In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Ex Input:
Stochastic gradient descent (SGD), which dates back to the 1950s, is one of the most popular and effective approaches for performing stochastic optimization. Research on SGD resurged recently in machine learning for optimizing convex loss functions and training nonconvex deep neural networks. The theory assumes that one can easily compute an unbiased gradient estimator, which is usually the case due to the sample average nature of empirical risk minimization. There exist, however, many scenarios (e.g., graphs) where an unbiased estimator may be as expensive to compute as the full gradient because training examples are interconnected. Recently, Chen et al. (2018) proposed using a consistent gradient estimator as an economic alternative. Encouraged by empirical success, we show, in a general setting, that consistent estimators result in the same convergence behavior as do unbiased ones. Our analysis covers strongly convex, convex, and nonconvex objectives. We verify the results with illustrative experiments on synthetic and real-world data. This work opens several new research directions, including the development of more efficient SGD updates with consistent estimators and the design of efficient training algorithms for large-scale graphs.

Ex Output:
Convergence theory for biased (but consistent) gradient estimators in stochastic optimization and application to graph convolutional networks

Ex Input:
Generative Adversarial Networks (GANs), when trained on large datasets with diverse modes, are known to produce conflated images which do not distinctly belong to any of the modes. We hypothesize that this problem occurs due to the interaction between two facts: (1) For datasets with large variety, it is likely that the modes lie on separate manifolds. (2) The generator (G) is formulated as a continuous function, and the input noise is derived from a connected set, due to which G's output is a connected set. If G covers all modes, then there must be some portion of G's output which connects them. This corresponds to undesirable, conflated images. We develop theoretical arguments to support these intuitions. We propose a novel method to break the second assumption via learnable discontinuities in the latent noise space. Equivalently, it can be viewed as training several generators, thus creating discontinuities in the G function. We also augment the GAN formulation with a classifier C that predicts which noise partition/generator produced the output images, encouraging diversity between each partition/generator. We experiment on MNIST, celebA, STL-10, and a difficult dataset with clearly distinct modes, and show that the noise partitions correspond to different modes of the data distribution, and produce images of superior quality.

Ex Output:
We introduce theory to explain the failure of GANs on complex datasets and propose a solution to fix it.

Ex Input:
The reparameterization trick has become one of the most useful tools in the field of variational inference. However, the reparameterization trick is based on the standardization transformation which restricts the scope of application of this method to distributions that have tractable inverse cumulative distribution functions or are expressible as deterministic transformations of such distributions. In this paper, we generalized the reparameterization trick by allowing a general transformation. Unlike other similar works, we develop the generalized transformation-based gradient model formally and rigorously. We discover that the proposed model is a special case of control variate indicating that the proposed model can combine the advantages of CV and generalized reparameterization. Based on the proposed gradient model, we propose a new polynomial-based gradient estimator which has better theoretical performance than the reparameterization trick under certain condition and can be applied to a larger class of variational distributions. In studies of synthetic and real data, we show that our proposed gradient estimator has a significantly lower gradient variance than other state-of-the-art methods thus enabling a faster inference procedure.

Ex Output:
We propose a novel generalized transformation-based gradient model and propose a polynomial-based gradient estimator based upon the model.