Analyzing Differentiable Fuzzy Implications

TL;DR

This paper investigates the behavior of fuzzy implications in differentiable settings, revealing limitations of existing implications, introducing sigmoidal implications, and demonstrating their effectiveness in semi-supervised learning.

Contribution

It analyzes formal properties of fuzzy implications in differentiable contexts, identifies issues, and proposes a new family of implications that improve semi-supervised learning performance.

Findings

Existing fuzzy implications are often unsuitable for differentiable learning.

Sigmoidal implications address gradient imbalance issues.

Sigmoidal implications outperform other fuzzy implications in semi-supervised tasks.

Abstract

Combining symbolic and neural approaches has gained considerable attention in the AI community, as it is often argued that the strengths and weaknesses of these approaches are complementary. One such trend in the literature are weakly supervised learning techniques that employ operators from fuzzy logics. In particular, they use prior background knowledge described in such logics to help the training of a neural network from unlabeled and noisy data. By interpreting logical symbols using neural networks (or grounding them), this background knowledge can be added to regular loss functions, hence making reasoning a part of learning. In this paper, we investigate how implications from the fuzzy logic literature behave in a differentiable setting. In such a setting, we analyze the differences between the formal properties of these fuzzy implications. It turns out that various fuzzy implications, including some of the most well-known, are highly unsuitable for use in a differentiable learning setting. A further finding shows a strong imbalance between gradients driven by the antecedent and the consequent of the implication. Furthermore, we introduce a new family of fuzzy implications (called sigmoidal implications) to tackle this phenomenon. Finally, we empirically show that it is possible to use Differentiable Fuzzy Logics for semi-supervised learning, and show that sigmoidal implications outperform other choices of fuzzy implications.