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

Susmit Sarkar; Abhinav Raghuvanshi; Kushal Chakrabarti; Mayank Baranwal

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arXiv · 2606.11339

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

Susmit Sarkar·Abhinav Raghuvanshi·Kushal Chakrabarti·Mayank Baranwal

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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.

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Susmit Sarkar, Abhinav Raghuvanshi, Kushal Chakrabarti, Mayank Baranwal. 2026-06-09. Quantized Stochastic Primal-Dual Methods for Distributed Optimization under Relaxed Global Geometry. https://arxiv.org/abs/2606.11339

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