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

• URL: http://arxiv.org/abs/2606.11339v1
• Date: Tue, 09 Jun 2026 18:18:41 GMT
• Title: Quantized Stochastic Primal-Dual Methods for Distributed Optimization under Relaxed Global Geometry
• Abstract summary: We study distributed optimization with gradients and finite-bit communication modeled by random (unbiased) quantization.<n>Under restricted secant inequality (RSI), a constant step-size yields a linear contraction to an explicit neighborhood determined by gradient noise, quantization distortion, and network connectivity.<n>Under PolyakLojasiewicz (PL) inequality, we obtain linear-to-neighborhood convergence in the same quantized setting.
• Score: 2.15242029196761
• License: http://creativecommons.org/licenses/by/4.0/
• 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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