This work introduces DADAO: the first decentralized, accelerated, asynchronous, primal, first-order algorithm to minimize a sum of $L$-smooth and $\mu$-strongly convex functions distributed over a given network of size $n$. Our key insight is based on modeling the local gradient updates and gossip communication procedures with separate independent Poisson Point Processes. This allows us to decouple the computation and communication steps, which can be run in parallel, while making the whole approach completely asynchronous. This leads to communication acceleration compared to synchronous approaches. Our new 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 of the Laplacian matrix $\chi_1$ and the maximal resistance $\chi_2\leq \chi_1$ of the graph to a sufficient minimal communication rate between the nodes of the network, we show that our algorithm requires $\mathcal{O}(n\sqrt{\frac{L}{\mu}}\log(\frac{1}{\epsilon}))$ local gradients and only $\mathcal{O}(n\sqrt{\chi_1\chi_2}\sqrt{\frac{L}{\mu}}\log(\frac{1}{\epsilon}))$ communications to reach a precision $\epsilon$, up to logarithmic terms. Thus, we simultaneously obtain an accelerated rate for both computations and communications, leading to an improvement over state-of-the-art works, our simulations further validating the strength of our relatively unconstrained method.

翻译：本文提出DADAO：首个同时具备去中心化、加速、异步、原始一阶特性的优化算法，用于最小化分布在给定规模为$n$的网络上的$L$-光滑且$\mu$-强凸函数之和。我们的核心洞见在于用相互独立的泊松点过程分别对局部梯度更新和八卦通信过程进行建模。这使得计算与通信步骤解耦并能够并行运行，同时确保整个方法完全异步。与同步方法相比，这带来了通信加速。新方法采用原始梯度，无需多共识内循环或误差反馈、梯度追踪、近端算子等额外机制。通过将拉普拉斯矩阵最小正特征值倒数$\chi_1$和图的极大电阻$\chi_2\leq \chi_1$与网络节点间充足最小通信速率相关联，我们证明该算法达到精度$\epsilon$所需本地梯度次数为$\mathcal{O}(n\sqrt{\frac{L}{\mu}}\log(\frac{1}{\epsilon}))$，通信次数仅为$\mathcal{O}(n\sqrt{\chi_1\chi_2}\sqrt{\frac{L}{\mu}}\log(\frac{1}{\epsilon}))$（忽略对数项）。因此，我们同时获得了计算与通信的加速率，优于现有最优方法，仿真结果进一步验证了这种相对无约束方法的有效性。