Low-bit quantization-aware training of neural networks typically relies on the straight-through estimator (STE) to learn both quantized weights and their associated scales or effective bit-widths. Building on earlier analysis of gradual differentiable noise-scale quantization, we first recall that the classical STE noise-scale gradient can be interpreted as an asymptotically optimal estimator of the rounding-noise gradient obtained by averaging over mini-batches. This viewpoint justifies replacing the exact rounding residual by any zero-mean proxy noise with matched variance, such as symmetric Bernoulli (Rademacher), Uniform or Gaussian perturbations. In this paper we make the connection between such STE-based scale updates and simultaneous perturbation stochastic approximation (SPSA) precise. We show that injecting noise into the learnable quantization scale implements a one-measurement SPSA scheme in the low-dimensional noise-scale space, and we verify the standard stochastic-approximation assumptions for a realistic quantization-aware loss with a bit-width penalty. Experiments with one-bit weight and activation quantization on ResNet-20 for CIFAR-10/100 classification, and four-bit quantization on RFDN for standard super-resolution (SR) benchmarks, demonstrate that the stochastic scale update consistently accelerates convergence of the effective bit-width while preserving final accuracy compared to empirical rounding. In a PyTorch implementation this modification requires only one additional line on top of a standard LSQ-type quantization-aware training loop, adding negligible computational and memory overhead.