Learning and T-Norms Theory

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.

An Unsupervised Character-Aware Neural Approach to Word and Context Representation Learning

In the last few years, neural networks have been intensively used to develop meaningful distributed representations of words and contexts around them. When these representations, also known as "embeddings", are learned from unsupervised large corpora, they can be transferred to different tasks with positive effects in terms of performances, especially when only a few supervisions are available. In this work, we further extend this concept, and we present an unsupervised neural architecture that jointly learns word and context embeddings, processing words as sequences of characters. This allows our model to spot the regularities that are due to the word morphology, and to avoid the need of a fixed-sized input vocabulary of words. We show that we can learn compact encoders that, despite the relatively small number of parameters, reach high-level performances in downstream tasks, comparing them with related state-of-the-art approaches or with fully supervised methods.

On the relation between Loss Functions and T-Norms

Deep learning has been shown to achieve impressive results in several domains like computer vision and natural language processing. A key element of this success has been the development of new loss functions, like the popular cross-entropy loss, which has been shown to provide faster convergence and to reduce the vanishing gradient problem in very deep structures. While the cross-entropy loss is usually justified from a probabilistic perspective, this paper shows an alternative and more direct interpretation of this loss in terms of t-norms and their associated generator functions, and derives a general relation between loss functions and t-norms. In particular, the presented work shows intriguing results leading to the development of a novel class of loss functions. These losses can be exploited in any supervised learning task and which could lead to faster convergence rates that the commonly employed cross-entropy loss.

Neural Markov Logic Networks

We introduce Neural Markov Logic Networks (NMLNs), a statistical relational learning system that borrows ideas from Markov logic. Like Markov Logic Networks (MLNs), NMLNs are an exponential-family model for modelling distributions over possible worlds, but unlike MLNs, they do not rely on explicitly specified first-order logic rules. Instead, NMLNs learn an implicit representation of such rules as a neural network that acts as a potential function on fragments of the relational structure. Interestingly, any MLN can be represented as an NMLN. Similarly to recently proposed Neural theorem provers (NTPs) [Rockt\"aschel and Riedel, 2017], NMLNs can exploit embeddings of constants but, unlike NTPs, NMLNs work well also in their absence. This is extremely important for predicting in settings other than the transductive one. We showcase the potential of NMLNs on knowledge-base completion tasks and on generation of molecular (graph) data.

LYRICS: a General Interface Layer to Integrate AI and Deep Learning

In spite of the amazing results obtained by deep learning in many applications, a real intelligent behavior of an agent acting in a complex environment is likely to require some kind of higher-level symbolic inference. Therefore, there is a clear need for the definition of a general and tight integration between low-level tasks, processing sensorial data that can be effectively elaborated using deep learning techniques, and the logic reasoning that allows humans to take decisions in complex environments. This paper presents LYRICS, a generic interface layer for AI, which is implemented in TersorFlow (TF). LYRICS provides an input language that allows to define arbitrary First Order Logic (FOL) background knowledge. The predicates and functions of the FOL knowledge can be bound to any TF computational graph, and the formulas are converted into a set of real-valued constraints, which participate to the overall optimization problem. This allows to learn the weights of the learners, under the constraints imposed by the prior knowledge. The framework is extremely general as it imposes no restrictions in terms of which models or knowledge can be integrated. In this paper, we show the generality of the approach showing some use cases of the presented language, including generative models, logic reasoning, model checking and supervised learning.

Integrating Learning and Reasoning with Deep Logic Models

Deep learning is very effective at jointly learning feature representations and classification models, especially when dealing with high dimensional input patterns. Probabilistic logic reasoning, on the other hand, is capable to take consistent and robust decisions in complex environments. The integration of deep learning and logic reasoning is still an open-research problem and it is considered to be the key for the development of real intelligent agents. This paper presents Deep Logic Models, which are deep graphical models integrating deep learning and logic reasoning both for learning and inference. Deep Logic Models create an end-to-end differentiable architecture, where deep learners are embedded into a network implementing a continuous relaxation of the logic knowledge. The learning process allows to jointly learn the weights of the deep learners and the meta-parameters controlling the high-level reasoning. The experimental results show that the proposed methodology overtakes the limitations of the other approaches that have been proposed to bridge deep learning and reasoning.

Constraint-Based Visual Generation

In the last few years the systematic adoption of deep learning to visual generation has produced impressive results that, amongst others, definitely benefit from the massive exploration of convolutional architectures. In this paper, we propose a general approach to visual generation that combines learning capabilities with logic descriptions of the target to be generated. The process of generation is regarded as a constrained satisfaction problem, where the constraints describe a set of properties that characterize the target. Interestingly, the constraints can also involve logic variables, while all of them are converted into real-valued functions by means of the t-norm theory. We use deep architectures to model the involved variables, and propose a computational scheme where the learning process carries out a satisfaction of the constraints. We propose some examples in which the theory can naturally be used, including the modeling of GAN and auto-encoders, and report promising results in problems with the generation of handwritten characters and face transformations.

Learning Neuron Non-Linearities with Kernel-Based Deep Neural Networks

The effectiveness of deep neural architectures has been widely supported in terms of both experimental and foundational principles. There is also clear evidence that the activation function (e.g. the rectifier and the LSTM units) plays a crucial role in the complexity of learning. Based on this remark, this paper discusses an optimal selection of the neuron non-linearity in a functional framework that is inspired from classic regularization arguments. It is shown that the best activation function is represented by a kernel expansion in the training set, that can be effectively approximated over an opportune set of points modeling 1-D clusters. The idea can be naturally extended to recurrent networks, where the expressiveness of kernel-based activation functions turns out to be a crucial ingredient to capture long-term dependencies. We give experimental evidence of this property by a set of challenging experiments, where we compare the results with neural architectures based on state of the art LSTM cells.

Backpropagation and Biological Plausibility

By and large, Backpropagation (BP) is regarded as one of the most important neural computation algorithms at the basis of the progress in machine learning, including the recent advances in deep learning. However, its computational structure has been the source of many debates on its arguable biological plausibility. In this paper, it is shown that when framing supervised learning in the Lagrangian framework, while one can see a natural emergence of Backpropagation, biologically plausible local algorithms can also be devised that are based on the search for saddle points in the learning adjoint space composed of weights, neural outputs, and Lagrangian multipliers. This might open the doors to a truly novel class of learning algorithms where, because of the introduction of the notion of support neurons, the optimization scheme also plays a fundamental role in the construction of the architecture.