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，将带有背景知识（以一组一阶逻辑子句形式表示）的有监督与无监督示例整合到核机器中。具体而言，我们考虑一种多任务学习方案：对定义在一组对象上的多个谓词，通过学习示例并强制执行关于其值可接受配置的一组FOL约束，实现联合学习。这些谓词由输入对象所在的特征空间定义，既可以是先验已知的，也可以通过适当的基于核的学习器进行近似。我们提出一种通用方法，将FOL子句转换为连续实现，从而处理基于核的谓词计算得到的输出。该学习问题被形式化为半监督任务，需要在原问题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，以及强制对监督与无监督示例约束的惩罚项。尽管惩罚项非凸，可能阻碍优化过程，但通过两阶段学习模式（先学习监督示例，再施加约束）可避免陷入不良解。