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 provide a theoretical algorithm for checking local optimality and escaping saddles at nondifferentiable points of empirical risks of two-layer ReLU networks. Our algorithm receives any parameter value and returns: local minimum, second-order stationary point, or a strict descent direction. The presence of M data points on the nondifferentiability of the ReLU divides the parameter space into at most 2^M regions, which makes analysis difficult. By exploiting polyhedral geometry, we reduce the total computation down to one convex quadratic program (QP) for each hidden node, O(M) (in)equality tests, and one (or a few) nonconvex QP. For the last QP, we show that our specific problem can be solved efficiently, in spite of nonconvexity. In the benign case, we solve one equality constrained QP, and we prove that projected gradient descent solves it exponentially fast. In the bad case, we have to solve a few more inequality constrained QPs, but we prove that the time complexity is exponential only in the number of inequality constraints. Our experiments show that either benign case or bad case with very few inequality constraints occurs, implying that our algorithm is efficient in most cases.
[EX A]: A theoretical algorithm for testing local optimality and extracting descent directions at nondifferentiable points of empirical risks of one-hidden-layer ReLU networks.

[EX Q]: In recent years, softmax together with its fast approximations has become the de-facto loss function for deep neural networks with multiclass predictions. However, softmax is used in many problems that do not fully fit the multiclass framework and where the softmax assumption of mutually exclusive outcomes can lead to biased results. This is often the case for applications such as language modeling, next event prediction and matrix factorization, where many of the potential outcomes are not mutually exclusive, but are more likely to be independent conditionally on the state. To this end, for the set of problems with positive and unlabeled data, we propose a relaxation of the original softmax formulation, where, given the observed state, each of the outcomes are conditionally independent but share a common set of negatives. Since we operate in a regime where explicit negatives are missing, we create an adversarially-trained model of negatives and derive a new negative sampling and weighting scheme which we denote as Cooperative Importance Sampling (CIS). We show empirically the advantages of our newly introduced negative sampling scheme by pluging it in the Word2Vec algorithm and benching it extensively against other negative sampling schemes on both language modeling and matrix factorization tasks and show large lifts in performance.
[EX A]: Defining a partially mutual exclusive softmax loss for postive data and implementing a cooperative based sampling scheme

[EX Q]: Training with larger number of parameters while keeping fast iterations is an increasingly
 adopted strategy and trend for developing better performing Deep Neural
 Network (DNN) models. This necessitates increased memory footprint and
 computational requirements for training. Here we introduce a novel methodology
 for training deep neural networks using 8-bit floating point (FP8) numbers.
 Reduced bit precision allows for a larger effective memory and increased computational
 speed. We name this method Shifted and Squeezed FP8 (S2FP8). We
 show that, unlike previous 8-bit precision training methods, the proposed method
 works out of the box for representative models: ResNet50, Transformer and NCF.
 The method can maintain model accuracy without requiring fine-tuning loss scaling
 parameters or keeping certain layers in single precision. We introduce two
 learnable statistics of the DNN tensors - shifted and squeezed factors that are used
 to optimally adjust the range of the tensors in 8-bits, thus minimizing the loss in
 information due to quantization.
[EX A]:
We propose a novel 8-bit format that eliminates the need for loss scaling, stochastic rounding, and other low precision techniques