# Learning Optimized Risk Scores

arXiv: https://arxiv.org/abs/1610.00168 · HTML: https://mlsanity.com/paper/1610.00168/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Oct 2016.

## Built on by (most cited first, 30 of 47)

- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [Framework for Evaluating Faithfulness of Local Explanations](https://mlsanity.com/paper/2202.00734/) (2022)
- [Deep Cox Mixtures for Survival Regression](https://mlsanity.com/paper/2101.06536/) (2021)
- [Deep learning for temporal data representation in electronic health records: A systematic review of challenges and methodologies](https://mlsanity.com/paper/2107.09951/) (2021)
- [FasterRisk: Fast and Accurate Interpretable Risk Scores](https://mlsanity.com/paper/2210.05846/) (2022)
- [Fast Sparse Classification for Generalized Linear and Additive Models](https://mlsanity.com/paper/2202.11389/) (2022)
- [Connecting Interpretability and Robustness in Decision Trees through Separation](https://mlsanity.com/paper/2102.07048/) (2021)
- [A Holistic Approach to Interpretability in Financial Lending: Models, Visualizations, and Summary-Explanations](https://mlsanity.com/paper/2106.02605/) (2021)
- [TimberTrek: Exploring and Curating Sparse Decision Trees with Interactive Visualization](https://mlsanity.com/paper/2209.09227/) (2022)
- [Interpretable and Fair Boolean Rule Sets via Column Generation](https://mlsanity.com/paper/2111.08466/) (2021)
- [Learning Optimal Predictive Checklists](https://mlsanity.com/paper/2112.01020/) (2021)
- [AutoScore-Survival: Developing interpretable machine learning-based time-to-event scores with right-censored survival data](https://mlsanity.com/paper/2106.06957/) (2021)
- [Shapley variable importance clouds for interpretable machine learning](https://mlsanity.com/paper/2110.02484/) (2021)
- [Perspectives on Incorporating Expert Feedback into Model Updates](https://mlsanity.com/paper/2205.06905/) (2022)
- [When Personalization Harms: Reconsidering the Use of Group Attributes in Prediction](https://mlsanity.com/paper/2206.02058/) (2022)
- [Discretionary Trees: Understanding Street-Level Bureaucracy via Machine Learning](https://mlsanity.com/paper/2312.10694/) (2023)
- [Fair Decision Rules for Binary Classification](https://mlsanity.com/paper/2107.01325/) (2021)
- [Improved Policy Evaluation for Randomized Trials of Algorithmic Resource Allocation](https://mlsanity.com/paper/2302.02570/) (2023)
- [Fast and Interpretable Mortality Risk Scores for Critical Care Patients](https://mlsanity.com/paper/2311.13015/) (2023)
- [MINTY: Rule-based Models that Minimize the Need for Imputing Features with Missing Values](https://mlsanity.com/paper/2311.14108/) (2023)
- [Fairness in Contextual Resource Allocation Systems: Metrics and Incompatibility Results](https://mlsanity.com/paper/2212.01725/) (2022)
- [Ethical Implementation of Artificial Intelligence to Select Embryos in In Vitro Fertilization](https://mlsanity.com/paper/2105.00060/) (2021)
- [An Interpretable Loan Credit Evaluation Method Based on Rule Representation Learner](https://mlsanity.com/paper/2304.00731/) (2023)
- [Towards Explainable and Interpretable Musical Difficulty Estimation: A Parameter-efficient Approach](https://mlsanity.com/paper/2408.00473/) (2024)
- [Scalable First-order Method for Certifying Optimal k-Sparse GLMs](https://mlsanity.com/paper/2502.09502/) (2025)
- [Leo Breiman, the Rashomon Effect, and the Occam Dilemma](https://mlsanity.com/paper/2507.03884/) (2025)
- [Joint Score-Threshold Optimization for Interpretable Risk Assessment](https://mlsanity.com/paper/2510.21934/) (2025)
- [Automatic Construction of Clinical Scoring Systems with LLM Agents](https://mlsanity.com/paper/2601.22324/) (2026)
- [Learning an Interpretable Risk Scoring System for Maximizing Decision Net Benefit](https://mlsanity.com/paper/2604.04241/) (2026)
- [On the Importance of Application-Grounded Experimental Design for Evaluating Explainable ML Methods](https://mlsanity.com/paper/2206.13503/) (2022)

## Abstract (arXiv)

Risk scores are simple classification models that let users make quick risk predictions by adding and subtracting a few small numbers. These models are widely used in medicine and criminal justice, but are difficult to learn from data because they need to be calibrated, sparse, use small integer coefficients, and obey application-specific operational constraints. In this paper, we present a new machine learning approach to learn risk scores. We formulate the risk score problem as a mixed integer nonlinear program, and present a cutting plane algorithm for non-convex settings to efficiently recover its optimal solution. We improve our algorithm with specialized techniques to generate feasible solutions, narrow the optimality gap, and reduce data-related computation. Our approach can fit risk scores in a way that scales linearly in the number of samples, provides a certificate of optimality, and obeys real-world constraints without parameter tuning or post-processing. We benchmark the performance benefits of this approach through an extensive set of numerical experiments, comparing to risk scores built using heuristic approaches. We also discuss its practical benefits through a real-world application where we build a customized risk score for ICU seizure prediction in collaboration with the Massachusetts General Hospital.