It is of increasing importance to develop learning methods for ranking. In contrast to many learning objectives, however, the ranking problem presents difficulties due to the fact that the space of permutations is not smooth. In this paper, we examine the class of rank-linear objective functions, which includes popular metrics such as precision and discounted cumulative gain. In particular, we observe that expectations of these gains are completely characterized by the marginals of the corresponding distribution over permutation matrices. Thus, the expectations of rank-linear objectives can always be described through locations in the Birkhoff polytope, i.e., doubly-stochastic matrices (DSMs). We propose a technique for learning DSM-based ranking functions using an iterative projection operator known as Sinkhorn normalization. Gradients of this operator can be computed via backpropagation, resulting in an algorithm we call Sinkhorn propagation, or SinkProp. This approach can be combined with a wide range of gradient-based approaches to rank learning. We demonstrate the utility of SinkProp on several information retrieval data sets.
Although this work represents the first approach to ranking that has directly incorporated Sinkhorn normalization into the training procedure, previously-developed methods have also found it useful. The SoftRank algorithm @cite_17 , for example, uses Sinkhorn balancing at test-time, as a post-processing step for the approximated ranking matrix. Unlike the approach proposed here, however, it does not take this step into account when optimizing the objective function. SmoothRank uses half a step of Sinkhorn balancing, normalizing only the scores within each column of the matrix, to produce a distribution over items at a particular rank.
@cite_17
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Permutations and matchings are core building blocks in a variety of latent variable models, as they allow us to align, canonicalize, and sort data. Learning in such models is difficult, however, because exact marginalization over these combinatorial objects is intractable. In response, this paper introduces a collection of new methods for end-to-end learning in such models that approximate discrete maximum-weight matching using the continuous Sinkhorn operator. Sinkhorn iteration is attractive because it functions as a simple, easy-to-implement analog of the softmax operator. With this, we can define the Gumbel-Sinkhorn method, an extension of the Gumbel-Softmax method ( 2016, 2016) to distributions over latent matchings. We demonstrate the effectiveness of our method by outperforming competitive baselines on a range of qualitatively different tasks: sorting numbers, solving jigsaw puzzles, and identifying neural signals in worms. We use convex relaxation techniques to provide a sequence of regularized low-rank solutions for large-scale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm SOFT-IMPUTE iteratively replaces the missing elements with those obtained from a soft-thresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter. The computationally intensive part of our algorithm is in computing a low-rank SVD of a dense matrix. Exploiting the problem structure, we show that the task can be performed with a complexity of order linear in the matrix dimensions. Our semidefinite-programming algorithm is readily scalable to large matrices; for example SOFT-IMPUTE takes a few hours to compute low-rank approximations of a 106 X 106 incomplete matrix with 107 observed entries, and fits a rank-95 approximation to the full Netflix training set in 3.3 hours. Our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state-of-the-art techniques. We address the problem of learning large complex ranking functions. Most IR applications use evaluation metrics that depend only upon the ranks of documents. However, most ranking functions generate document scores, which are sorted to produce a ranking. Hence IR metrics are innately non-smooth with respect to the scores, due to the sort. Unfortunately, many machine learning algorithms require the gradient of a training objective in order to perform the optimization of the model parameters,and because IR metrics are non-smooth,we need to find a smooth proxy objective that can be used for training. We present a new family of training objectives that are derived from the rank distributions of documents, induced by smoothed scores. We call this approach SoftRank. We focus on a smoothed approximation to Normalized Discounted Cumulative Gain (NDCG), called SoftNDCG and we compare it with three other training objectives in the recent literature. We present two main results. First, SoftRank yields a very good way of optimizing NDCG. Second, we show that it is possible to achieve state of the art test set NDCG results by optimizing a soft NDCG objective on the training set with a different discount function In this paper, we propose a rank minimization method to fuse the predicted confidence scores of multiple models, each of which is obtained based on a certain kind of feature. Specifically, we convert each confidence score vector obtained from one model into a pairwise relationship matrix, in which each entry characterizes the comparative relationship of scores of two test samples. Our hypothesis is that the relative score relations are consistent among component models up to certain sparse deviations, despite the large variations that may exist in the absolute values of the raw scores. Then we formulate the score fusion problem as seeking a shared rank-2 pairwise relationship matrix based on which each original score matrix from individual model can be decomposed into the common rank-2 matrix and sparse deviation errors. A robust score vector is then extracted to fit the recovered low rank score relation matrix. We formulate the problem as a nuclear norm and l 1 norm optimization objective function and employ the Augmented Lagrange Multiplier (ALM) method for the optimization. Our method is isotonic (i.e., scale invariant) to the numeric scales of the scores originated from different models. We experimentally show that the proposed method achieves significant performance gains on various tasks including object categorization and video event detection.