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：本文提出一个通用框架，将带有一阶逻辑子句形式背景知识的监督与无监督示例整合到核机器中。具体而言，我们考虑一个多任务学习方案，其中定义在一组对象上的多个谓词需要从示例中联合学习，并强制执行对它们可取值配置的一组FOL约束。这些谓词定义在输入对象的特征空间上，既可以通过先验知识已知，也可以通过适当的基于核的学习器近似得到。我们提出一种通用方法，将FOL子句转换为能够处理基于核的谓词输出结果的连续实现形式。该学习问题被表述为半监督任务，需要在原始空间中优化一个损失函数，该函数结合了监督示例上的拟合损失度量、正则化项以及强制作用于监督和无监督示例约束的惩罚项。遗憾的是，该惩罚项非凸且可能阻碍优化过程。然而，通过采用两阶段学习范式（先学习监督示例，再强制执行约束），可以避免陷入较差的解。