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.

[Q]: Computations for the softmax function in neural network models are expensive when the number of output classes is large. This can become a significant issue in both training and inference for such models. In this paper, we present Doubly Sparse Softmax (DS-Softmax), Sparse Mixture of Sparse of Sparse Experts, to improve the efficiency for softmax inference. During training, our method learns a two-level class hierarchy by dividing entire output class space into several partially overlapping experts. Each expert is responsible for a learned subset of the output class space and each output class only belongs to a small number of those experts. During inference, our method quickly locates the most probable expert to compute small-scale softmax. Our method is learning-based and requires no knowledge of the output class partition space a priori. We empirically evaluate our method on several real-world tasks and demonstrate that we can achieve significant computation reductions without loss of
[A]: We present doubly sparse softmax, the sparse mixture of sparse of sparse experts, to improve the efficiency for softmax inference through exploiting the two-level overlapping hierarchy. 


[Q]: Neural networks exhibit good generalization behavior in the
 over-parameterized regime, where the number of network parameters
 exceeds the number of observations. Nonetheless,
 current generalization bounds for neural networks fail to explain this
 phenomenon. In an attempt to bridge this gap, we study the problem of
 learning a two-layer over-parameterized neural network, when the data is generated by a linearly separable function. In the case where the network has Leaky
 ReLU activations, we provide both optimization and generalization guarantees for over-parameterized networks.
 Specifically, we prove convergence rates of SGD to a global
 minimum and provide generalization guarantees for this global minimum
 that are independent of the network size. 
 Therefore, our result clearly shows that the use of SGD for optimization both finds a global minimum, and avoids overfitting despite the high capacity of the model. This is the first theoretical demonstration that SGD can avoid overfitting, when learning over-specified neural network classifiers.
[A]: We show that SGD learns two-layer over-parameterized neural networks with Leaky ReLU activations that provably generalize on linearly separable data.


[Q]: Weight pruning has proven to be an effective method in reducing the model size and computation cost while not sacrificing the model accuracy. Conventional sparse matrix formats, however, involve irregular index structures with large storage requirement and sequential reconstruction process, resulting in inefficient use of highly parallel computing resources. Hence, pruning is usually restricted to inference with a batch size of one, for which an efficient parallel matrix-vector multiplication method exists. In this paper, a new class of sparse matrix representation utilizing Viterbi algorithm that has a high, and more importantly, fixed index compression ratio regardless of the pruning rate, is proposed. In this approach, numerous sparse matrix candidates are first generated by the Viterbi encoder, and then the one that aims to minimize the model accuracy degradation is selected by the Viterbi algorithm. The model pruning process based on the proposed Viterbi encoder and Viterbi algorithm is highly parallelizable, and can be implemented efficiently in hardware to achieve low-energy, high-performance index decoding process. Compared with the existing magnitude-based pruning methods, index data storage requirement can be further compressed by 85.2% in MNIST and 83.9% in AlexNet while achieving similar pruning rate. Even compared with the relative index compression technique, our method can still reduce the index storage requirement by 52.7% in MNIST and 35.5% in AlexNet.
[A]:
We present a new pruning method and sparse matrix format to enable high index compression ratio and parallel index decoding process.