Quantized Stochastic Primal-Dual Methods for Distributed Optimization under Relaxed Global Geometry

2026-06-09

AI Summary

This work addresses the convergence challenges in distributed optimization under finite-bit communication and stochastic gradients by proposing a quantized stochastic primal-dual gradient descent method (q-PDGD). The algorithm achieves effective optimization under mild geometric conditions without requiring the common assumption of shared minimizers. It integrates unbiased stochastic quantization with stochastic primal-dual updates and, for the first time, establishes linear neighborhood convergence and an $O(1/k)$ global convergence rate under the restricted secant inequality and Polyak–Łojasiewicz (PL) conditions. Theoretical analysis shows that its oracle complexity matches that of optimal centralized stochastic algorithms. Empirical results further validate the trade-offs among quantization precision, stepsize, and network topology, highlighting both the practical utility and theoretical novelty of the proposed method.

Abstract

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.

Problem

• distributed optimization
• stochastic gradients
• quantization
• relaxed global geometry
• primal-dual methods

Innovation

• quantized stochastic primal-dual
• distributed optimization
• relaxed global geometry
• restricted secant inequality
• Polyak-Lojasiewicz inequality