406.06044/) (2024)
- [TPC: Test-time Procrustes Calibration for Diffusion-based Human Image Animation](https://mlsanity.com/paper/2410.24037/) (2024)
- [ConfPO: Exploiting Policy Model Confidence for Critical Token Selection in Preference Optimization](https://mlsanity.com/paper/2506.08712/) (2025)
- [Calibrating Bayesian Learning via Regularization, Confidence Minimization, and Selective Inference](https://mlsanity.com/paper/2404.11350/) (2024)
- [Probabilistic Calibration by Design for Neural Network Regression](https://mlsanity.com/paper/2403.11964/) (2024)
- [Multivariate Latent Recalibration for Conditional Normalizing Flows](https://mlsanity.com/paper/2505.16636/) (2025)
- [A-TPT: Angular Diversity Calibration Properties for Test-Time Prompt Tuning of Vision-Language Models](https://mlsanity.com/paper/2510.26441/) (2025)
- [h-calibration: Rethinking Classifier Recalibration with Probabilistic Error-Bounded Objective](https://mlsanity.com/paper/2506.17968/) (2025)
- [Improving Calibration in Test-Time Prompt Tuning for Vision-Language Models via Data-Free Flatness-Aware Prompt Pretraining](https://mlsanity.com/paper/2604.27715/) (2026)
- [Rethinking the Test-Time Prompt Tuning Objective from the Perspective of Calibration](https://mlsanity.com/paper/2608.30230/) (2026)

## Abstract (arXiv)

Studies have shown that modern neural networks tend to be poorly calibrated due to over-confident predictions. Traditionally, post-processing methods have been used to calibrate the model after training. In recent years, various trainable calibration measures have been proposed to incorporate them directly into the training process. However, these methods all incorporate internal hyperparameters, and the performance of these calibration objectives relies on tuning these hyperparameters, incurring more computational costs as the size of neural networks and datasets become larger. As such, we present Expected Squared Difference (ESD), a tuning-free (i.e., hyperparameter-free) trainable calibration objective loss, where we view the calibration error from the perspective of the squared difference between the two expectations. With extensive experiments on several architectures (CNNs, Transformers) and datasets, we demonstrate that (1) incorporating ESD into the training improves model calibration in various batch size settings without the need for internal hyperparameter tuning, (2) ESD yields the best-calibrated results compared with previous approaches, and (3) ESD drastically improves the computational costs required for calibration during training due to the absence of internal hyperparameter. The code is publicly accessible at https://github.com/hee-suk-yoon/ESD.