Optimal Brain Restoration for Joint Quantization and Sparsification of LLMs

Recent advances in Large Language Model (LLM) compression, such as quantization and pruning, have achieved notable success. However, as these techniques gradually approach their limits, relying on a single method for further compression has become increasingly challenging. In this work, we explore an alternative solution by combining quantization and sparsity. This joint approach, though promising, introduces new difficulties due to the inherently conflicting requirements on weight distributions: quantization favors compact ranges, while pruning benefits from high variance. To attack this problem, we propose Optimal Brain Restoration (OBR), a general and training-free framework that aligns pruning and quantization by error compensation between both. OBR minimizes performance degradation on downstream tasks by building on a second-order Hessian objective, which is then reformulated into a tractable problem through surrogate approximation and ultimately reaches a closed-form solution via group error compensation. Experiments show that OBR incurs only a 1.4 perplexity degradation on Llama2-7B to enable aggressive W4A4KV4 quantization with 50\% sparsity, delivering up to 4.72x speedup and 6.4x memory reduction compared to the FP16-dense baseline.