Interpretable and Fair Boolean Rule Sets via Column Generation

Abstract

This paper considers the learning of Boolean rules in either disjunctive normal form (DNF, ORof-ANDs, equivalent to decision rule sets) or conjunctive normal form (CNF, AND-of-ORs) as an interpretable model for classification. An integer program is formulated to optimally trade classification accuracy for rule simplicity. We also consider the fairness setting and extend the formulation to include explicit constraints on two different measures of classification parity: equality of opportunity and equalized odds. Column generation (CG) is used to efficiently search over an exponential number of candidate clauses (conjunctions or disjunctions) without the need for heuristic rule mining. This approach also bounds the gap between the selected rule set and the best possible rule set on the training data. To handle large datasets, we propose an approximate CG algorithm using randomization. Compared to three recently proposed alternatives, the CG algorithm dominates the accuracy-simplicity trade-off in 8 out of 16 datasets. When maximized for accuracy, CG is competitive with rule learners designed for this purpose, sometimes finding significantly simpler solutions that are no less accurate. Compared to other fair and interpretable classifiers, our method is able to find rule sets that meet stricter notions of fairness with a modest trade-off in accuracy.

Explainable Artificial Intelligence Using Expressive Boolean Formulas

“…There is an existing body of work on using specific Boolean formulas as ML models. For example, learning conjunctive/disjunctive normal form (CNF/DNF) rules using a MaxSAT (Maximum Satisfiability) solver [22,23], an ILP solver [24][25][26], or via LP relaxation [27].…”

“…The prevalence of methods for finding optimal CNF (or equivalently, DNF) rules using MaxSAT solvers [22,23] or ILP solvers [24][25][26] suggests that one might use such a formula as the rule for the classifier. However, in this case, it is easy to construct simple datasets that require complicated rules.…”

Explaining with Greater Support: Weighted Column Sampling Optimization for q-Consistent Summary-Explanations

“…With the advances in machine learning and the explosion of artificial intelligence applications, the transparency of machine learning models in many contexts has become increasingly important. Some domains that require critical decision making, such as healthcare and criminal justice, have begun incorporating machine learning systems as support [6,15,22,25]. However, models with high accuracies are often black boxes, which are opaque and have their decision logic hidden.…”

“…al. [5,15], boolean decision rules are generated using an approximate column generation algorithm. Wang et.…”