On the O(1/k) Convergence of Distributed Gradient Methods Under Random Quantization

We revisit the so-called distributed two-time-scale stochastic gradient method for solving a strongly convex optimization problem over a network of agents in a bandwidth-limited regime. In this setting, the agents can only exchange the quantized values of their local variables using a limited number of communication bits. Due to quantization errors, the existing best-known convergence results of this method can only achieve a suboptimal rate $\mathcal {O}$ ( $1/\sqrt {k}$ ), while the optimal rate is $\mathcal {O}$ ( $1/k$ ) under no quantization, where k is the time iteration. The main contribution of this letter is to address this theoretical gap, where we study a sufficient condition and develop an innovative analysis and step-size selection to achieve the optimal convergence rate $\mathcal {O}$ ( $1/k$ ) for the distributed gradient methods given any number of quantization bits. We provide numerical simulations to illustrate the effectiveness of our theoretical results.