Neuro-Symbolic Entropy Regularization (2201.11250v1)

Published 25 Jan 2022 in cs.LG, cs.AI, cs.LO, and stat.ML

Abstract: In structured prediction, the goal is to jointly predict many output variables that together encode a structured object -- a path in a graph, an entity-relation triple, or an ordering of objects. Such a large output space makes learning hard and requires vast amounts of labeled data. Different approaches leverage alternate sources of supervision. One approach -- entropy regularization -- posits that decision boundaries should lie in low-probability regions. It extracts supervision from unlabeled examples, but remains agnostic to the structure of the output space. Conversely, neuro-symbolic approaches exploit the knowledge that not every prediction corresponds to a valid structure in the output space. Yet, they does not further restrict the learned output distribution. This paper introduces a framework that unifies both approaches. We propose a loss, neuro-symbolic entropy regularization, that encourages the model to confidently predict a valid object. It is obtained by restricting entropy regularization to the distribution over only valid structures. This loss is efficiently computed when the output constraint is expressed as a tractable logic circuit. Moreover, it seamlessly integrates with other neuro-symbolic losses that eliminate invalid predictions. We demonstrate the efficacy of our approach on a series of semi-supervised and fully-supervised structured-prediction experiments, where we find that it leads to models whose predictions are more accurate and more likely to be valid.

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Summary

The paper introduces a neuro-symbolic entropy regularization framework that restricts entropy to valid output structures, improving prediction confidence and constraint adherence.
It employs an efficient algorithm leveraging circuit compilations to compute conditional entropy in linear time relative to circuit size, overcoming NP-hard challenges.
Empirical evaluations on tasks like entity-relation extraction and grid path prediction demonstrate superior accuracy and compliance compared to traditional methods.

Neuro-Symbolic Entropy Regularization

This essay explores the key contributions and evaluations presented in the paper "Neuro-Symbolic Entropy Regularization." The work introduces a novel framework that combines entropy regularization with neuro-symbolic approaches to improve structured prediction tasks. The framework focuses on leveraging logical constraints to guide the predictive distribution of neural networks, ensuring both accuracy and validity of output structures.

Introduction

The paper addresses the challenges of structured prediction, where the goal is to predict interdependent output variables that represent structured objects. These tasks often require large quantities of labeled data, which are not always readily available. Traditional entropy regularization minimizes class overlap by reducing predictive distribution entropy, assuming data clusters naturally. However, it often neglects the inherent structure of output spaces, potentially leading to invalid predictions despite high confidence.

Conversely, neuro-symbolic methods leverage symbolic logic to ensure predictions adhere to valid structures but do not constrain the model's confidence in its predictions. This paper proposes a unified approach, neuro-symbolic entropy regularization, which restricts entropy computation to only valid structures. This integration helps models confidently predict valid structures, improving accuracy and adherence to constraints.

Neuro-Symbolic Entropy Loss

Background

Logical constraints are represented as Boolean formulas over variables. Neuro-symbolic reasoning uses these constraints to guide predictions. Consider a logical sentence $\alpha$ over variables $Y = \{Y_1,...,Y_n\}$ ; a neural network outputs probabilities for each variable, inducing a distribution over the possible states of $\alpha$ .

Motivation and Definition

The method restricts entropy regularization to valid output structures characterized by logical constraints. The resulting loss encourages neural networks to allocate probability mass among valid structures, increasing both predictive certainty and compliance with constraints.

Figure 1: Warcraft dataset demonstrating input, output, and valid shortest path prediction.

Computing the Loss

The neuro-symbolic entropy loss is computationally challenging because it involves computing conditional entropy over logical constraints—an NP-hard problem. The paper proposes an efficient algorithm leveraging tractable circuit compilations of logical constraints, allowing linear time computation relative to the circuit's size.

Algorithm

The algorithm decomposes expectation over a distribution by recursively partitioning query variables and distribution support. For conjunctions, entropy is computed as the sum of individual entropies. For disjunctions, decomposition yields a sum of child entropies weighted by probabilities, effectively managing complexity with modular computation.

Experimental Evaluation

The paper evaluates the approach across various tasks, including entity-relation extraction and preference learning, in both semi-supervised and fully-supervised settings. Results demonstrate improved model accuracy and compliance with constraints, showing the superiority of neuro-symbolic entropy over traditional and other neuro-symbolic methods.

Semi-Supervised Learning

On datasets like ACE05 and SciERC, integrating entropy regularization with neuro-symbolic approaches achieved higher accuracy than baseline semantic loss, self-training, and fuzzy logic alternatives. The restricted entropy approach showed substantial improvements in coherence and constraint adherence.

Fully-Supervised Learning

In structured prediction tasks such as grid path prediction, preference learning, and Warcraft shortest path, the proposed method outperformed baselines by improving the validity of predictions and maintaining high accuracy despite full supervision, demonstrating the method's robustness across learning settings.

Neuro-symbolic reasoning has gained attention for integrating neural networks with logical reasoning for structured prediction. Existing methods, such as fuzzy logic and logic programming extensions (e.g., DeepProbLog), address components of this integration but fall short in achieving efficient, confident, and valid predictions concurrently.

The paper advances this field by offering a systematic framework that unifies entropy regularization with logical structure-oriented predictions, backed by compelling empirical evidence. Future development could involve scaling this approach to more complex logical representations and diverse application domains. The framework sets a new standard for ensuring prediction integrity in machine learning models through principled integration of symbolic and statistical learning.