This paper investigates nonlinear vector quantization strategies for efficient gradient compression in deep neural network training, based on Helmert-domain decorrelation combined with A-law and \(\mu \) -law companding. By pairing gradient coefficients and applying the pairwise Helmert transform prior to semilogarithmic companding, the proposed method reduces precision requirements while preserving convergence stability. An adaptive switching mechanism selects between low-bit and high-bit quantization regions based on local gradient statistics, without additional hyperparameter tuning. We evaluate the approach on a multi-layer perceptron (MLP) for tabular prediction and a convolutional neural network (CNN) on CIFAR-10. Both companding schemes closely track the full-precision (FP32) loss and accuracy trajectories, while achieving substantial communication reduction and low gradient root mean square error (RMSE). Additional comparisons against quantized stochastic gradient descent (QSGD), Top-k sparsification, error-feedback sign SGD (EFSignSGD), and PowerSGD show that the proposed Helmert-based companding framework offers a favorable trade-off between accuracy and compression, particularly in bandwidth-constrained distributed training. These results indicate that proposed Helmert-based companding in the decorrelated gradient domain provides an effective and scalable alternative to conventional gradient compression techniques. In particular, it achieves stable convergence at moderate compression ratios, while maintaining lower gradient distortion and reduced communication cost, making it especially suitable for bandwidth-constrained distributed and federated training environments.
