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 Q]: We propose an approach to training machine learning models that are fair in the sense that their performance is invariant under certain perturbations to the features. For example, the performance of a resume screening system should be invariant under changes to the name of the applicant. We formalize this intuitive notion of fairness by connecting it to the original  notion of individual fairness put forth by Dwork et al and show that the proposed approach achieves this notion of fairness. We also demonstrate the effectiveness of the approach on two machine learning tasks that are susceptible to gender and racial biases.
[EX A]: Algorithm for training individually fair classifier using adversarial robustness

[EX Q]: We extend the recent results of (Arora et al., 2019) by a spectral analysis of representations corresponding to kernel and neural embeddings. They showed that in a simple single layer network, the alignment of the labels to the eigenvectors of the corresponding Gram matrix determines both the convergence of the optimization during training as well as the generalization properties. We generalize their result to kernel and neural representations and show that these extensions improve both optimization and generalization of the basic setup studied in (Arora et al., 2019).
[EX A]: Spectral analysis for understanding how different representations can improve optimization and generalization.

[EX Q]: Generalization error (also known as the out-of-sample error) measures how well the hypothesis learned from training data generalizes to previously unseen data. Proving tight generalization error bounds is a central question in statistical learning  theory.    In  this  paper,  we  obtain  generalization  error  bounds  for  learning general  non-convex  objectives,  which  has  attracted  significant  attention  in  recent years.    We develop a new framework,  termed Bayes-Stability,  for proving algorithm-dependent generalization error bounds.   The new framework combines ideas from both the PAC-Bayesian theory and the notion of algorithmic stability.   Applying the Bayes-Stability method, we obtain new data-dependent generalization bounds for stochastic gradient Langevin dynamics (SGLD) and several other noisy gradient methods (e.g., with momentum, mini-batch and acceleration, Entropy-SGD). Our result recovers (and is typically tighter than) a recent result in Mou et al. (2018) and improves upon the results in Pensia et al. (2018).   Our experiments demonstrate that our data-dependent bounds can distinguish randomly labelled data from normal data, which provides an explanation to the intriguing phenomena observed in Zhang et al. (2017a). We also study the setting where the total loss is the sum of a bounded loss and an additiona l`2 regularization term. We obtain new generalization bounds for the continuous Langevin dynamic in this setting by developing a new Log-Sobolev inequality for the parameter distribution at any time. Our new bounds are more desirable when the noise level of the processis not very small, and do not become vacuous even when T tends to infinity.
[EX A]:
We give some generalization error bounds of noisy gradient methods such as SGLD, Langevin dynamics, noisy momentum and so forth.