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.

Example Input: Adaptive gradient algorithms perform gradient-based updates using the history of gradients and are ubiquitous in training deep neural networks. While adaptive gradient methods theory is well understood for minimization problems, the underlying factors driving their empirical success in min-max problems such as GANs remain unclear. In this paper, we aim at bridging this gap from both theoretical and empirical perspectives. First, we analyze a variant of Optimistic Stochastic Gradient (OSG) proposed in~\citep{daskalakis2017training} for solving a class of non-convex non-concave min-max problem and establish $O(\epsilon^{-4})$ complexity for finding $\epsilon$-first-order stationary point, in which the algorithm only requires invoking one stochastic first-order oracle while enjoying state-of-the-art iteration complexity achieved by stochastic extragradient method by~\citep{iusem2017extragradient}. Then we propose an adaptive variant of OSG named Optimistic Adagrad (OAdagrad) and reveal an \emph{improved} adaptive complexity $\widetilde{O}\left(\epsilon^{-
rac{2}{1-lpha}}
ight)$~
ootnote{Here $\widetilde{O}(\cdot)$ compresses a logarithmic factor of $\epsilon$. }, where $lpha$ characterizes the growth rate of the cumulative stochastic gradient and $0\leq lpha\leq 1/2$. To the best of our knowledge, this is the first work for establishing adaptive complexity in non-convex non-concave min-max optimization. Empirically, our experiments show that indeed adaptive gradient algorithms outperform their non-adaptive counterparts in GAN training. Moreover, this observation can be explained by the slow growth rate of the cumulative stochastic gradient, as observed empirically.
Example Output: This paper provides novel analysis of adaptive gradient algorithms for solving non-convex non-concave min-max problems as GANs, and explains the reason why adaptive gradient methods outperform its non-adaptive counterparts by empirical studies.

Example Input: State of the art computer vision models have been shown to be vulnerable to small adversarial perturbations of the input. In other words, most images in the data distribution are both correctly classified by the model and are very close to a visually similar misclassified image. Despite substantial research interest, the cause of the phenomenon is still poorly understood and remains unsolved. We hypothesize that this counter intuitive behavior is a naturally occurring result of the high dimensional geometry of the data manifold. As a first step towards exploring this hypothesis, we study a simple synthetic dataset of classifying between two concentric high dimensional spheres. For this dataset we show a fundamental tradeoff between the amount of test error and the average distance to nearest error. In particular, we prove that any model which misclassifies a small constant fraction of a sphere will be vulnerable to adversarial perturbations of size $O(1/\sqrt{d})$. Surprisingly, when we train several different architectures on this dataset, all of their error sets naturally approach this theoretical bound. As a result of the theory, the vulnerability of neural networks to small adversarial perturbations is a logical consequence of the amount of test error observed. We hope that our theoretical analysis of this very simple case will point the way forward to explore how the geometry of complex real-world data sets leads to adversarial examples.
Example Output: We hypothesize that the vulnerability of image models to small adversarial perturbation is a naturally occurring result of the high dimensional geometry of the data manifold. We explore and theoretically prove this hypothesis for a simple synthetic dataset.

Example Input: We present Generative Adversarial Privacy and Fairness (GAPF), a data-driven framework for learning private and fair representations of the data. GAPF leverages recent advances in adversarial learning to allow a data holder to learn "universal" representations that decouple a set of sensitive attributes from the rest of the dataset. Under GAPF, finding the optimal decorrelation scheme is formulated as a constrained minimax game between a generative decorrelator and an adversary. We show that for appropriately chosen adversarial loss functions, GAPF provides privacy guarantees against strong information-theoretic adversaries and enforces demographic parity. We also evaluate the performance of GAPF on multi-dimensional Gaussian mixture models and real datasets, and show how a designer can certify that representations learned under an adversary with a fixed architecture perform well against more complex adversaries.
Example Output:
We present Generative Adversarial Privacy and Fairness (GAPF), a data-driven framework for learning private and fair representations with certified privacy/fairness guarantees