Value-Monotonicity Matters: A Concordance Loss for Deep Survival Prediction
Machine LearningComputer Vision and Pattern Recognition
Summary
The authors point out that deep survival models are usually evaluated with a measure called the C-index, but trained using different likelihood-based losses that don't always reflect this measure well during training. They show that these likelihood losses can decrease even when the C-index doesn't improve, making it hard to know if the model is actually getting better. To fix this, they propose a new loss function called sigmoid concordance loss (SCL) that better matches the C-index, so the loss value more reliably indicates model performance. Testing on many datasets, they find SCL performs similarly or better in ranking patients correctly and keeps a strong link between the loss value and the C-index during training.
Authors
Meixu Chen, Kai Wang, Jing Wang
Abstract
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.