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# On Regularization and Inference with Label Constraints

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:35740-35762, 2023.

#### Abstract

Prior knowledge and symbolic rules in machine learning are often expressed in the form of label constraints, especially in structured prediction problems. In this work, we compare two common strategies for encoding label constraints in a machine learning pipeline,

*regularization with constraints*and*constrained inference*, by quantifying their impact on model performance. For regularization, we show that it narrows the generalization gap by precluding models that are inconsistent with the constraints. However, its preference for small violations introduces a bias toward a suboptimal model. For constrained inference, we show that it reduces the population risk by correcting a model’s violation, and hence turns the violation into an advantage. Given these differences, we further explore the use of two approaches together and propose conditions for constrained inference to compensate for the bias introduced by regularization, aiming to improve both the model complexity and optimal risk.