End-Cut Preference in Survival Trees

2025-09-22

AI Summary

Maximizing the log-rank statistic in survival trees induces an endpoint-cut preference (ECP), wherein split points concentrate near covariate boundaries, yielding highly imbalanced nodes, obscuring weak signals, and compromising model stability. This work provides the first systematic characterization of ECP in survival tree learning. We propose a novel, theoretically grounded framework that replaces the conventional hard-threshold indicator function with a smooth, differentiable sigmoid function. This reformulation mitigates ECP by enabling continuous, gradient-informed optimization of split criteria. Within the greedy splitting procedure, our method integrates the smoothed splitting criterion with the log-rank test to achieve more robust cutpoint selection. Empirical evaluation demonstrates substantial improvements in node balance, enhanced detection of weak prognostic effects, increased structural stability of the resulting trees, and improved interpretability—establishing a more reliable nonparametric tool for survival analysis.

Technology Category

• Machine Learning: Ensemble Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

The end-cut preference (ECP) problem, referring to the tendency to favor split points near the boundaries of a feature's range, is a well-known issue in CART (Breiman et al., 1984). ECP may induce highly imbalanced and biased splits, obscure weak signals, and lead to tree structures that are both unstable and difficult to interpret. For survival trees, we show that ECP also arises when using greedy search to select the optimal cutoff point by maximizing the log-rank test statistic. To address this issue, we propose a smooth sigmoid surrogate (SSS) approach, in which the hard-threshold indicator function is replaced by a smooth sigmoid function. We further demonstrate, both theoretically and through numerical illustrations, that SSS provides an effective remedy for mitigating or avoiding ECP.

Problem

• Addresses end-cut preference bias in survival tree algorithms
• Mitigates imbalanced splits from greedy log-rank optimization
• Proposes smooth sigmoid surrogate to improve tree stability

Innovation

• Uses smooth sigmoid surrogate for log-rank test
• Replaces hard-threshold indicator function
• Mitigates end-cut preference in survival trees