On the Complexity of Decentralized Smooth Nonconvex Finite-Sum Optimization
Luo Luo, Yunyan Bai, Lesi Chen, Yuxing Liu, Haishan Ye
TL;DR
This work tackles decentralized smooth nonconvex finite-sum optimization by formulating a global objective $f(x)$ as the mean of local finite-sum components and introducing DEAREST$^+$, a stochastic variance-reduced, gradient-tracking algorithm with multi-consensus. The authors derive sharp complexity bounds that depend on the global smoothness $L$, the mean-squared smoothness $\bar{L}$, and the network spectral gap $\gamma$, showing near-optimal communication, computation, and LIFO complexities. They further extend the approach to the Polyak-Łojasiewicz setting, obtaining matching complexity guarantees and establishing lower bounds for both general nonconvex and PL cases. Numerical experiments on regression problems and PL-laden scenarios demonstrate that DEAREST$^+$ consistently outperforms strong decentralized baselines across the key metrics of communication, LIFO, and computation costs, highlighting its practical impact for distributed learning and optimization on networks.
Abstract
We study the decentralized optimization problem $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{m}\sum_{i=1}^m f_i({\bf x})$, where the local function on the $i$-th agent has the form of $f_i({\bf x})\triangleq \frac{1}{n}\sum_{j=1}^n f_{i,j}({\bf x})$ and every individual $f_{i,j}$ is smooth but possibly nonconvex. We propose a stochastic algorithm called DEcentralized probAbilistic Recursive gradiEnt deScenT (DEAREST) method, which achieves an $ε$-stationary point at each agent with the communication rounds of $\tilde{\mathcal O}(Lε^{-2}/\sqrtγ\,)$, the computation rounds of $\tilde{\mathcal O}(n+(L+\min\{nL, \sqrt{n/m}\bar L\})ε^{-2})$, and the local incremental first-oracle calls of ${\mathcal O}(mn + {\min\{mnL, \sqrt{mn}\bar L\}}{ε^{-2}})$, where $L$ is the smoothness parameter of the objective function, $\bar L$ is the mean-squared smoothness parameter of all individual functions, and $γ$ is the spectral gap of the mixing matrix associated with the network. We then establish the lower bounds to show that the proposed method is near-optimal. Notice that the smoothness parameters $L$ and $\bar L$ used in our algorithm design and analysis are global, leading to sharper complexity bounds than existing results that depend on the local smoothness. We further extend DEAREST to solve the decentralized finite-sum optimization problem under the Polyak-Łojasiewicz condition, also achieving the near-optimal complexity bounds.