Deep learning has been shown to achieve impressive results in several domains like computer vision and natural language processing. Deep architectures are typically trained following a supervised scheme and, therefore, they rely on the availability of a large amount of labeled training data to effectively learn their parameters. Neuro-symbolic approaches have recently gained popularity to inject prior knowledge into a deep learner without requiring it to induce this knowledge from data. These approaches can potentially learn competitive solutions with a significant reduction of the amount of supervised data. A large class of neuro-symbolic approaches is based on First-Order Logic to represent prior knowledge, that is relaxed to a differentiable form using fuzzy logic. This paper shows that the loss function expressing these neuro-symbolic learning tasks can be unambiguously determined given the selection of a t-norm generator. When restricted to simple supervised learning, the presented theoretical apparatus provides a clean justification to the popular cross-entropy loss, that has been shown to provide faster convergence and to reduce the vanishing gradient problem in very deep structures. One advantage of the proposed learning formulation is that it can be extended to all the knowledge that can be represented by a neuro-symbolic method, and it allows the development of a novel class of loss functions, that the experimental results show to lead to faster convergence rates than other approaches previously proposed in the literature.
From a learning point of view, different quantifier conversions can be taken into account and validated, as well. For instance, the arithmetic mean and maximum operator have been used to convert the universal and existential quantifiers in @cite_28 , respectively. Different possibilities have been considered for the universal quantifier in @cite_14 , while the existential quantifier depends on this choice via the application of the strong negation using the DeMorgan law. The Arithmetic mean operator has been shown to achieve better performances in the conversion of the universal quantifier @cite_14 , with the existential quantifier implemented by Skolemization. However, the universal and existential quantifiers can be thought of as a generalized AND and OR, respectively. Therefore, converting the quantifiers using a mean operator has no direct justification inside a logic theory.
@cite_28 @cite_14
2201744460 2963277062
Abstract This paper proposes a unified approach to learning from constraints, which integrates the ability of classical machine learning techniques to learn from continuous feature-based representations with the ability of reasoning using higher-level semantic knowledge typical of Statistical Relational Learning. Learning tasks are modeled in the general framework of multi-objective optimization, where a set of constraints must be satisfied in addition to the traditional smoothness regularization term. The constraints translate First Order Logic formulas, which can express learning-from-example supervisions and general prior knowledge about the environment by using fuzzy logic. By enforcing the constraints also on the test set, this paper presents a natural extension of the framework to perform collective classification. Interestingly, the theory holds for both the case of data represented by feature vectors and the case of data simply expressed by pattern identifiers, thus extending classic kernel machines and graph regularization, respectively. This paper also proposes a probabilistic interpretation of the proposed learning scheme, and highlights intriguing connections with probabilistic approaches like Markov Logic Networks. Experimental results on classic benchmarks provide clear evidence of the remarkable improvements that are obtained with respect to related approaches. Semantic Image Interpretation (SII) is the task of extracting structured semantic descriptions from images. It is widely agreed that the combined use of visual data and background knowledge is of great importance for SII. Recently, Statistical Relational Learning (SRL) approaches have been developed for reasoning under uncertainty and learning in the presence of data and rich knowledge. Logic Tensor Networks (LTNs) are a SRL framework which integrates neural networks with first-order fuzzy logic to allow (i) efficient learning from noisy data in the presence of logical constraints, and (ii) reasoning with logical formulas describing general properties of the data. In this paper, we develop and apply LTNs to two of the main tasks of SII, namely, the classification of an image's bounding boxes and the detection of the relevant part-of relations between objects. To the best of our knowledge, this is the first successful application of SRL to such SII tasks. The proposed approach is evaluated on a standard image processing benchmark. Experiments show that background knowledge in the form of logical constraints can improve the performance of purely data-driven approaches, including the state-of-theart Fast Region-based Convolutional Neural Networks (Fast R-CNN). Moreover, we show that the use of logical background knowledge adds robustness to the learning system when errors are present in the labels of the training data.