# 🧠 Reproducing “Super Weights” in Large Language Models

**Paper:** *The Super Weight in Large Language Models*  
**Authors:** Mengxia Yu, De Wang, Qi Shan, Colorado J. Reed, Alvin Wan  
**Affiliation:** Apple & University of Notre Dame  
**arXiv:** [2411.07191v2 (July 2025)](https://arxiv.org/abs/2411.07191)

---

## 🧩 1. Background

Large Language Models (LLMs) often exhibit *outlier weights and activations* — values with extremely large magnitudes that strongly influence model quality.  
This paper identifies a **single scalar parameter**, termed a **Super Weight (SW)**, whose removal alone can **destroy a model’s ability to generate text**.

### Key findings

- Pruning **one scalar** in Llama-7B causes zero-shot accuracy to drop → random guessing.  
- The same weight induces a **Super Activation (SA)** — a huge activation spike that persists across layers.  
- Both SW and SA can be found **data-free**, with a single forward pass.  
- Preserving them dramatically improves **quantization quality**.

---

## 🧠 2. Conceptual Overview

| Term | Description |
|------|--------------|
| **Super Weight (SW)** | A single extremely important weight (scalar) in `mlp.down_proj` of an early transformer block. |
| **Super Activation (SA)** | The corresponding massive activation value generated by SW; propagates via skip connections. |
| **Effect of Pruning SW** | Model generates gibberish output, perplexity ↑ ×1000, zero-shot accuracy ↓ ≈ 35 points. |
| **Effect of Restoring SA** | Restores ≈ 40 % of performance loss → shows SW works partly through SA. |

---

## ⚙️ 3. How to Find Super Weights (Data-Free Method)

### Step 1 — Locate MLP Layers
In each Transformer block, focus on the **MLP down-projection** (`mlp.down_proj`) module.

### Step 2 — Forward Pass
Run one forward pass with any prompt ( no dataset required ):

```python
prompt = "My favorite food is"
inputs = tokenizer(prompt, return_tensors="pt").to(model.device)
outputs = model(**inputs)
```

### Step 3 — Record Activations
Hook the input and output of each `down_proj` layer to capture activations:

```python
activations = {}

def hook_fn(module, inp, out):
    activations[module] = (inp[0].detach(), out.detach())

for i, layer in enumerate(model.model.layers):
    layer.mlp.down_proj.register_forward_hook(hook_fn)
model(**inputs)
```

### Step 4 — Find Activation Spikes
For each layer, compute maximum absolute values per channel:

```python
max_in = torch.max(torch.abs(inp), dim=0).values
max_out = torch.max(torch.abs(out), dim=0).values
```

Plot or inspect their peaks across layers (Figure 3 of the paper).  
A layer with a **sharp activation spike** indicates presence of a Super Weight.

### Step 5 — Determine Coordinates
- Row index  = channel of max output (`out.argmax()` → row)  
- Column index = channel of max input (`inp.argmax()` → col)  
- The Super Weight is:  
  ```
  model.layers[layer_id].mlp.down_proj.weight[row, col]
  ```

Example (Llama-7B): `layer[2].mlp.down_proj.weight[3968, 7003]`.

---

## 🧮 4. Mathematical Explanation

For down-projection layer:
\[
Y = X W^T
\]
If a super activation \( Y_{ij} \) is dominant,  
then it is mainly produced by one large input–weight pair \((X_{ik}, W_{jk})\).  
Detecting the indices of extreme \( X_{ik} \) and \( Y_{ij} \) reveals the Super Weight \( W_{jk} \).

---

## 📋 5. Known Super Weight Coordinates (Table 2)

| Model | Layer | Type | Coordinates [row, col] |
|:------|:------|:------|:------|
| **Llama-7B** | 2 | mlp.down_proj | [3968, 7003] |
| **Llama-13B** | 2 | mlp.down_proj | [2231, 2278], [2231, 6939] |
| **Llama-30B** | 3 / 10 | mlp.down_proj | [5633, 12817], [5633, 17439], [5633, 14386] |
| **Llama-2 7B** | 1 | mlp.down_proj | [2533, 7890] |
| **Mistral-7B** | 1 | mlp.down_proj | [2070, 7310] |
| **OLMo-7B** | 1 / 2 / 7 / 24 | mlp.down_proj | [269, 7467], [269, 8275], [269, 453], [269, 2300] |
| **Phi-3 mini-4k-instruct** | 2 / 4 | mlp.down_proj | [525, 808], [1113, 2723], … |

---

## 🧪 6. Verification Procedure

### ✅ Step A — Pruning Test
```python
row, col = 3968, 7003
model.model.layers[2].mlp.down_proj.weight[row, col] = 0
```
Then generate text:
```python
print(model.generate(**tokenizer("My favorite condiment is", return_tensors="pt")))
```
→ If output becomes gibberish → found SW successfully.

### ✅ Step B — Super Activation Restoration
Record that activation value before pruning, restore it manually after pruning to verify partial recovery.

---

## ⚡ 7. Practical PyTorch Snippet

```python
import torch
from transformers import AutoModelForCausalLM, AutoTokenizer

model_name = "meta-llama/Llama-2-7b-hf"
model = AutoModelForCausalLM.from_pretrained(model_name, torch_dtype=torch.float16)
tokenizer = AutoTokenizer.from_pretrained(model_name)

prompt = "Hello world"
inputs = tokenizer(prompt, return_tensors="pt").to(model.device)

def find_super_weight(model, inputs, threshold=3.0):
    for i, layer in enumerate(model.model.layers):
        x = layer.mlp.gate_proj(inputs['input_ids'].float())
        y = layer.mlp.down_proj(x)
        max_in, idx_in = torch.max(torch.abs(x), dim=1)
        max_out, idx_out = torch.max(torch.abs(y), dim=1)
        if max_in.max() > threshold and max_out.max() > threshold:
            print(f"[Layer {i}] Super weight candidate at ({idx_out.item()}, {idx_in.item()})")

find_super_weight(model, inputs)
```

---

## 📈 8. Interpretation & Use Cases

| Use Case | Effect of Preserving SW/SA |
|:--|:--|
| **Quantization** | Enhances simple round-to-nearest (INT4/INT8) to ≈ 70–80 % of SmoothQuant quality. |
| **Model Compression** | Allows larger block sizes (e.g., 512×512) with less degradation. |
| **Explainability** | Reveals that a few weights govern semantic token probabilities (stopword suppression). |

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## 🧭 9. Summary

- 🧩 Super Weights exist — a few scalars dominate LLM behavior.  
- ⚙️ They can be found with a single forward pass.  
- ⚡ Preserving them is vital for model compression and quantization.  
- 📊 Author released a directory of SW coordinates for open LLMs.

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## 📚 10. References

Yu et al., **“The Super Weight in Large Language Models,”** arXiv:2411.07191v2, 2025.  
Sun et al., *Massive Activations in Large Language Models*, ICLR Workshop 2024.  
Dettmers et al., *GPTQ / AWQ / SmoothQuant* (2022–2024).

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