Neural Information Processing Systems http://nips.cc/

Multilabel classification is a challenging problem arising in applications ranging from information retrieval to image tagging. A popular approach to this problem is to employ a reduction to a suitable series of binary or multiclass problems (e.g., computing a softmax based cross-entropy over the relevant labels). While such methods have seen empirical success, less is understood about how well they approximate two fundamental performance measures: precision@k and recall@k. In this paper, we study five commonly used reductions, including the one-versus-all reduction, a reduction to multiclass classification, and normalised versions of the same, wherein the contribution of each instance is normalised by the number of relevant labels. Our main result is a formal justification of each reduction: we explicate their underlying risks, and show they are each consistent with respect to either precision or recall. Further, we show that in general no reduction can be optimal for both measures. We empirically validate our results, demonstrating scenarios where normalised reductions yield recall gains over unnormalised counterparts.

However, it has been recently demonstrated that such methods can fail to converge even in simple convex optimization settings. In this work, we provide a new analysis of such methods applied to nonconvex stochastic optimization problems, characterizing the effect of increasing minibatch size. Our analysis shows that under this scenario such methods do converge to stationarity up to the statistical limit of variance in the stochastic gradients (scaled by a constant factor). In particular, our result implies that increasing minibatch sizes enables convergence, thus providing a way to circumvent the nonconvergence issues.

Neural Information Processing Systems http://nips.cc/

Distributed stochastic gradient descent is an important subroutine in distributed learning. A setting of particular interest is when the clients are mobile devices, where two important concerns are communication efficiency and the privacy of the clients. Several recent works have focused on reducing the communication cost or introducing privacy guarantees, but none of the proposed communication efficient methods are known to be privacy preserving and none of the known privacy mechanisms are known to be communication efficient. To this end, we study algorithms that achieve both communication efficiency and differential privacy. For d variables and n d clients, the proposed method uses O(log log(nd)) bits of communication per client per coordinate and ensures constant privacy. We also improve previous analysis of the Binomial mechanism showing that it achieves nearly the same utility as the Gaussian mechanism, while requiring fewer representation bits, which can be of independent interest.

The computational cost of training with softmax cross entropy loss grows linearly with the number of classes. For the settings where a large number of classes are involved, a common method to speed up training is to sample a subset of classes and utilize an estimate of the loss gradient based on these classes, known as the sampled softmax method. However, the sampled softmax provides a biased estimate of the gradient unless the samples are drawn from the exact softmax distribution, which is again expensive to compute. Therefore, a widely employed practical approach involves sampling from a simpler distribution in the hope of approximating the exact softmax distribution. In this paper, we develop the first theoretical understanding of the role that different sampling distributions play in determining the quality of sampled softmax. Motivated by our analysis and the work on kernel-based sampling, we propose the Random Fourier Softmax (RFsoftmax) method that utilizes the powerful Random Fourier Features to enable more efficient and accurate sampling from an approximate softmax distribution. We show that RF-softmax leads to low bias in estimation in terms of both the full softmax distribution and the full softmax gradient.

Multilabel classification is a challenging problem arising in applications ranging from information retrieval to image tagging. A popular approach to this problem is to employ a reduction to a suitable series of binary or multiclass problems (e.g., computing a softmax based cross-entropy over the relevant labels). While such methods have seen empirical success, less is understood about how well they approximate two fundamental performance measures: precision@k and recall@k. In this paper, we study five commonly used reductions, including the one-versus-all reduction, a reduction to multiclass classification, and normalised versions of the same, wherein the contribution of each instance is normalised by the number of relevant labels. Our main result is a formal justification of each reduction: we explicate their underlying risks, and show they are each consistent with respect to either precision or recall. Further, we show that in general no reduction can be optimal for both measures. We empirically validate our results, demonstrating scenarios where normalised reductions yield recall gains over unnormalised counterparts.

In extreme classification settings, embedding-based neural network models are currently not competitive with sparse linear and tree-based methods in terms of accuracy. Most prior works attribute this poor performance to the low-dimensional bottleneck in embedding-based methods. In this paper, we demonstrate that theoretically there is no limitation to using low-dimensional embedding-based methods, and provide experimental evidence that overfitting is the root cause of the poor performance of embedding-based methods. These findings motivate us to investigate novel data augmentation and regularization techniques to mitigate overfitting. To this end, we propose GLaS, a new regularizer for embedding-based neural network approaches. It is a natural generalization from the graph Laplacian and spread-out regularizers, and empirically it addresses the drawback of each regularizer alone when applied to the extreme classification setup. With the proposed techniques, we attain or improve upon the state-of-the-art on most widely tested public extreme classification datasets with hundreds of thousands of labels.

We present an intriguing discovery related to Random Fourier Features: in Gaussian kernel approximation, replacing the random Gaussian matrix by a properly scaled random orthogonal matrix significantly decreases kernel approximation error. We call this technique Orthogonal Random Features (ORF), and provide theoretical and empirical justification for this behavior. Motivated by this discovery, we further propose Structured Orthogonal Random Features (SORF), which uses a class of structured discrete orthogonal matrices to speed up the computation.

However, it has been recently demonstrated that such methods can fail to converge even in simple convex optimization settings. In this work, we provide a new analysis of such methods applied to nonconvex stochastic optimization problems, characterizing the effect of increasing minibatch size. Our analysis shows that under this scenario such methods do converge to stationarity up to the statistical limit of variance in the stochastic gradients (scaled by a constant factor). In particular, our result implies that increasing minibatch sizes enables convergence, thus providing a way to circumvent the nonconvergence issues.