Title: When Are Neural Pruning Approximation Bounds Useful?. Abstract:    Approximation bounds for neural network pruning attempt to predict the trade-off between sparsity and fidelity while shrinking neural networks. In the first half of this paper, we empirically evaluate the predictive power of two recently proposed methods based on coreset algorithms. We identify several circumstances in which the bounds are loose or impractical to use and provide a brief analysis of the components of the bounds that contribute to those short-comings. In the second half, we examine the role of fine-tuning in prunability and observe that even tight approximation bounds would be poor predictors of accuracy after fine-tuning. This is because fine-tuning can recover large amounts of accuracy while simultaneously maintaining or increasing approximation error. We discuss the implications of these finding on the application of coreset-based pruning methods in practice and the role of approximation in the pruning community. Our code is available in the attached supplementary material.