Optimizing for the decision not the prediction: an exploration of Smooth Net Benefit as a training objective
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Computer Science > Machine Learning
arXiv:2609.12752 (cs)
[Submitted on 11 Sep 2026]
Title:Optimizing for the decision not the prediction: an exploration of Smooth Net Benefit as a training objective
Authors:Koen M.F. Gorgels, Lasai Barreñada, Maarten van Smeden, Ben Van Calster, Ewout W. Steyerberg, Wouter A.C. van Amsterdam
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Abstract: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.
| Comments: | 22 pages, 5 figures, 2 tables. Code and supplementary data available |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2609.12752 [cs.LG] |
| (or arXiv:2609.12752v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2609.12752
arXiv-issued DOI via DataCite (pending registration)
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