DADAO: Decoupled Accelerated Decentralized Asynchronous Optimization

Adel Nabli, Edouard Oyallon

Keywords: Decentralized Asynchronous Optimization, Convex Optimization, Time-Varying Networks

TL;DR: We introduce a novel decentralized asynchronous accelerated stochastic first order algorithm to minimize a sum of smooth and strongly convex functions over a time-varying connectivity network.

Abstract: DADAO is a novel decentralized asynchronous stochastic first order algorithm to minimize a sum of $L$-smooth and $\mu$-strongly convex functions distributed over a time-varying connectivity network of size $n$. We model the local gradient updates and gossip communication procedures with separate independent Poisson Point Processes, decoupling the computation and communication steps in addition to making the whole approach completely asynchronous. Our method employs primal gradients and does not use a multi-consensus inner loop nor other ad-hoc mechanisms such as Error Feedback, Gradient Tracking, or a Proximal operator. By relating the inverse of the smallest positive eigenvalue $\chi^*_1$ and the effective resistance $\chi_2^*$ of our graph to a necessary minimal communication rate between nodes of the network, we show that our algorithm requires $\mathcal{O}(n\sqrt{\frac{L}{\mu}}\log \epsilon)$ local gradients and only $\mathcal{O}(n\sqrt{\chi_1^*\chi_2^*}\sqrt{\frac{L}{\mu}}\log \epsilon)$ communications to reach a precision $\epsilon$. If SGD with uniform noise $\sigma^2$ is used, we reach a precision $\epsilon$ with same speed, up to a bias term in $\mathcal{O}(\frac{\sigma^2}{\sqrt{\mu L}})$. This improves upon the bounds obtained with current state-of-the-art approaches, our simulations validating the strength of our relatively unconstrained method.

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