A Learnability Analysis on Neuro-Symbolic Learning

TL;DR

The paper develops a principled learnability framework for neuro-symbolic learning by recasting NeSy tasks as derived constraint satisfaction problems (DCSPs). It proves that a NeSy task is learnable if and only if the DCSP has a unique solution, with a concrete sample-complexity bound N ≥ (1/κ) log(|B|/ε) ensuring small concept error; when the DCSP admits multiple solutions (disagreement d > 0), the task is unlearnable and the asymptotic error scales as E* ≤ d/L. The authors further show that the NeSy risk and concept risk align under a unique DCSP solution, and that ensembles of unlearnable tasks can become learnable by enforcing mutual constraints that reduce solution-space ambiguity. Empirical validation on MNIST-like and real-world datasets demonstrates the surrogate risks (PNL/ABL) effectively minimize the NeSy risk for learnable tasks, and showcases how DCSP disagreement and ensemble configurations impact learnability. The work offers a framework to diagnose learnability and guides the design of new NeSy algorithms and task ensembles with practical impact for hybrid AI systems.

Abstract

This paper analyzes the learnability of neuro-symbolic (NeSy) tasks within hybrid systems. We show that the learnability of NeSy tasks can be characterized by their derived constraint satisfaction problems (DCSPs). Specifically, a task is learnable if the corresponding DCSP has a unique solution; otherwise, it is unlearnable. For learnable tasks, we establish error bounds by exploiting the clustering property of the hypothesis space. Additionally, we analyze the asymptotic error for general NeSy tasks, showing that the expected error scales with the disagreement among solutions. Our results offer a principled approach to determining learnability and provide insights into the design of new algorithms.

Figures (13)

• Figure 1: A typical inference process of hybrid neuro-symbolic system. Shadowed circles denote observed variables, ${\bm{x}}$ is raw input data, $\hat{{\bm{z}}}$ is intermediate concepts, $\hat{y}$ is the final answer inferred by $\mathtt{KB}$, and $y$ denotes the true final answer. The goal is to learn the model $f$.
• Figure 2: Accuracies versus sample size for different NeSy tasks (top MNIST and bottom KMNIST). The shadowed area denotes the standard error. The number of the DCSP solutions (#Sols) is shown at the top left of each plot. The asymptotic bound (green line) from \ref{['thm: average error bound']} indicates that concept accuracy should exceed this bound as the sample size grows.
• Figure 3: Accuracies on the learnable tasks.
• Figure 4: Ensemble of unlearnable NeSy tasks. The left shows confusion matrices and the right displays accuracy curves. (a) The top row illustrates an unlearnable case, where combining the tasks still results in multiple DCSP solutions. (b) The bottom row illustrates a learnable case, where combining the tasks reduces the DCSP solutions to a single one.
• Figure 5: Example of addition knowledge base with Python program form.

Theorems & Definitions (28)

• Theorem 1.1: Informal
• Example 1: Addition
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Definition 4.1
• Proposition 4.1
• Definition 4.2
• Definition 4.3
• Definition 4.4