# Learning in Probabilistic Boolean Networks via Structural Policy Gradients

## TL;DR

This paper introduces a new way to train interpretable rule-based models that perform as well as neural networks on various tasks.

## Contribution

The novel approach uses structural policy gradients to train Probabilistic Boolean Networks with interpretable logic.

## Key findings

- Learning PBNs achieve ANN-level performance on classification, regression, and reinforcement learning tasks.
- The models maintain interpretability while achieving high accuracy and low error rates.
- Learned logic becomes stable during training and performs well in noisy/tabular data regimes.

## Abstract

We revisit Probabilistic Boolean Networks as trainable function approximators. The key obstacle, non-differentiable structural choices (which predictors to read and which Boolean operators to apply), is addressed by casting the PBN’s structure as a stochastic policy whose parameters are optimized with score-function (REINFORCE) gradients. Continuous output heads (logistic/linear/softmax or policy logits) are trained with ordinary gradients. We call the resulting model a Learning PBN. We formalize the Learning Probabilistic Boolean Network, derive unbiased structural gradients with variance reduction, and prove a universal approximation property over discretized inputs. Empirically, Learning Probabilistic Boolean Networks approach ANN performance across classification (accuracy ↑), regression (RMSE ↓), representation quality via clustering (ARI ↑), and reinforcement learning (return ↑) while yielding interpretable, rule-like internal units. We analyze the effect of binning resolution, operator sets, and unit counts, and show how the learned logic stabilizes as training progresses. Our results indicate that PBNs can serve as general-purpose learners, competitive with ANNs in tabular/noisy regimes, without sacrificing interpretability.