Optimizing for the decision not the prediction: an exploration of Smooth Net Benefit as a training objective

RESUMO

Objective Prediction models are commonly trained using objectives such as Bernoulli negative log-likelihood (NLL), although downstream clinical decisions may depend on specific risk thresholds. We introduce Smooth Net Benefit ($\sigma$NB), a differentiable approximation of Net Benefit designed to align model training with threshold-specific clinical utility. Materials and Methods We evaluated $\sigma$NB as a training objective for logistic regression, generalized additive models (GAMs), and XGBoost with three Hessian implementations. Experiments used the Framingham cardiovascular risk dataset and 44 TabZilla datasets comprising 72 dataset-threshold combinations. Results $\sigma$NB training did not consistently improve Net Benefit in Framingham. Across the TabZilla benchmark, mean standardized Net Benefit for logistic regression increased from 0.5669 with NLL to 0.5765 with $\sigma$NB (mean difference 0.0096, 95% CI -0.0001 to 0.0193). For GAMs, mean standardized Net Benefit decreased from 0.5921 to 0.5625 (mean difference -0.0296, 95% CI -0.0721 to 0.0129). For XGBoost, NLL achieved 0.6745 compared with 0.6723--0.6735 across $\sigma$NB implementations. In logistic regression, $\sigma$NB gains were positively associated with the performance advantage of XGBoost over NLL-trained logistic regression. Discussion The effect of $\sigma$NB was context dependent, with modest gains concentrated in logistic regression and little benefit for more flexible model classes. This suggests that decision-focused optimization may be most useful when limited model flexibility leaves greater scope for improvement. Conclusion Our results do not support $\sigma$NB as a general replacement for NLL training, but support further investigation of decision-focused objectives in settings where conventional likelihood-based training may not adequately capture decision-relevant structure.