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On the Complexity of Decentralized Smooth Nonconvex Finite-Sum Optimization

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 $\epsilon$-stationary point at each agent with the communication rounds of $\tilde{\mathcal O}(L\epsilon^{-2}/\sqrt{\gamma}\,)$, the computation rounds of $\tilde{\mathcal O}(n+(L+\min\{nL, \sqrt{n/m}\bar L\})\epsilon^{-2})$, and the local incremental first-oracle calls of ${\mathcal O}(mn + {\min\{mnL, \sqrt{mn}\bar L\}}{\epsilon^{-2}})$, where $L$ is the smoothness parameter of the objective function, $\bar L$ is the mean-squared smoothness parameter of all individual functions, and $\gamma$ 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-{\L}ojasiewicz condition, also achieving the near-optimal complexity bounds.

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