Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow these measures with the 2-Wasserstein metric, with two important benefits: (i) For such measures, the squared 2-Wasserstein metric has a closed form, equal to a weighted sum of the squared Euclidean distance between means and the squared Bures metric between covariance matrices. The latter is a Riemannian metric between positive semi-definite matrices, which turns out to be Euclidean on a suitable factor representation of such matrices, which is valid on the entire geodesic between these matrices. (ii) The 2-Wasserstein distance boils down to the usual Euclidean metric when comparing Diracs, and therefore provides a natural framework to extend point embeddings. We show that for these reasons Wasserstein elliptical embeddings are more intuitive and yield tools that are better behaved numerically than the alternative choice of Gaussian embeddings with the Kullback-Leibler divergence. In particular, and unlike previous work based on the KL geometry, we learn elliptical distributions that are not necessarily diagonal. We demonstrate the advantages of elliptical embeddings by using them for visualization, to compute embeddings of words, and to reflect entailment or hypernymy.

In this work, we present new theoretical results on convolutional generative neural networks, in particular their invertibility (i.e., the recovery of input latent code given the network output). The study of network inversion problem is motivated by image inpainting and the mode collapse problem in training GAN. Network inversion is highly non-convex, and thus is typically computationally intractable and without optimality guarantees. However, we rigorously prove that, under some mild technical assumptions, the input of a two-layer convolutional generative network can be deduced from the network output efficiently using simple gradient descent. This new theoretical finding implies that the mapping from the low- dimensional latent space to the high-dimensional image space is bijective (i.e., one-to-one). In addition, the same conclusion holds even when the network output is only partially observed (i.e., with missing pixels). Our theorems hold for 2-layer convolutional generative network with ReLU as the activation function, but we demonstrate empirically that the same conclusion extends to multi-layer networks and networks with other activation functions, including the leaky ReLU, sigmoid and tanh.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space the time-discretization error can dominate when we are near the boundary of the space. We demonstrate that because of this, current SGMCMC methods for the simplex struggle with sparse simplex spaces; when many of the components are close to zero. Unfortunately, many popular large-scale Bayesian models, such as network or topic models, require inference on sparse simplex spaces. To avoid the biases caused by this discretization error, we propose the stochastic Cox-Ingersoll-Ross process (SCIR), which removes all discretization error and we prove that samples from the SCIR process are asymptotically unbiased. We discuss how this idea can be extended to target other constrained spaces. Use of the SCIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.

Converting an n-dimensional vector to a probability distribution over n objects is a commonly used component in many machine learning tasks like multiclass classification, multilabel classification, attention mechanisms etc. For this, several probability mapping functions have been proposed and employed in literature such as softmax, sum-normalization, spherical softmax, and sparsemax, but there is very little understanding in terms how they relate with each other. Further, none of the above formulations offer an explicit control over the degree of sparsity. To address this, we develop a unified framework that encompasses all these formulations as special cases. This framework ensures simple closed-form solutions and existence of sub-gradients suitable for learning via backpropagation. Within this framework, we propose two novel sparse formulations, sparsegen-lin and sparsehourglass, that seek to provide a control over the degree of desired sparsity. We further develop novel convex loss functions that help induce the behavior of aforementioned formulations in the multilabel classification setting, showing improved performance. We also demonstrate empirically that the proposed formulations, when used to compute attention weights, achieve better or comparable performance on standard seq2seq tasks like neural machine translation and abstractive summarization.

Model compression is essential for serving large deep neural nets on devices with limited resources or applications that require real-time responses. For advanced NLP problems, a neural language model usually consists of recurrent layers (e.g., using LSTM cells), an embedding matrix for representing input tokens, and a softmax layer for generating output tokens. For problems with a very large vocabulary size, the embedding and the softmax matrices can account for more than half of the model size. For instance, the bigLSTM model achieves state-of-the-art performance on the One-Billion-Word (OBW) dataset with around 800k vocabulary, and its word embedding and softmax matrices use more than 6GBytes space, and are responsible for over 90\% of the model parameters. In this paper, we propose GroupReduce, a novel compression method for neural language models, based on vocabulary-partition (block) based low-rank matrix approximation and the inherent frequency distribution of tokens (the power-law distribution of words). We start by grouping words into $c$ blocks based on their frequency, and then refine the clustering iteratively by constructing weighted low-rank approximation for each block, where the weights are based the frequencies of the words in the block. The experimental results show our method can significantly outperform traditional compression methods such as low-rank approximation and pruning. On the OBW dataset, our method achieved 6.6x compression rate for the embedding and softmax matrices, and when combined with quantization, our method can achieve 26x compression rate without losing prediction accuracy.

Observational data is increasingly used as a means for making individual-level causal predictions and intervention recommendations. The foremost challenge of causal inference from observational data is hidden confounding, whose presence cannot be tested in data and can invalidate any causal conclusion. Experimental data does not suffer from confounding but is usually limited in both scope and scale. We introduce a novel method of using limited experimental data to correct the hidden confounding in causal effect models trained on larger observational data, even if the observational data does not fully overlap with the experimental data. Our method makes strictly weaker assumptions than existing approaches, and we prove conditions under which it yields a consistent estimator. We demonstrate our method's efficacy using real-world data from a large educational experiment.

We consider a generalization of mixed regression where the response is an additive combination of several mixture components. Standard mixed regression is a special case where each response is generated from exactly one component. Typical approaches to the mixture regression problem employ local search methods such as Expectation Maximization (EM) that are prone to spurious local optima. On the other hand, a number of recent theoretically-motivated \emph{Tensor-based methods} either have high sample complexity, or require the knowledge of the input distribution, which is not available in most of practical situations. In this work, we study a novel convex estimator \emph{MixLasso} for the estimation of generalized mixed regression, based on an atomic norm specifically constructed to regularize the number of mixture components. Our algorithm gives a risk bound that trades off between prediction accuracy and model sparsity without imposing stringent assumptions on the input/output distribution, and can be easily adapted to the case of non-linear functions. In our numerical experiments on mixtures of linear as well as nonlinear regressions, the proposed method yields high-quality solutions in a wider range of settings than existing approaches.

Algorithms for clustering points in metric spaces is a long-studied area of research. Clustering has seen a multitude of work both theoretically, in understanding the approximation guarantees possible for many objective functions such as k-median and k-means clustering, and experimentally, in finding the fastest algorithms and seeding procedures for Lloyd's algorithm. The performance of a given clustering algorithm depends on the specific application at hand, and this may not be known up front. For example, a "typical instance" may vary depending on the application, and different clustering heuristics perform differently depending on the instance. In this paper, we define an infinite family of algorithms generalizing Lloyd's algorithm, with one parameter controlling the the initialization procedure, and another parameter controlling the local search procedure. This family of algorithms includes the celebrated k-means++ algorithm, as well as the classic farthest-first traversal algorithm. We design efficient learning algorithms which receive samples from an application-specific distribution over clustering instances and learn a near-optimal clustering algorithm from the class. We show the best parameters vary significantly across datasets such as MNIST, CIFAR, and mixtures of Gaussians. Our learned algorithms never perform worse than k-means++, and on some datasets we see significant improvements.

We study the implicit regularization properties of optimization techniques by explicitly connecting their optimization paths to the regularization paths of ``corresponding'' regularized problems. This surprising connection shows that iterates of optimization techniques such as gradient descent and mirror descent are \emph{pointwise} close to solutions of appropriately regularized objectives. While such a tight connection between optimization and regularization is of independent intellectual interest, it also has important implications for machine learning: we can port results from regularized estimators to optimization, and vice versa. We investigate one key consequence, that borrows from the well-studied analysis of regularized estimators, to then obtain tight excess risk bounds of the iterates generated by optimization techniques.

We present computationally efficient algorithms to test various combinatorial structures of large-scale graphical models. In order to test the hypotheses on their topological structures, we propose two adjacency matrix sketching frameworks: neighborhood sketching and subgraph sketching. The neighborhood sketching algorithm is proposed to test the connectivity of graphical models. This algorithm randomly subsamples vertices and conducts neighborhood regression and screening. The global sketching algorithm is proposed to test the topological properties requiring exponential computation complexity, especially testing the chromatic number and the maximum clique. This algorithm infers the corresponding property based on the sampled subgraph. Our algorithms are shown to substantially accelerate the computation of existing methods.