Stochastic Optimization of Sorting Networks via Continuous Relaxations
This addresses a fundamental bottleneck for researchers and practitioners in ML who need end-to-end differentiable pipelines involving sorting, offering a novel solution rather than an incremental improvement.
The paper tackles the problem of non-differentiable sorting operators in machine learning by proposing NeuralSort, a continuous relaxation that enables gradient-based optimization, resulting in a fully differentiable k-nearest neighbors algorithm and other applications.
Sorting input objects is an important step in many machine learning pipelines. However, the sorting operator is non-differentiable with respect to its inputs, which prohibits end-to-end gradient-based optimization. In this work, we propose NeuralSort, a general-purpose continuous relaxation of the output of the sorting operator from permutation matrices to the set of unimodal row-stochastic matrices, where every row sums to one and has a distinct arg max. This relaxation permits straight-through optimization of any computational graph involve a sorting operation. Further, we use this relaxation to enable gradient-based stochastic optimization over the combinatorially large space of permutations by deriving a reparameterized gradient estimator for the Plackett-Luce family of distributions over permutations. We demonstrate the usefulness of our framework on three tasks that require learning semantic orderings of high-dimensional objects, including a fully differentiable, parameterized extension of the k-nearest neighbors algorithm.