The Convergence of Sparsified Gradient Methods

Distributed training of massive machine learning models, in particular deep neural networks, via Stochastic Gradient Descent (SGD) is becoming commonplace. Several families of communication-reduction methods, such as quantization, large-batch methods, and gradient sparsification, have been proposed. To date, gradient sparsification methods - where each node sorts gradients by magnitude, and only communicates a subset of the components, accumulating the rest locally - are known to yield some of the largest practical gains. Such methods can reduce the amount of communication per step by up to three orders of magnitude, while preserving model accuracy. Yet, this family of methods currently has no theoretical justification. This is the question we address in this paper. We prove that, under analytic assumptions, sparsifying gradients by magnitude with local error correction provides convergence guarantees, for both convex and non-convex smooth objectives, for data-parallel SGD. The main insight is that sparsification methods implicitly maintain bounds on the maximum impact of stale updates, thanks to selection by magnitude. Our analysis and empirical validation also reveal that these methods do require analytical conditions to converge well, justifying existing heuristics.Comment: NIPS 2018 - Advances in Neural Information Processing Systems; Authors in alphabetic orde

L0L_0-ARM: Network Sparsification via Stochastic Binary Optimization

We consider network sparsification as an $L_0$-norm regularized binary optimization problem, where each unit of a neural network (e.g., weight, neuron, or channel, etc.) is attached with a stochastic binary gate, whose parameters are jointly optimized with original network parameters. The Augment-Reinforce-Merge (ARM), a recently proposed unbiased gradient estimator, is investigated for this binary optimization problem. Compared to the hard concrete gradient estimator from Louizos et al., ARM demonstrates superior performance of pruning network architectures while retaining almost the same accuracies of baseline methods. Similar to the hard concrete estimator, ARM also enables conditional computation during model training but with improved effectiveness due to the exact binary stochasticity. Thanks to the flexibility of ARM, many smooth or non-smooth parametric functions, such as scaled sigmoid or hard sigmoid, can be used to parameterize this binary optimization problem and the unbiasness of the ARM estimator is retained, while the hard concrete estimator has to rely on the hard sigmoid function to achieve conditional computation and thus accelerated training. Extensive experiments on multiple public datasets demonstrate state-of-the-art pruning rates with almost the same accuracies of baseline methods. The resulting algorithm $L_0$-ARM sparsifies the Wide-ResNet models on CIFAR-10 and CIFAR-100 while the hard concrete estimator cannot. The code is public available at https://github.com/leo-yangli/l0-arm.Comment: Published as a conference paper at ECML 201