Beyond NLL: Pathwise Cross-Entropy Loss for Discriminative and Calibrated Event-Time Survival Prediction.

Abstract

Deep survival models are increasingly used for time-to-event prediction under censoring, yet training objectives remain a bottleneck. The widely used discrete-time negative log-likelihood (NLL) supervises hazards and can suffer from temporal information imbalance and gradient attenuation, yielding early-dominated probability mass and degraded late-horizon calibration, especially under heavy censoring and competing risks. We introduce Pathwise Cross-Entropy (PCE), which utilizes a symmetric, full-path objective that directly learns the occurred-by-t trajectory as a Cumulative Incidence Function (CIF). This direct approach seamlessly yields a normalized Probability Mass Function (PMF) for predicting event times, unlike NLL, where the derived PMF is structurally biased toward monotonic decrease, hindering its predictive utility. In a counting-process view, PCE supplies bidirectional gradients and constitutes a strictly proper scoring rule on counting paths. We extend PCE to competing risks with cause-specific supervision that avoids the multinomial coupling in NLL under competing risks. Empirically, across the tabular SEER and a WSI-derived kidney dataset and multiple backbones, PCE consistently improves discrimination (C-index, AUC) and calibration (IBS), produces calibration plots (ECE and PP plots) that are closer to observation, and enables ordinal first-hit time prediction directly with minimal practical monotonicity violations. These results indicate that PCE is a reliable and interpretable objective for single and competing-risk survival.