Training deep neural networks (DNNs) efficiently is a challenge due to the associated highly nonconvex optimization. The backpropagation (backprop) algorithm has long been the most widely used algorithm for gradient computation of parameters of DNNs and is used along with gradient descent-type algorithms for this optimization task. Recent work have shown the efficiency of block coordinate descent (BCD) type methods empirically for training DNNs. In view of this, we propose a novel algorithm based on the BCD method for training DNNs and provide its global convergence results built upon the powerful framework of the Kurdyka-Lojasiewicz (KL) property. Numerical experiments on standard datasets demonstrate its competitive efficiency against standard optimizers with backprop.
@cite_9 and @cite_7 also suggest the use of BCD for training DNNs and observe empirically the per epoch efficiency where the training loss drops much faster than SGD. Multiple related work consider a similar scheme to ours. A very recent piece of work implements proximal steps for model parameter updates only but keep gradient steps for updating the activation parameters and the output layer parameters, whilst we also apply proximal steps for updating these parameters. In the problem formulation of @cite_7 , bias vectors are not used in the layers. They concatenate all weight matrices in all hidden layers and all activation vectors (defined below) respectively into two separate blocks and update together with the weight matrix of the output layer so that these three blocks are updated alternately instead of an overall backward order as in backprop. @cite_9 consider a specific DNN using squared loss and sigmoid activation function, and propose the so-called method of auxiliary coordinate (MAC). Our problem formulation is similar, but is further simplified described in the next section.
@cite_9 @cite_7
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By lifting the ReLU function into a higher dimensional space, we develop a smooth multi-convex formulation for training feed-forward deep neural networks (DNNs). This allows us to develop a block coordinate descent (BCD) training algorithm consisting of a sequence of numerically well-behaved convex optimizations. Using ideas from proximal point methods in convex analysis, we prove that this BCD algorithm will converge globally to a stationary point with R-linear convergence rate of order one. In experiments with the MNIST database, DNNs trained with this BCD algorithm consistently yielded better test-set error rates than identical DNN architectures trained via all the stochastic gradient descent (SGD) variants in the Caffe toolbox. Training deep neural networks (DNNs) efficiently is a challenge due to the associated highly nonconvex optimization. Recently, the efficiency of the block coordinate descent (BCD) type methods has been empirically illustrated for DNN training. The main idea of BCD is to decompose the highly composite and nonconvex DNN training problem into several almost separable simple subproblems. However, their convergence property has not been thoroughly studied. In this paper, we establish some unified global convergence guarantees of BCD type methods for a wide range of DNN training models, including but not limited to multilayer perceptrons (MLPs), convolutional neural networks (CNNs) and residual networks (ResNets). This paper nontrivially extends the existing convergence results of nonconvex BCD from the smooth case to the nonsmooth case. Our convergence analysis is built upon the powerful Kurdyka- ojasiewicz (KL) framework but some new techniques are introduced, including the establishment of the KL property of the objective functions of many commonly used DNNs, where the loss function can be taken as squared, hinge and logistic losses, and the activation function can be taken as rectified linear units (ReLUs), sigmoid and linear link functions. The efficiency of BCD method is also demonstrated by a series of exploratory numerical experiments. With the fast development of deep learning, it has become common to learn big neural networks using massive training data. Asynchronous Stochastic Gradient Descent (ASGD) is widely adopted to fulfill this task for its efficiency, which is, however, known to suffer from the problem of delayed gradients. That is, when a local worker adds its gradient to the global model, the global model may have been updated by other workers and this gradient becomes "delayed". We propose a novel technology to compensate this delay, so as to make the optimization behavior of ASGD closer to that of sequential SGD. This is achieved by leveraging Taylor expansion of the gradient function and efficient approximation to the Hessian matrix of the loss function. We call the new algorithm Delay Compensated ASGD (DC-ASGD). We evaluated the proposed algorithm on CIFAR-10 and ImageNet datasets, and the experimental results demonstrate that DC-ASGD outperforms both synchronous SGD and asynchronous SGD, and nearly approaches the performance of sequential SGD. It is common practice to decay the learning rate. Here we show one can usually obtain the same learning curve on both training and test sets by instead increasing the batch size during training. This procedure is successful for stochastic gradient descent (SGD), SGD with momentum, Nesterov momentum, and Adam. It reaches equivalent test accuracies after the same number of training epochs, but with fewer parameter updates, leading to greater parallelism and shorter training times. We can further reduce the number of parameter updates by increasing the learning rate @math and scaling the batch size @math . Finally, one can increase the momentum coefficient @math and scale @math , although this tends to slightly reduce the test accuracy. Crucially, our techniques allow us to repurpose existing training schedules for large batch training with no hyper-parameter tuning. We train Inception-ResNet-V2 on ImageNet to @math validation accuracy in under 2500 parameter updates, efficiently utilizing training batches of 65536 images.