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Publication# Stochastic distributed learning with gradient quantization and double-variance reduction

Résumé

We consider distributed optimization over several devices, each sending incremental model updates to a central server. This setting is considered, for instance, in federated learning. Various schemes have been designed to compress the model updates in order to reduce the overall communication cost. However, existing methods suffer from a significant slowdown due to additional variance omega > 0 coming from the compression operator and as a result, only converge sublinearly. What is needed is a variance reduction technique for taming the variance introduced by compression. We propose the first methods that achieve linear convergence for arbitrary compression operators. For strongly convex functions with condition number kappa, distributed among n machines with a finite-sum structure, each worker having less than in components, we also (i) give analysis for the weakly convex and the non-convex cases and (ii) verify in experiments that our novel variance reduced schemes are more efficient than the baselines. Moreover, we show theoretically that as the number of devices increases, higher compression levels are possible without this affecting the overall number of communications in comparison with methods that do not perform any compression. This leads to a significant reduction in communication cost. Our general analysis allows to pick the most suitable compression for each problem, finding the rig ht balance between additional variance and communication savings. Finally, we also (iii) give analysis for arbitrary quantized updates.

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Martin Jaggi, Anastasiia Koloskova, Sebastian Urban Stich

We consider decentralized stochastic optimization with the objective function (e.g. data samples for machine learning task) being distributed over n machines that can only communicate to their neighbors on a fixed communication graph. To reduce the communication bottleneck, the nodes compress (e.g. quantize or sparsify) their model updates. We cover both unbiased and biased compression operators with quality denoted by \omega 0. This is (up to our knowledge) the first gossip algorithm that supports arbitrary compressed messages for \omega > 0 and still exhibits linear convergence. We (iii) show in experiments that both of our algorithms do outperform the respective state-of-the-art baselines and CHOCO-SGD can reduce communication by at least two orders of magnitudes.

2019Introduction of optimisation problems in which the objective function is black box or obtaining the gradient is infeasible, has recently raised interest in zeroth-order optimisation methods. As an example finding adversarial examples for Deep Learning models (Chen et al. (2017); Moosavi-Dezfooli et al. (2016)) is one of the most common applications in which zeroth-order methods could be used. These optimisation methods use only function values at certain points to estimate the gradient. Most current approaches iteratively sample a random search direction along which they compute an estimation of the gradient (Nesterov and Spokoiny (2017); Conn et al. (2009); Wibisono et al. (2012)). However, due to the high variance in the search direction, these methods usually need d times more iterations than the standard gradient methods, where d is the dimensionality of the problem. So it seems that the main effort for improving the zeroth-order methods should be in reducing the variance of the gradient estimate. In this work we will analyse the gradient-free oracle which uses random directions sampled form a Gaussian distribution. Our analysis shows that in smooth and strongly convex setting, we have a convergence rate of O( d/T) which clearly shows the dependency to the dimension of the problem. Furthermore we propose some variance reduction methods to make the zeroth-order optimisation faster. We experiment our proposed methods in Python to compare their convergence in stochastic and non-stochastic setting. Our empirical results show that in a setting that number of allowed function evaluation is fixed, using a variance reduction method (e.g. momentum) can make the convergence of zeroth-order methods happen faster.

2019The interest for distributed stochastic optimization has raised to train complex Machine Learning models with more data on distributed systems. Increasing the computation power speeds up the training but it faces a communication bottleneck between workers which hurts the scale-up of these distributed algorithms. Previous work tried to address this issue through quantization by broadcasting low precision updates (Seide et al., 2014; Alistarh et al., 2017) or through sparsification by sharing only the most important coordinates to update (Aji and Heafield, 2017; Lin et al., 2018). Even though the sparsifica- tion method works well in practice, it lacked a theoretical proof until now. We propose a sparsification scheme for SGD where only a small constant number of coordinates are applied at each iteration. The amount of data to be communicated is drastically reduced while the O(1/T ) convergence rate of vanilla SGD is preserved. Our concise proof extends to parallel setting and gains a linear speed up in the number of workers. We exper- iment with sparsified SGD with memory in C++ and Python, and show the excellent convergence properties for top-k and random-k operators. Our scheme outperforms QSGD in progress per number of bits sent. It also opens the path to using lock-free asynchronous parallelization (e.g. Hogwild! Niu et al. (2011)) on dense problems as the sparsity of the gradient updates is enforced.

2018