Neuro-symbolic Learning Yielding Logical Constraints

TL;DR

To bridge the gap between the continuous neural network and the discrete logical constraint, this paper introduces a difference-of-convex programming technique to relax the logical constraints while maintaining their precision and employs cardinality constraints as the language for logical constraint learning.

Abstract

Neuro-symbolic systems combine the abilities of neural perception and logical reasoning. However, end-to-end learning of neuro-symbolic systems is still an unsolved challenge. This paper proposes a natural framework that fuses neural network training, symbol grounding, and logical constraint synthesis into a coherent and efficient end-to-end learning process. The capability of this framework comes from the improved interactions between the neural and the symbolic parts of the system in both the training and inference stages. Technically, to bridge the gap between the continuous neural network and the discrete logical constraint, we introduce a difference-of-convex programming technique to relax the logical constraints while maintaining their precision. We also employ cardinality constraints as the language for logical constraint learning and incorporate a trust region method to avoid the degeneracy of logical constraint in learning. Both theoretical analyses and empirical evaluations substantiate the effectiveness of the proposed framework.

Figures (8)

• Figure 1: An example of neuro-symbolic learning for visual SudoKu solving. In this task, the neural network is employed to transform the puzzle image (strawberry etc.) into its corresponding symbols, while symbolic reasoning is utilized to produce the puzzle's solution. Importantly, the neuro-symbolic learning task is framed in a weakly supervised setting, where only the raw input (the puzzle image $\mathbf{x}$) and the final output (the puzzle solution $\mathbf{y}$, but without numbers in $\mathbf{z}$) is observed.
• Figure 2: Training curves of accuracy (left) and rank (right). Our method significantly boosts the efficiency of symbol grounding, and accurately converges to ground-truth constraints.
• Figure 3: Avoid degeneracy by trust region method. In logical constraint learning, the imposition of the Boolean constraints and the implicit bias of the stochastic gradient descent cause $\bm{w}_1, \dots, \bm{w}_4$ to converge to the same result (left figure), while the trust region constraints guarantee that they can sufficiently indicate different rules (right figure).
• Figure 4: Results (%) of chained XOR task, including accuracy and F$_1$ score (of class $0$). The sequence length ranges from $20$ to $200$, showing that our method stably outperforms competitors.
• Figure 5: An example of nonograms.

Theorems & Definitions (8)

• Proposition 1
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• Theorem 1
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