Robust Training of Neural Networks at Arbitrary Precision and Sparsity

The discontinuous operations inherent in quantization and sparsification introduce a long-standing obstacle to backpropagation, particularly in ultra-low precision and sparse regimes. The standard Straight-Through Estimator (STE) is widely used to address this, but the well-understood mismatch between its quantization-aware forward pass and quantization-oblivious backward pass leads to unmanaged error that can corrupt the learning process. We solve this by introducing a denoising dequantization transform derived from a principled ridge regression objective. This transform makes the entire learning process aware of and robust to the quantization error that STE's surrogate gradient bypasses, by creating an explicit, corrective gradient path. We extend this principle to sparsification by viewing it as a special form of quantization that maps insignificant values to zero. Our unified framework allows existing models to be trained at a wide spectrum of precisions and sparsity levels with off-the-shelf recipes, achieving stable training of fully binary (A1W1) and sparse sub-1-bit networks where other methods falter. This approach yields state-of-the-art results and provides a theoretically-grounded path to hyper-efficient neural networks.