Understanding Boolean Function Learnability on Deep Neural Networks: PAC Learning Meets Neurosymbolic Models (2009.05908v3)

Abstract: Computational learning theory states that many classes of boolean formulas are learnable in polynomial time. This paper addresses the understudied subject of how, in practice, such formulas can be learned by deep neural networks. Specifically, we analyze boolean formulas associated with model-sampling benchmarks, combinatorial optimization problems, and random 3-CNFs with varying degrees of constrainedness. Our experiments indicate that: (i) neural learning generalizes better than pure rule-based systems and pure symbolic approach; (ii) relatively small and shallow neural networks are very good approximators of formulas associated with combinatorial optimization problems; (iii) smaller formulas seem harder to learn, possibly due to the fewer positive (satisfying) examples available; and (iv) interestingly, underconstrained 3-CNF formulas are more challenging to learn than overconstrained ones. Such findings pave the way for a better understanding, construction, and use of interpretable neurosymbolic AI methods.

• The paper shows that shallow deep neural networks achieve superior generalization by learning complex boolean functions with high accuracy on model sampling benchmarks.
• It employs a structured methodology using multi-layer perceptrons, training on datasets with positive and negative examples for rigorous cross-validation.
• The findings reveal that learning efficiency varies: smaller and underconstrained boolean formulas pose greater challenges compared to larger or overconstrained ones.

Exploring the Learnability of Boolean Functions by Deep Neural Networks

This blog post explores a paper that evaluates the effectiveness of deep neural networks in learning boolean functions, which are pivotal in various domains such as symbolic reasoning and combinatorial optimization. The paper investigates neural learning over boolean formulas linked to robust model-sampling benchmarks, combinatorial problems, and random 3-CNFs with differing levels of constrainedness. The experiments delineate that shallow neural networks can outperform traditional rule-based systems in generalization capabilities and that these networks can adeptly learn complex boolean functions which encode practical problem domains.

Methodological Overview

The research follows a structured methodology to evaluate deep learning performance on boolean formulas. The process starts by generating datasets composed of both positive and negative examples for given boolean formulas. These datasets then serve as the foundation for training and validating the performance of multi-layer perceptrons (MLPs) through cross-validation accuracy metrics. The MLPs investigated are relatively simple, comprising a few hidden layers, which suffices to demonstrate the capability of shallow networks to approximate complex boolean functions effectively.

Key Experimental Insights

Generalization Superiority

Neural network models were shown to generalize better than pure rule-based systems and symbolic approaches. This is particularly noted in large boolean formulas where MLPs achieved perfect learning scores on model sampling benchmarks, outperforming decision trees significantly.

Efficiency with Constrained Formulas

Smaller and simpler neural networks are quite effective in learning various classes of boolean functions that encode combinatorial optimization problems. The paper shows that even with increased complexity in formulas, the neural networks maintain high accuracy, suggesting a robust capability to handle intricate logical structures inherent in combinatorial problems.

Challenge with Smaller Formulas

The findings interestingly suggest that smaller formulas are more challenging to learn. This could be attributed to a smaller number of satisfying examples available in smaller formulas, which impacts the training efficiency of neural models.

Performance Across Constrainedness

The paper also explores the dependency of learnability on the constrainedness of random 3-CNF formulas. It reveals an intriguing pattern where underconstrained formulas are tougher to learn compared to overconstrained ones. This goes against the typical intuition that a higher clause-to-variable ratio, which indicates higher complexity, would pose more difficulty in learning.

Implications and Future Directions

The implications of this research are multifaceted. Practically, it paves the way for the integration of neural learning methods in applications requiring robust reasoning capabilities, like AI planning and decision making in uncertain environments. Theoretically, it contributes to the ongoing discourse on the utility of shallow neural networks in learning logical constructs, a foundational element in many AI systems.

The prospects for future work are ripe and varied. One avenue is to deepen the empirical evaluation by scaling the complexity of the neural networks or by integrating recent advancements in neural architecture design. Another interesting direction could be the exploration of the interplay between dataset coverage and the intrinsic properties of boolean functions, which might shed light on why smaller formulas are harder to learn. Additionally, extracting and examining the learned models could provide insights into how these neural networks internalize logical rules and constraints, advancing our understanding of neural-symbolic integration.

By systematically analyzing the learnability of boolean functions through deep neural networks, this paper contributes valuable insights that could inspire the next wave of innovations in both theoretical and applied machine learning.