In this paper we propose a general framework to integrate supervised and unsupervised examples with background knowledge expressed by a collection of first-order logic clauses into kernel machines. In particular, we consider a multi-task learning scheme where multiple predicates defined on a set of objects are to be jointly learned from examples, enforcing a set of FOL constraints on the admissible configurations of their values. The predicates are defined on the feature spaces, in which the input objects are represented, and can be either known a priori or approximated by an appropriate kernel-based learner. A general approach is presented to convert the FOL clauses into a continuous implementation that can deal with the outputs computed by the kernel-based predicates. The learning problem is formulated as a semi-supervised task that requires the optimization in the primal of a loss function that combines a fitting loss measure on the supervised examples, a regularization term, and a penalty term that enforces the constraints on both the supervised and unsupervised examples. Unfortunately, the penalty term is not convex and it can hinder the optimization process. However, it is possible to avoid poor solutions by using a two stage learning schema, in which the supervised examples are learned first and then the constraints are enforced.

翻译：本文提出了一种通用框架，将监督与无监督示例及由一阶逻辑子句集合表达的背景知识整合到核机中。具体而言，我们考虑一种多任务学习机制，其中定义在一组对象上的多个谓词需从示例中联合学习，并对其值域的可行配置施加一组一阶逻辑约束。这些谓词定义在表示输入对象的特征空间上，既可以是先验已知的，也可以通过基于核的学习器近似获得。我们提出了一种通用方法，将一阶逻辑子句转换为连续实现形式，以处理由基于核的谓词计算出的输出。该学习问题被表述为半监督任务，需要在原始空间中对损失函数进行优化，该损失函数结合了监督示例的拟合损失度量、正则化项以及对监督与无监督示例施加约束的惩罚项。然而，惩罚项非凸，可能阻碍优化过程。通过采用两阶段学习策略——先学习监督示例，再施加约束——可避免劣质解。