agop: AGOP matrices of Pythia-410M and Pythia-1B over pretraining

The average gradient outer products (AGOP) of the NeurIPS 2026 paper Geometrically Disentangling Concept Learning from the Language Modeling Loss (Wang, Mallinar, Belkin, Warstadt), used for every Optimization Engagement in the paper. Code: https://github.com/YpLarryWang/concept-energy. They are part of the disentangle-concept-learning release, which also holds one dataset of probes for each of the paper's 12 tasks (cls-*, reg-*).

Language models

Pythia-410M (24 layers, hidden size 1024) and Pythia-1B (16 layers, hidden size 2048), at 19 checkpoints: steps 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1000, 2000, 4000, 8000, 16000, 32000, 64000, 128000 and 143000. Layer L is the output of transformer block L, before the final layer norm. Layers are numbered from 0 (0-23 for Pythia-410M, 0-15 for Pythia-1B); the paper's figures number them from 1, so layer 8 here is layer 9 there.

Definition

For each layer, with ${\mathrm{\ell }}_s\backslash ell\_s$ the next-token cross-entropy at a sampled position $ss$ and $h_{s}h\_s$ the layer's output at that position,

$$G=\frac{1}{N}\sum _{s=1}^N{g}_s{g}_s^{\mathrm{\top }},g_s=\frac{\mathrm{\partial }{\mathrm{\ell }}_s}{\mathrm{\partial }{h}_s}\mathrm{.}G\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{s=1\}^\{N\}\; g\_s\; g\_s^\backslash top,\; \backslash qquad\; g\_s\; =\; \backslash frac\{\backslash partial\; \backslash ell\_s\}\{\backslash partial\; h\_s\}.$$

All $NN$ positions have equal weight, and $GG$ is symmetrized.

Data

1,000 lines of WikiText-103 (wikitext-103-v1), drawn with random.Random(42) from the lines with at least 4 tokens, at least 2 of them with a letter, and no URL. Each line is truncated to 128 tokens, and up to eight positions per line (fewer in lines shorter than 9 tokens) are drawn with np.random.RandomState(42). This gives $N=7765N\; =\; 7765$ positions, the same for every checkpoint and layer.

Hyperparameters

Lines are processed in batches of 4 (Pythia-1B; Pythia-410M layers 0-22) or 8 (Pythia-410M layer 23; Pythia-410M layers 0-22 at step 128000), with right padding.

Precision

All matrices are stored in float64. In the paper's experiments, they were computed at the following precisions:

• Pythia-410M layers 0-22: float32 model, except step 128000 (float16 model, as below);
• Pythia-410M layer 23 and Pythia-1B: float16 model, with the cross-entropy on float32 logits, outer products in float32 and the sum in float64.

PROVENANCE.json lists the precision, batch size and the SHA-256 of the original file of every matrix. The code in the GitHub repository computes the AGOP in float32.

Files

pythia-410m/step<checkpoint>.safetensors     layer0 ... layer23: float64 [1024, 1024]
pythia-1b/step<checkpoint>.safetensors       layer0 ... layer15: float64 [2048, 2048]
PROVENANCE.json
SHA256SUMS

The metadata of each file gives the format version (schema).

Usage

from huggingface_hub import hf_hub_download
from safetensors.numpy import load_file

agop = load_file(hf_hub_download('disentangle-concept-learning/agop', 'pythia-1b/step143000.safetensors', repo_type='dataset'))
G = agop['layer15']              # float64 [2048, 2048]

The probe repos of the same organization (cls-*, reg-*) show how to compute a probe's Optimization Engagement with it.

License and attribution

Computed from WikiText-103 (text from Wikipedia, CC BY-SA 3.0) with the Pythia models (Apache-2.0). We release the matrices under CC BY-SA 4.0.

Please cite:

@inproceedings{wang2026geometrically,
  title     = {Geometrically Disentangling Concept Learning from the Language Modeling Loss},
  author    = {Wang, Yupei and Mallinar, Neil and Belkin, Mikhail and Warstadt, Alex},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2026}
}