This paper studies a deep learning (DL) framework to solve distributed non-convex constrained optimizations in wireless networks where multiple computing nodes, interconnected via backhaul links, desire to determine an efficient assignment of their states based on local observations. Two different configurations are considered: First, an infinite-capacity backhaul enables nodes to communicate in a lossless way, thereby obtaining the solution by centralized computations. Second, a practical finite-capacity backhaul leads to the deployment of distributed solvers equipped along with quantizers for communication through capacity-limited backhaul. The distributed nature and the nonconvexity of the optimizations render the identification of the solution unwieldy. To handle them, deep neural networks (DNNs) are introduced to approximate an unknown computation for the solution accurately. In consequence, the original problems are transformed to training tasks of the DNNs subject to non-convex constraints where existing DL libraries fail to extend straightforwardly. A constrained training strategy is developed based on the primal-dual method. For distributed implementation, a novel binarization technique at the output layer is developed for quantization at each node. Our proposed distributed DL framework is examined in various network configurations of wireless resource management. Numerical results verify the effectiveness of our proposed approach over existing optimization techniques.
To overcome the drawbacks of traditional optimization techniques, deep learning (DL) frameworks have been recently investigated in wireless resource management @cite_19 @cite_18 @cite_23 @cite_35 @cite_5 @cite_8 and end-to-end communication system design @cite_17 @cite_4 @cite_31 @cite_24 @cite_33 . In particular, learning to optimize'' approaches in @cite_19 @cite_18 @cite_23 @cite_35 @cite_5 @cite_8 have received considerable attention for their potential to replace traditional optimization algorithms with neural network computations. Power control problems are addressed in multi-user interference channels (IFCs) via deep neural networks (DNNs) to maximize the sum rate performance @cite_19 . A supervised learning technique is used to learn a locally optimal solution produced by the weighted minimum-mean-square-error (WMMSE) algorithm @cite_7 . Its real-time computational complexity is shown to be much smaller than the original WMMSE, albeit involving intensive computations with numerous samples in training. However, the supervised learning task needs the generation of numerous training labels, i.e., the power control solution of the WMMSE algorithm, which would be a bottleneck for the DNN training step. In addition, the average sum rate performance achieved by the DNN in @cite_19 is normally lower than the WMMSE algorithm due to the nature of the supervised learning framework.
@cite_35 @cite_18 @cite_4 @cite_33 @cite_7 @cite_8 @cite_24 @cite_19 @cite_23 @cite_5 @cite_31 @cite_17
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Numerical optimization has played a central role in addressing key signal processing (SP) problems. Highly effective methods have been developed for a large variety of SP applications such as communications, radar, filter design, and speech and image analytics, just to name a few. However, optimization algorithms often entail considerable complexity, which creates a serious gap between theoretical design analysis and real-time processing. In this paper, we aim at providing a new learning-based perspective to address this challenging issue. The key idea is to treat the input and output of an SP algorithm as an unknown nonlinear mapping and use a deep neural network (DNN) to approximate it. If the nonlinear mapping can be learned accurately by a DNN of moderate size, then SP tasks can be performed effectively—since passing the input through a DNN only requires a small number of simple operations. In our paper, we first identify a class of optimization algorithms that can be accurately approximated by a fully connected DNN. Second, to demonstrate the effectiveness of the proposed approach, we apply it to approximate a popular interference management algorithm, namely, the WMMSE algorithm. Extensive experiments using both synthetically generated wireless channel data and real DSL channel data have been conducted. It is shown that, in practice, only a small network is sufficient to obtain high approximation accuracy, and DNNs can achieve orders of magnitude speedup in computational time compared to the state-of-the-art interference management algorithm. We propose a new framework for parallelizing deep neural network training that maximize the amount of data that is ingested by the training algorithm. Our proposed framework called Livermore Tournament Fast Batch Learning (LTFB) targets large-scale data problems. The LTFB approach creates a set of Deep Neural Network (DNN) models and trains each instance of these models independently and in parallel. Periodically each model selects another model to pair with, exchanges models, and then run a local tournament against held-out tournament datasets. The winning model is will continue training on the local training datasets. This new approach maximizes computation and minimizes amount of synchronization required in training deep neural network, a major bottleneck in existing synchronous deep learning algorithms. We evaluate our proposed algorithm on two HPC machines at Lawrence Livermore National Laboratory including an early access IBM Power8+ with NVIDIA Tesla P100 GPUs machine. Experimental evaluations of the LTFB framework on two popular image classification benchmark: CIFAR10 [18] and ImageNet [19], show significant speed up compared to the sequential baseline. Optimizing deep neural networks (DNNs) often suffers from the ill-conditioned problem. We observe that the scaling-based weight space symmetry property in rectified nonlinear network will cause this negative effect. Therefore, we propose to constrain the incoming weights of each neuron to be unit-norm, which is formulated as an optimization problem over Oblique manifold. A simple yet efficient method referred to as projection based weight normalization (PBWN) is also developed to solve this problem. PBWN executes standard gradient updates, followed by projecting the updated weight back to Oblique manifold. This proposed method has the property of regularization and collaborates well with the commonly used batch normalization technique. We conduct comprehensive experiments on several widely-used image datasets including CIFAR-10, CIFAR-100, SVHN and ImageNet for supervised learning over the state-of-the-art convolutional neural networks, such as Inception, VGG and residual networks. The results show that our method is able to improve the performance of DNNs with different architectures consistently. We also apply our method to Ladder network for semi-supervised learning on permutation invariant MNIST dataset, and our method outperforms the state-of-the-art methods: we obtain test errors as 2.52 , 1.06 , and 0.91 with only 20, 50, and 100 labeled samples, respectively. Previous theoretical work on deep learning and neural network optimization tend to focus on avoiding saddle points and local minima. However, the practical observation is that, at least in the case of the most successful Deep Convolutional Neural Networks (DCNNs), practitioners can always increase the network size to fit the training data (an extreme example would be [1]). The most successful DCNNs such as VGG and ResNets are best used with a degree of "overparametrization". In this work, we characterize with a mix of theory and experiments, the landscape of the empirical risk of overparametrized DCNNs. We first prove in the regression framework the existence of a large number of degenerate global minimizers with zero empirical error (modulo inconsistent equations). The argument that relies on the use of Bezout theorem is rigorous when the RELUs are replaced by a polynomial nonlinearity (which empirically works as well). As described in our Theory III [2] paper, the same minimizers are degenerate and thus very likely to be found by SGD that will furthermore select with higher probability the most robust zero-minimizer. We further experimentally explored and visualized the landscape of empirical risk of a DCNN on CIFAR-10 during the entire training process and especially the global minima. Finally, based on our theoretical and experimental results, we propose an intuitive model of the landscape of DCNN's empirical loss surface, which might not be as complicated as people commonly believe.