Finding the k largest or smallest elements from a collection of scores, i.e., top-k operation, is an important model component widely used in information retrieval, machine learning, and data mining. However, if the top-k operation is implemented in an algorithmic way, e.g., using bubble algorithm, the resulted model cannot be trained in an end-to-end way using prevalent gradient descent algorithms. This is because these implementations typically involve swapping indices, whose gradient cannot be computed. Moreover, the corresponding mapping from the input scores to the indicator vector of whether this element belongs to the top-k set is essentially discontinuous. To address the issue, we propose a smoothed approximation, namely SOFT (Scalable Optimal transport-based diFferenTiable) top-k operator. Specifically, our SOFT top-k operator approximates the output of top-k operation as the solution of an Entropic Optimal Transport (EOT) problem. The gradient of the SOFT operator can then be efficiently approximated based on the optimality conditions of EOT problem. We then apply the proposed operator to k-nearest neighbors algorithm and beam search algorithm. The numerical experiment demonstrates their achieve improved performance.

Additional Feedback: Some comments: l. 117 The entropic OT is surely not more computationally friendly than a a top-k operator that simply sorts the vector. Same for the beam-search method, the present work seems to be a sequence of ad-hoc definitions rather than a principled objective. In particular it is important to make the optimization objective clear to enable future comparisons. Can the authors clearly distinguish their contributions from the ones of Cuturi et al, 2019? It seems that the implementation of the authors is potentially faster, which should be highlighted.

Finding the k largest or smallest elements from a collection of scores, i.e., top-k operation, is an important model component widely used in information retrieval, machine learning, and data mining. However, if the top-k operation is implemented in an algorithmic way, e.g., using bubble algorithm, the resulted model cannot be trained in an end-to-end way using prevalent gradient descent algorithms. This is because these implementations typically involve swapping indices, whose gradient cannot be computed. Moreover, the corresponding mapping from the input scores to the indicator vector of whether this element belongs to the top-k set is essentially discontinuous. To address the issue, we propose a smoothed approximation, namely SOFT (Scalable Optimal transport-based diFferenTiable) top-k operator.