# Value-Monotonicity Matters: A Concordance Loss for Deep Survival Prediction

arXiv: https://arxiv.org/abs/2607.16802 · HTML: https://mlsanity.com/paper/2607.16802/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 46 references that are in the map.
- First posted on arXiv 18 Jul 2026.

## Builds on (most cited first, 11 of 11)

- [DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network](https://mlsanity.com/paper/1606.00931/) (2016)
- [Time-to-Event Prediction with Neural Networks and Cox Regression](https://mlsanity.com/paper/1907.00825/) (2019)
- [Multimodal Whole Slide Foundation Model for Pathology](https://mlsanity.com/paper/2411.19666/) (2024)
- [Modeling Dense Multimodal Interactions Between Biological Pathways and Histology for Survival Prediction](https://mlsanity.com/paper/2304.06819/) (2023)
- [A Scalable Discrete-Time Survival Model for Neural Networks](https://mlsanity.com/paper/1805.00917/) (2018)
- [Deep Survival Machines: Fully Parametric Survival Regression and Representation Learning for Censored Data with Competing Risks](https://mlsanity.com/paper/2003.01176/) (2020)
- [The University of California San Francisco Preoperative Diffuse Glioma MRI (UCSF-PDGM) Dataset](https://mlsanity.com/paper/2109.00356/) (2021)
- [SODEN: A Scalable Continuous-Time Survival Model through Ordinary Differential Equation Networks](https://mlsanity.com/paper/2008.08637/) (2020)
- [A Multimodal and Multi-centric Head and Neck Cancer Dataset for Segmentation, Diagnosis and Outcome Prediction](https://mlsanity.com/paper/2509.00367/) (2025)
- [HEad and neCK TumOR (HECKTOR) 2025: Benchmark of Segmentation, Diagnosis, and Prognosis in Multimodal PET/CT](https://mlsanity.com/paper/2606.20143/) (2026)
- [A Statistical Learning Take on the Concordance Index for Survival Analysis](https://mlsanity.com/paper/2302.12059/) (2023)

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

- [HERMES: A Hybrid Ensemble for Head-and-Neck Tumor Segmentation, TN Staging, and Recurrence-Free Survival on PET/CT](https://mlsanity.com/paper/2607.26498/) (2026)

## Abstract (arXiv)

Deep survival models are evaluated almost exclusively by the concordance index (C-index), yet they are commonly trained using likelihood objectives such as the Cox partial likelihood, discrete-time negative log-likelihood, and DeepHit likelihood. This mismatch is usually considered acceptable because the C-index can be recomputed on validation data during training. However, for end-to-end training of high-capacity encoders on small, heavily censored oncology cohorts, frequent C-index evaluation is computationally expensive, making the loss value itself an important signal for monitoring, early stopping, and model selection. We show that likelihood losses are unreliable for this purpose and propose a value-monotone concordance loss. We prove that every strictly proper survival likelihood admits directions where the loss decreases while the C-index remains unchanged, causing the loss value to decouple from ranking performance. We then study a sigmoid concordance loss (SCL), whose value approximates one minus the C-index up to a temperature term, ensuring that lower loss corresponds to higher C-index during optimization. The loss is architecture agnostic and reduces to a convex survival ranking support vector machine for linear models. Across eighteen datasets from four modalities using a unified five-fold cross-validation protocol, SCL achieves discrimination comparable to standard likelihood losses and is the best or within one standard deviation of the best C-index. Unlike likelihood losses, SCL maintains a strong correlation between loss value and C-index during training, with rank correlations of 0.96 to 0.99 compared with -0.03 to 0.53 for likelihood losses. Calibration measured by the integrated Brier score is comparable. SCL provides a value-monotone optimization objective whose value can serve as a reliable surrogate for the C-index during expensive end-to-end training.