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Regularization is an effective way to promote the generalization performance of machine learning models. In this paper, we focus on label smoothing, a form of output distribution regularization that prevents overfitting of a neural network by softening the ground-truth labels in the training data in an attempt to penalize overconfident outputs. Existing approaches typically use cross-validation to impose this smoothing, which is uniform across all training data. In this paper, we show that such label smoothing imposes a quantifiable bias in the Bayes error rate of the training data, with regions of the feature space with high overlap and low marginal likelihood having a lower bias and regions of low overlap and high marginal likelihood having a higher bias. These theoretical results motivate a simple objective function for data-dependent smoothing to mitigate the potential negative consequences of the operation while maintaining its desirable properties as a regularizer. We call this approach Structural Label Smoothing (SLS). We implement SLS and empirically validate on synthetic, Higgs, SVHN, CIFAR-10, and CIFAR-100 datasets. The results confirm our theoretical insights and demonstrate the effectiveness of the proposed method in comparison to traditional label smoothing.