We study distributed optimization with stochastic gradients and finite-bit communication modeled by random (unbiased) quantization. We propose q-PDGD, a quantized stochastic primal-dual method, and analyze it under relaxed global geometry. Under restricted secant inequality (RSI), a constant step-size yields linear contraction to an explicit neighborhood determined by gradient noise, quantization distortion, and network connectivity, while a diminishing step-size achieves O(1/k) convergence without shared-minimizer assumptions. Under Polyak-Lojasiewicz (PL) inequality, we obtain linear-to-neighborhood convergence in the same stochastic quantized setting. Our results match the best-known centralized stochastic rates in oracle complexity, and are supported by experiments demonstrating the predicted tradeoffs between quantization level, step-size choice, and graph structure.

翻译：我们研究带随机梯度与有限比特通信（由随机无偏量化建模）的分布式优化问题。提出q-PDGD——一种量化随机原始-对偶方法，并在放宽的全局几何条件下对其进行分析。在限制割线不等式（RSI）条件下，恒定步长能实现向由梯度噪声、量化畸变及网络连通性共同决定的显式邻域线性收缩；而递减步长可在无需共享极小化假设的前提下达到O(1/k)收敛速度。在Polyak-Lojasiewicz（PL）不等式条件下，我们在相同的随机量化场景中获得邻域线性收敛。本方法的oracle复杂度与最先进的集中式随机优化速率相匹配，实验验证了量化等级、步长选择与图结构之间预期的权衡关系。