Supernodes and Halos: Loss-Critical Hubs in LLM Feed-Forward Layers
Authors: Audrey Cherilyn, Houman Safaai
Organizations: Kempner Institute at Harvard University
Abstract
We study the organization of channel-level importance in transformer feed-forward networks (FFNs). Using a Fisher-style loss proxy (LP) based on activation-gradient second moments, we show that loss sensitivity is concentrated in a small set of channels within each layer. In Llama-3.1-8B, the top 1% of channels per layer accounts for a median of 58.7% of LP mass, with a range of 33.0% to 86.1%. We call these loss-critical channels supernodes. Although FFN layers also contain strong activation outliers, LP-defined supernodes overlap only weakly with activation-defined outliers and are not explained by activation power or weight norms alone. Around this core, we find a weaker but consistent halo structure: some non-supernode channels share the supernodes' write support and show stronger redundancy with the protected core. We use one-shot structured FFN pruning as a diagnostic test of this organization. At 50% FFN sparsity, baselines that prune many supernodes degrade sharply, whereas our SCAR variants explicitly protect the supernode core; the strongest variant, SCAR-Prot, reaches perplexity 54.8 compared with 989.2 for Wanda-channel. The LP-concentration pattern appears across Mistral-7B, Llama-2-7B, and Qwen2-7B, remains visible in targeted Llama-3.1-70B experiments, and increases during OLMo-2-7B pretraining. These results suggest that LLM FFNs develop a small learned core of loss-critical channels, and that preserving this core is important for reliable structured pruning.
One-shot pruning methods like Wanda and SparseGPT apply the same sparsity ratio to every layer of a transformer, ignoring known variation in layer importance. We propose PALS (Percentile-Aware Layerwise Sparsity), which adjusts per-layer sparsity based on the 99th percentile of activation magnitudes, bounded to ±5% around the target ratio. On LLaMA-2-7B at 50% sparsity, PALS achieves 10.96 WikiText-2 perplexity versus 12.92 for uniform Wanda (mean over 9 runs, p<0.001). The benefit is architecture-dependent: LLaMA-3-8B shows marginal gains and Mistral-7B shows none. We also find that gradient-based allocation -- the seemingly more principled approach -- produces results worse than random, suggesting that gradient magnitude does not predict the impact of discrete weight removal. PALS adds negligible cost to the pruning pipeline and requires no fine-tuning.
Structured pruning compresses large language models (LLMs) by removing whole computational units, such as attention heads and feed-forward (FFN) channel groups. Most training-free methods, however, rank these units independently, implicitly treating the loss from pruning a set as the sum of its individual losses. This view fails for Transformers, whose sublayers are coupled through a shared residual stream. Two individually weak units can thus be jointly indispensable, yet independent scoring is blind to such dependence and removes them together. We introduce CoCurve (Cross-Module Co-Pruning Curvature), a calibration-only, fine-tuning-free method that prunes attention and FFN units jointly. A second-order Taylor expansion of the token-level KL between the frozen model and its masked copy yields a single Fisher matrix whose diagonal is classical node saliency and whose off-diagonal entries are co-pruning curvature edges: the extra damage of removing two units together. Under a single-ablation additivity approximation this matrix reduces to a Gram product of single-unit ablation features, so the full M x M interaction is recovered from M forward passes, with no pairwise sweeps or gradients. Pruning then reduces to one budgeted quadratic program, solved in a single shot under a shared attention--FFN budget, with no labels, fine-tuning, or recovery.
Feed-forward networks (FFNs) dominate memory traffic and computation in large language model (LLM) inference, making them a primary target for activation sparsification. However, existing training-free methods suffer substantial model-quality degradation at high sparsity due to limitations in their channel-selection strategies. We observe that the SwiGLU intermediate state provides a highly effective channel-selection signal, but obtaining it requires costly dense computation. To address this, we present \emph{Prox}, a two-stage training-free framework for sparse SwiGLU FFNs. Prox hinges on the key insight: sparse execution requires only the channel mask induced by the intermediate state, which can be constructed from the magnitude ranking of its entries rather than their exact values. Specifically, Stage 1 uses input sparsity and quantized proxy weights to construct a shared mask; Stage 2 computes the selected channels exactly, enabling sparse execution of all three projections. Across ten LLMs from six model families, Prox outperforms training-free baselines at all sparsity levels, achieves up to a 1.99× end-to-end decoding speedup at 70% FFN sparsity, and is compatible with quantization and sparse attention.