Exact solutions to the nonlinear dynamics of learning in deep linear neural networks

Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap between the theory and practice of deep learning by systematically analyzing learning dynamics for the restricted case of deep linear neural networks. Despite the linearity of their input-output map, such networks have nonlinear gradient descent dynamics on weights that change with the addition of each new hidden layer. We show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions. We provide an analytical description of these phenomena by finding new exact solutions to the nonlinear dynamics of deep learning. Our theoretical analysis also reveals the surprising finding that as the depth of a network approaches infinity, learning speed can nevertheless remain finite: for a special class of initial conditions on the weights, very deep networks incur only a finite, depth independent, delay in learning speed relative to shallow networks. We show that, under certain conditions on the training data, unsupervised pretraining can find this special class of initial conditions, while scaled random Gaussian initializations cannot. We further exhibit a new class of random orthogonal initial conditions on weights that, like unsupervised pre-training, enjoys depth independent learning times. We further show that these initial conditions also lead to faithful propagation of gradients even in deep nonlinear networks, as long as they operate in a special regime known as the edge of chaos.Comment: Submission to ICLR2014. Revised based on reviewer feedbac

Model Accuracy and Runtime Tradeoff in Distributed Deep Learning:A Systematic Study

This paper presents Rudra, a parameter server based distributed computing framework tuned for training large-scale deep neural networks. Using variants of the asynchronous stochastic gradient descent algorithm we study the impact of synchronization protocol, stale gradient updates, minibatch size, learning rates, and number of learners on runtime performance and model accuracy. We introduce a new learning rate modulation strategy to counter the effect of stale gradients and propose a new synchronization protocol that can effectively bound the staleness in gradients, improve runtime performance and achieve good model accuracy. Our empirical investigation reveals a principled approach for distributed training of neural networks: the mini-batch size per learner should be reduced as more learners are added to the system to preserve the model accuracy. We validate this approach using commonly-used image classification benchmarks: CIFAR10 and ImageNet.Comment: Accepted by The IEEE International Conference on Data Mining 2016 (ICDM 2016

Spectral Analysis of Kernel and Neural Embeddings: Optimization and Generalization

We extend the recent results of (Arora et al. 2019). by spectral analysis of the representations corresponding to the 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 the kernel and neural representations and show these extensions improve both optimization and generalization of the basic setup studied in (Arora et al. 2019). In particular, we first extend the setup with the Gaussian kernel and the approximations by random Fourier features as well as with the embeddings produced by two-layer networks trained on different tasks. We then study the use of more sophisticated kernels and embeddings, those designed optimally for deep neural networks and those developed for the classification task of interest given the data and the training labels, independent of any specific classification model