Communication-efficient distributed quantile regression neural networks via convolution smoothing and poisson sampling

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Quantile regression neural networks (QRNNs) provide flexible nonlinear conditional quantile estimation, but training is hindered by the non-differentiability of the check-loss, and existing methods do not scale to large datasets stored non-randomly across distributed machines. We propose a convolution-smoothed QRNN (CS-QRNN) that replaces the check-loss with a twice-differentiable kernel-convolved surrogate, and a distributed extension (DCS-QRNN) that combines Poisson pilot sampling with a gradient-corrected surrogate loss. After a one-time pilot-sample collection and pilot-estimator broadcast, DCS-QRNN requires only a single round of gradient aggregation, with no further communication during optimization. Under mild regularity conditions, we establish consistency, asymptotic normality, and first-order equivalence to the centralized full-data estimator. In simulations across three nonlinear regression surfaces and three error distributions, DCS-QRNN achieves 80–100 × speedup at a 1% pilot ratio while preserving centralized accuracy: on a challenging multi-peak surface with t (3) errors, DCS-QRNN attains a mean absolute error (MAE) of 0.3176 versus 0.3097 for the full-data estimator. Counting all three DCS-QRNN communication stages, the simulated wall-clock time at a network bandwidth of 100 Mbit/s and a latency of 100 ms per one-way stage is 38.0 seconds, versus 140.3 seconds for the fixed-budget FedAvg-QRNN protocol with 𝑅 = 1 0 . On the Household Electric Power Consumption and Beijing Multi-Site Air Quality datasets, DCS-QRNN preserves near-centralized accuracy while reducing running time by roughly an order of magnitude, confirming that its primary practical advantage is computational efficiency together with a substantially smaller number of synchronization stages.