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. While the community has long viewed quantization as unfriendly to gradient descent due to its lack of smoothness, we pinpoint—for the first time—that the key issue is the absence of a proper gradient path that allows training to learn robustness to quantization noise. The standard Straight-Through Estimator (STE) exacerbates this with its well-understood mismatch: a quantization-aware forward pass but oblivious backward pass, leading to unmanaged error and instability. We solve this by explicitly modeling quantization as additive noise, making the full forward-backward path well-defined without heuristic gradient estimation. As one natural solution, we introduce a denoising dequantization transform derived from a principled ridge regression objective, creating an explicit, corrective gradient path that makes learning robust to the noise STE bypasses. We extend this to sparsification by treating it as a special form of quantization that zeros out small values. Our unified framework trains models at arbitrary precisions and sparsity levels with off-the-shelf recipes, enabling stable A1W1 and sub-1-bit networks where others falter. It yields state-of-the-art results, mapping efficiency frontiers for modern LLMs and providing a theoretically grounded path to hyper-efficient neural networks.