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Title: Stochastic Optimization of Sorting Networks via Continuous Relaxations

Paper content:
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 argmax.
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
Learning to automatically sort objects is useful in many machine learning applications, such as topk multi-class classification BID5 , ranking documents for information retrieval (Liu et al., 2009) , and multi-object target tracking in computer vision BID3 .
Such algorithms typically require learning informative representations of complex, high-dimensional data, such as images, before sorting and subsequent downstream processing.
For instance, the k-nearest neighbors image classification algorithm, which orders the neighbors based on distances in the canonical pixel basis, can be highly suboptimal for classification (Weinberger et al., 2006) .
Deep neural networks can instead be used to learn representations, but these representations cannot be optimized end-to-end for a downstream sorting-based objective, since the sorting operator is not differentiable with respect to its input.In this work, we seek to remedy this shortcoming by proposing NeuralSort, a continuous relaxation to the sorting operator that is differentiable almost everywhere with respect to the inputs.
The output of any sorting algorithm can be viewed as a permutation matrix, which is a square matrix with entries in {0, 1} such that every row and every column sums to 1.
Instead of a permutation matrix, NeuralSort returns a unimodal row-stochastic matrix.
A unimodal row-stochastic matrix is defined as a square matrix with positive real entries, where each row sums to 1 and has a distinct arg max.
All permutation matrices are unimodal row-stochastic matrices.
NeuralSort has a temperature knob that controls the degree of approximation, such that in the limit of zero temperature, we recover a permutation matrix that sorts the inputs.
Even for a non-zero temperature, we can efficiently project any unimodal matrix to the desired permutation matrix via a simple row-wise arg max operation.
Hence, NeuralSort is also suitable for efficient straight-through gradient optimization BID4 , which requires "exact" permutation matrices to evaluate learning objectives.As the second primary contribution, we consider the use of NeuralSort for stochastic optimization over permutations.
In many cases, such as latent variable models, the permutations may be latent but directly influence observed behavior, e.g., utility and choice models are often expressed as distributions over permutations which govern the observed decisions of agents (Regenwetter et al., 2006; BID7 .
By learning distributions over unobserved permutations, we can account for the uncertainty in these permutations in a principled manner.
However, the challenge with stochastic optimization over discrete distributions lies in gradient estimation with respect to the distribution parameters.
Vanilla REINFORCE estimators are impractical for most cases, or necessitate custom control variates for low-variance gradient estimation (Glasserman, 2013) .In
this regard, we consider the Plackett-Luce (PL) family of distributions over permutations (Plackett, 1975; Luce, 1959) . A
common modeling choice for ranking models, the PL distribution is parameterized by n scores, with its support defined over the symmetric group consisting of n! permutations. We
derive a reparameterizable sampler for stochastic optimization with respect to this distribution, based on Gumbel perturbations to the n (log-)scores. However
, the reparameterized sampler requires sorting these perturbed scores, and hence the gradients of a downstream learning objective with respect to the scores are not defined. By using
NeuralSort instead, we can approximate the objective and obtain well-defined reparameterized gradient estimates for stochastic optimization.Finally, we apply NeuralSort to tasks that require us to learn semantic orderings of complex, highdimensional input data. First, we
consider sorting images of handwritten digits, where the goal is to learn to sort images by their unobserved labels. Our second
task extends the first one to quantile regression, where we want to estimate the median (50-th percentile) of a set of handwritten numbers. In addition
to identifying the index of the median image in the sequence, we need to learn to map the inferred median digit to its scalar representation. In the third
task, we propose an algorithm that learns a basis representation for the k-nearest neighbors (kNN) classifier in an end-to-end procedure. Because the
choice of the k nearest neighbors requires a non-differentiable sorting, we use NeuralSort to obtain an approximate, differentiable surrogate. On all tasks
, we observe significant empirical improvements due to NeuralSort over the relevant baselines and competing relaxations to permutation matrices.
The problem of learning to rank documents based on relevance has been studied extensively in the context of information retrieval.
In particular, listwise approaches learn functions that map objects to scores.
Much of this work concerns the PL distribution: the RankNet algorithm BID6 can be interpreted as maximizing the PL likelihood of pairwise comparisons between items, while the ListMLE ranking algorithm in Xia et al. (2008) extends this with a loss that maximizes the PL likelihood of ground-truth permutations directly.
The differentiable pairwise approaches to ranking, such as Rigutini et al. FORMULA1 , learn to approximate the comparator between pairs of objects.
Our work considers a generalized setting where sorting based operators can be inserted anywhere in computation graphs to extend traditional pipelines e.g., kNN.Prior works have proposed relaxations of permutation matrices to the Birkhoff polytope, which is defined as the convex hull of the set of permutation matrices a.k.a. the set of doubly-stochastic matrices.
A doubly-stochastic matrix is a permutation matrix iff it is orthogonal and continuous relaxations based on these matrices have been used previously for solving NP-complete problems such as seriation and graph matching (Fogel et al., 2013; Fiori et al., 2013; Lim & Wright, 2014) .
BID1 proposed the use of the Sinkhorn operator to map any square matrix to the Birkhoff polytope.
They interpret the resulting doubly-stochastic matrix as the marginals of a distribution over permutations.
Mena et al. (2018) propose an alternate method where the square matrix defines a latent distribution over the doubly-stochastic matrices themselves.
These distributions can be sampled from by adding elementwise Gumbel perturbations.
Linderman et al. FORMULA1 propose a rounding procedure that uses the Sinkhorn operator to directly sample matrices near the Birkhoff polytope.
Unlike Mena et al. (2018) , the resulting distribution over matrices has a tractable density.
In practice, however, the approach of Mena et al. FORMULA1 performs better and will be the main baseline we will be comparing against in our experiments in Section 6.As discussed in Section 3, NeuralSort maps permutation matrices to the set of unimodal rowstochastic matrices.
For the stochastic setting, the PL distribution permits efficient sampling, exact and tractable density estimation, making it an attractive choice for several applications, e.g., variational inference over latent permutations.
Our reparameterizable sampler, while also making use of the Gumbel distribution, is based on a result unique to the PL distribution (Proposition 5).The
use of the Gumbel distribution for defining continuous relaxations to discrete distributions was first proposed concurrently by Jang et al. FORMULA1 and Maddison et al. (2017) for categorical variables, referred to as Gumbel-Softmax. The
number of possible permutations grow factorially with the dimension, and thus any distribution over n-dimensional permutations can be equivalently seen as a distribution over n! categories. Gumbel-softmax
does not scale to a combinatorially large number of categories (Kim et al., 2016; Mussmann et al., 2017) , necessitating the use of alternate relaxations, such as the one considered in this work.
In this paper, we proposed NeuralSort, a continuous relaxation of the sorting operator to the set of unimodal row-stochastic matrices.
Our relaxation facilitates gradient estimation on any computation graph involving a sort operator.
Further, we derived a reparameterized gradient estimator for the Plackett-Luce distribution for efficient stochastic optimization over permutations.
On three illustrative tasks including a fully differentiable k-nearest neighbors, our proposed relaxations outperform prior work in end-to-end learning of semantic orderings of high-dimensional objects.In the future, we would like to explore alternate relaxations to sorting as well as applications that extend widely-used algorithms such as beam search (Goyal et al., 2018) .
Both deterministic and stochastic NeuralSort are easy to implement.
We provide reference implementations in Tensorflow BID0 Proof.
For any value of λ, the following inequalities hold: DISPLAYFORM0 This finishes the proof.

assistant: text: We provide a continuous relaxation to the sorting operator, enabling end-to-end, gradient-based stochastic optimization.