Snap-Shot Decentralized Stochastic Gradient Tracking Methods

In decentralized optimization, m agents form a network and only communicate with their neighbors, which gives advantages in data ownership, privacy, and scalability. At the same time, decentralized stochastic gradient descent (textttSGD) methods, as popular decentralized algorithms for training large-scale machine learning models, have shown their superiority over centralized counterparts. Distributed stochastic gradient tracking~(textttDSGT)~citeppu2021distributed has been recognized as the popular and state-of-the-art decentralized textttSGD method due to its proper theoretical guarantees. However, the theoretical analysis of dsgt~citepkoloskova2021improved shows that its iteration complexity is tildeO left(fracbarσ2mμ varepsilon + frac√Lbarσμ(1 - λ2(W))1/2 CW √varepsilon right), where W is a double stochastic mixing matrix that presents the network topology and CW is a parameter that depends on W. Thus, it indicates that the convergence property of textttDSGT is heavily affected by the topology of the communication network. To overcome the weakness of textttDSGT, we resort to the snap-shot gradient tracking skill and propose two novel algorithms. We further justify that the proposed two algorithms are more robust to the topology of communication networks under similar algorithmic structures and the same communication strategy to dsgt~. Compared with dsgt, their iteration complexity are Oleft( fracbarσ2mμvarepsilon + frac√Lbarσμ (1 - λ2(W))√varepsilon right) and Oleft( fracbarσ2mμ varepsilon + frac√Lbarσμ (1 - λ2(W))1/2√varepsilon right) which reduce the impact on network topology (no CW).