Fully Stochastic Primal-dual Gradient Algorithm for Non-convex Optimization on Random Graphs

Stochastic decentralized optimization algorithms often suffer from issues such as synchronization overhead and intermittent communication. This paper proposes a $\underline{\rm F}$ully $\underline{\rm S}$tochastic $\underline{\rm P}$rimal $\underline{\rm D}$ual gradient $\underline{\rm A}$lgorithm (FSPDA) that suggests an asynchronous decentralized procedure with (i) sparsified non-blocking communication on random undirected graphs and (ii) local stochastic gradient updates. FSPDA allows multiple local gradient steps to accelerate convergence to stationarity while finding a consensual solution with stochastic primal-dual updates. For problems with smooth (possibly non-convex) objective function, we show that FSPDA converges to an $\mathrm{\mathcal{O}( {\it \sigma /\sqrt{nT}} )}$-stationary solution after $\mathrm{\it T}$ iterations without assuming data heterogeneity. The performance of FSPDA is on par with state-of-the-art algorithms whose convergence depend on static graph and synchronous updates. To our best knowledge, FSPDA is the first asynchronous algorithm that converges exactly under the non-convex setting. Numerical experiments are presented to show the benefits of FSPDA.

EMC$^2$: Efficient MCMC Negative Sampling for Contrastive Learning with Global Convergence

A key challenge in contrastive learning is to generate negative samples from a large sample set to contrast with positive samples, for learning better encoding of the data. These negative samples often follow a softmax distribution which are dynamically updated during the training process. However, sampling from this distribution is non-trivial due to the high computational costs in computing the partition function. In this paper, we propose an Efficient Markov Chain Monte Carlo negative sampling method for Contrastive learning (EMC$^2$). We follow the global contrastive learning loss as introduced in SogCLR, and propose EMC$^2$ which utilizes an adaptive Metropolis-Hastings subroutine to generate hardness-aware negative samples in an online fashion during the optimization. We prove that EMC$^2$ finds an $\mathcal{O}(1/\sqrt{T})$-stationary point of the global contrastive loss in $T$ iterations. Compared to prior works, EMC$^2$ is the first algorithm that exhibits global convergence (to stationarity) regardless of the choice of batch size while exhibiting low computation and memory cost. Numerical experiments validate that EMC$^2$ is effective with small batch training and achieves comparable or better performance than baseline algorithms. We report the results for pre-training image encoders on STL-10 and Imagenet-100.

DoCoM-SGT: Doubly Compressed Momentum-assisted Stochastic Gradient Tracking Algorithm for Communication Efficient Decentralized Learning

This paper proposes the Doubly Compressed Momentum-assisted Stochastic Gradient Tracking algorithm (DoCoM-SGT) for communication efficient decentralized learning. DoCoM-SGT utilizes two compression steps per communication round as the algorithm tracks simultaneously the averaged iterate and stochastic gradient. Furthermore, DoCoM-SGT incorporates a momentum based technique for reducing variances in the gradient estimates. We show that DoCoM-SGT finds a solution $\bar{\theta}$ in $T$ iterations satisfying $\mathbb{E} [ \| \nabla f(\bar{\theta}) \|^2 ] = {\cal O}( 1 / T^{2/3} )$ for non-convex objective functions; and we provide competitive convergence rate guarantees for other function classes. Numerical experiments on synthetic and real datasets validate the efficacy of our algorithm.