Two for the Price of One: Curvature-Aware Pruning Preserves Influence Functions Too

Post-training model pruning enables efficient deployment, but conventional methods do not explicitly preserve internal-gradient alignment between sparse and dense models. We study novel connections between downstream performance of compressed models and the ability to retain gradient information via influence-function preservation. Central to this connection is a curvature-aware objective that accounts for both layer activations and backpropagated gradients. Despite its promise, this objective remains underexplored in pruning due to a lack of efficient solvers. We introduce $\text{CACTUS}$, a novel ADMM-based optimizer supporting both pruning and sparse-plus-low-rank decompositions of pre-trained weights. Compared to LLM-Surgeon which targets the same objective, $\text{CACTUS}$ achieves over an order-of-magnitude speedup. At $50$% pruning of Llama-3.2-1B, $\text{CACTUS}$ reduces the C4 perplexity gap to the dense model by $24$%, improves compressed-to-dense per-sample gradient cosine similarity on WikiText-2 from $0.445$ to $0.512$, and runs over $15\times$ faster than LLM-Surgeon on substantially weaker hardware. By preserving influence functions, $\text{CACTUS}$ supports effective data selection. On Llama-3.2-3B, fine-tuning a fresh dense model on the $5$% of data selected by $\text{CACTUS}$ achieves $49.44$% accuracy on BIG-Bench Hard, compared with $47.22$% using ALPS and $49.54$% using the dense-mode selection. Finally, integrating $\text{CACTUS}$ with LoGRA, a recent gradient-projection strategy, on Qwen3-4B with MLP layers pruned using 2:4 sparsity and a rank-64 correction retains a $0.711$ cosine alignment with dense LoGRA-projected C4 gradients.