On the Hardness of Probabilistic Neurosymbolic Learning

TL;DR

This paper investigates the computational complexity of training probabilistic neurosymbolic models, introduces a new unbiased gradient estimator called WeightME, and evaluates its effectiveness compared to biased methods.

Contribution

It proves the intractability of gradient approximation in general, shows tractability during training, and introduces WeightME with probabilistic guarantees for efficient gradient estimation.

Findings

Approximate gradients are generally intractable but become tractable during training.

WeightME provides an unbiased gradient estimate with probabilistic guarantees.

Biased gradient approximations struggle to optimize even when exact solutions are feasible.

Abstract

The limitations of purely neural learning have sparked an interest in probabilistic neurosymbolic models, which combine neural networks with probabilistic logical reasoning. As these neurosymbolic models are trained with gradient descent, we study the complexity of differentiating probabilistic reasoning. We prove that although approximating these gradients is intractable in general, it becomes tractable during training. Furthermore, we introduce WeightME, an unbiased gradient estimator based on model sampling. Under mild assumptions, WeightME approximates the gradient with probabilistic guarantees using a logarithmic number of calls to a SAT solver. Lastly, we evaluate the necessity of these guarantees on the gradient. Our experiments indicate that the existing biased approximations indeed struggle to optimize even when exact solving is still feasible.