On the Influence of Bias-Correction on Distributed Stochastic Optimization

Various bias-correction methods such as EXTRA, gradient tracking methods, and exact diffusion have been proposed recently to solve distributed {\em deterministic} optimization problems. These methods employ constant step-sizes and converge linearly to the {\em exact} solution under proper conditions. However, their performance under stochastic and adaptive settings is less explored. It is still unknown {\em whether}, {\em when} and {\em why} these bias-correction methods can outperform their traditional counterparts (such as consensus and diffusion) with noisy gradient and constant step-sizes. This work studies the performance of exact diffusion under the stochastic and adaptive setting, and provides conditions under which exact diffusion has superior steady-state mean-square deviation (MSD) performance than traditional algorithms without bias-correction. In particular, it is proven that this superiority is more evident over sparsely-connected network topologies such as lines, cycles, or grids. Conditions are also provided under which exact diffusion method match or may even degrade the performance of traditional methods. Simulations are provided to validate the theoretical findings.