Stochastic Distributed Optimization under Average Second-order Similarity: Algorithms and Analysis

We study finite-sum distributed optimization problems with $n$-clients under popular $\delta$-similarity condition and $\mu$-strong convexity. We propose two new algorithms: SVRS and AccSVRS motivated by previous works. The non-accelerated SVRS method combines the techniques of gradient-sliding and variance reduction, which achieves superior communication complexity $\tilde{\gO}(n {+} \sqrt{n}\delta/\mu)$ compared to existing non-accelerated algorithms. Applying the framework proposed in Katyusha X, we also build a direct accelerated practical version named AccSVRS with totally smoothness-free $\tilde{\gO}(n {+} n^{3/4}\sqrt{\delta/\mu})$ communication complexity that improves upon existing algorithms on ill-conditioning cases. Furthermore, we show a nearly matched lower bound to verify the tightness of our AccSVRS method.