Asynchronous distributed methods are a popular way to reduce the communication and synchronization costs of large-scale optimization. Yet, for all their success, little is known about their convergence guarantees in the challenging case of general non-smooth, non-convex objectives, beyond cases where closed-form proximal operator solutions are available. This is all the more surprising since these objectives are the ones appearing in the training of deep neural networks.
In this paper, we introduce the first convergence analysis covering asynchronous methods in the case of general non-smooth, non-convex objectives. Our analysis applies to stochastic sub-gradient descent methods both with and without block variable partitioning, and both with and without momentum. It is phrased in the context of a general probabilistic model of asynchronous scheduling accurately adapted to modern hardware properties. We validate our analysis experimentally in the context of training deep neural network architectures. We show their overall successful asymptotic convergence as well as exploring how momentum, synchronization, and partitioning all affect performance.
This paper proposes a model to study asynchronous stochastic subgradient methods for minimizing a nonsmooth nonconvex function. Studying stochastic subgradient methods in the nonsmooth nonconvex setting is already very challenging. Throwing asynchronous updates into the mix further complicates the analysis. This is overcome by proposing a model for asynchrony which captures the salient features of computation platforms, while being amenable to analysis.
The main drawback of this paper is that the experiments seem to suggest that, although asynchronous methods do a good job optimizing training metrics (loss, accuracy), the models they train do not generalize as well as using synchronous SGD. This demotivates the need for the theory developed in this paper - rather, the theory is ahead of it's time. The paper could be significantly strengthened by reporting an example application where using an asynchronous stochastic subgradient method is advantageous for training deep networks.
The notation and discussion in Sec 3.1 is pretty heavy. Can you explain how this model differs from other models of asynchrony, e.g., as put forward in the well-known book by Bertsekas and Tsitsiklis on Parallel and Distributed Computation?
The main convergence result (Theorem 1) holds for a continuous-time process, following the typical analysis of stochastic approximation algorithms. However, practitioners rarely use a diminishing step size, especially for training deep networks. Is it possible to quantify the effect of using a constant step size? (E.g., in stochastic approximation analyses for smooth functions, one typically gets a "law of large numbers" which ensures convergence of the average, and a "central limit theorem" which bounds the distance of any realization of the process to its mean. Is there an analogous CLT for this setting?)
In the experiments, the SGD with DataParallel is effectively using a mini-batch size that is 3-5x larger than that of a single worker. In this case one would expect to be able to use a larger learning rate too (since there is less noise). Evidence for this is provided in Goyal et al., "ImageNet in 1 hour". Did you try tuning the learning rate separately for each method?
How were the parameters of the deep CNNs used in the experiments divided into blocks? This choice should affect the average time to evaluate a block gradient, right? Computing the gradient of parameters in the lowest layer requires doing a full forward and backward, while computing the gradient of higher layers should be faster since it doesn't require a full backward pass. Did you measure the average time to compute a block gradient, and could you report it in the paper? Did you divide into blocks based on layer? Are all parameters within the same block part of the same filter at a given layer?
The last paragraph of Sec 4 talks about the necessity of momentum, but it isn't clear what evidence this claim is based on.