Optimal binning: mathematical programming formulation (2001.08025v3)

Abstract: The optimal binning is the optimal discretization of a variable into bins given a discrete or continuous numeric target. We present a rigorous and extensible mathematical programming formulation for solving the optimal binning problem for a binary, continuous and multi-class target type, incorporating constraints not previously addressed. For all three target types, we introduce a convex mixed-integer programming formulation. Several algorithmic enhancements, such as automatic determination of the most suitable monotonic trend via a Machine-Learning-based classifier and implementation aspects are thoughtfully discussed. The new mathematical programming formulations are carefully implemented in the open-source python library OptBinning.

• The paper’s main contribution is the development of a convex MIP formulation that rigorously optimizes data binning with novel monotonicity and statistical constraints.
• It introduces automatic trend determination and flexible methods to handle binary, continuous, and multi-class targets in binning.
• Empirical results using OptBinning and solver comparisons demonstrate the model's efficiency in processing large, complex datasets.

An Overview of "Optimal Binning: Mathematical Programming Formulation"

The paper "Optimal Binning: Mathematical Programming Formulation" by Guillermo Navas-Palencia introduces a comprehensive mathematical programming approach for optimal binning, an essential data preprocessing technique extensively used in machine learning, particularly within credit risk modeling. Binning facilitates the discretization of continuous variables into groups or bins, enhancing model interpretability, handling missing values, outliers, and reducing noise, ultimately contributing to the simplification of complex models.

Core Contributions

Navas-Palencia develops a rigorous and extensible mathematical formulation to address the optimal binning problem, accommodating binary, continuous, and multi-class target variables while introducing new constraints not previously explored. He presents a convex mixed-integer programming (MIP) approach implemented in the open-source Python library OptBinning. This comprehensive formulation incorporates the constraints typically required for effective binning processes, alongside novel constraints addressing monotonicity and statistical significance in bin mergers.

Methodological Advancements

The paper advances the field by introducing several algorithmic enhancements:

1. Automatic Trend Determination: Leveraging a machine learning classifier, the proposed model autonomously identifies the most suitable monotonic trends for optimal binning.
2. Mixed-Integer Programming: By introducing a convex MIP formulation, the paper provides solutions ranging from integer linear programming for simpler scenarios to mixed-integer quadratic programming for more complex constraints.
3. Algorithmic Flexibility: The formulation is designed to handle various monotonic trends, including ascending, descending, concave, convex, peak, and valley trends, thus offering significant adaptability for diverse data scenarios.
4. Handling Special Cases: Special and missing values are incorporated effectively as separate bins post-optimization, ensuring comprehensive treatment of all potential data conditions.
5. Local and Heuristic Search Reformulation: For large instances, a reformulation reduces computational complexity by shifting the focus to specialized local search heuristics, which proves beneficial in handling substantial datasets.

Empirical Evaluation and Results

The paper includes several numerical experiments based on real-world datasets, showcasing the robustness and efficacy of the proposed approach. A comparison of different solvers demonstrates the model's capability to achieve optimal solutions efficiently, even under stringent constraints and large data volumes. The experiments highlight the performance of Google's OR-Tools solvers in contrast to the commercial solver LocalSolver, particularly under varying monotonic trends.

Practical Implications and Future Directions

The implications of this research are significant for credit risk modeling and other domains relying on categorical and continuous data transformations. The capability to rigorously address monotonicity and optimize binning under complex constraints enhances predictive modeling's robustness and reliability. Additionally, the development and open availability of the OptBinning tool encourage broader adoption and continuous improvement in practical settings.

Future research could explore extensions into multivariate and piecewise-linear binning. Moreover, further investigation into improving machine learning classifiers for automatic trend detection can contribute to augmented decision processes in the binning framework.

In conclusion, this paper provides a solid foundational approach to optimal binning, effectively bridging gaps between traditional heuristic methods and rigorous mathematical programming formulations. Its contributions are poised to influence various domains where data transformation and preprocessing are pivotal.