Title: Cost-Sensitive Hierarchical Classification through Layer-wise Abstentions. Abstract: We study the problem of cost-sensitive hierarchical classification where a label taxonomy has a cost-sensitive loss associated with it, which represents the cost of (wrong) predictions at different levels of the hierarchy. Directly optimizing the cost-sensitive hierarchical loss is hard, due to its non-convexity, especially when the size of the taxonomy is large. In this paper, we propose a \textbf{L}ayer-wise \textbf{A}bstaining Loss \textbf{M}inimization method (LAM),  a tractable method that breaks the hierarchical learning problem into layer-by-layer learning-to-abstain sub-problems.  We prove that there is a bijective mapping between the original hierarchical cost-sensitive loss and the set of layer-wise abstaining losses under symmetry assumptions. We employ the distributionally robust learning framework to solve the learning-to-abstain problems in each layer. We conduct experiments on large-scale bird dataset as well as on cell classification problems. Our results demonstrate that LAM achieves a lower hierarchical cost-sensitive loss in high accuracy regions, compared to previous methods and their modified versions for a fair comparison, even though they are not directly optimizing this loss. For each layer, we also achieve higher accuracy when the overall accuracy is kept fixed across different methods. Furthermore, we also show the flexibility of LAM by proposing a per-class loss-adjustment heuristic to achieve a performance profile. This can be used for cost design to translate user requirements into optimizable cost functions.