Source
Wang, Hoogland, van Wingerden, Furman and Murfet (2024, ICLR 2025), LIT-1535, read in full in NOTE-1188. The account rests on §3 (the definitions and limitations), §4.1, Figs. 1, 2 and 11, and appendices B (classification), F (model, SGLD settings, clustering) and G (seeds).
What was actually shown
The quantity. For a split of the weights W = U × V, with V one head’s weights, the weight-refined LLC is λ(w*; V) = −lim_{t→0⁺} log₂[vol(t/2)/vol(t)], where vol(t) is the volume of v near v* whose loss, with u held at u*, is within t of its minimum. It is estimated as nβ(E[ℓ_n] − ℓ_n(w*)) under a localized tempered posterior over V, sampled by SGLD. As a check on what the number means: for a regular model of d parameters near a nondegenerate minimum, vol(t) ∝ t^{d/2}, so λ = d/2; more degenerate directions give less.
The model. A two-layer attention-only transformer, 8 heads per layer, about 3M parameters, trained about 50,000 steps on a filtered Pile subset; the run analysed by Hoogland et al. (2024).
The types. Independently of the LLC, each head is mean-ablated at the end of training and its most affected tokens in context are matched to templates (induction, bracket matching, skip n-gram, n-gram), with manual inspection. Previous-token heads, a current-token head, two induction heads and multigram heads result.
The finding. The per-head curves are similar early and diverge with training. Coloured by type, same-type heads share scale, shape and the location of critical points, and K-means on the curves recovers the types. In layer 1, the induction and bracket-matching heads end lowest and the heads that memorize the most n-grams end highest (Fig. 11 plots the correlation). Clusters fitted to this run, applied to three further seeds, mostly recover their heads’ types, with exceptions the appendix lists.
What could have come out otherwise. The curves could have differed only in level, not in shape, or grouped heads by layer rather than function. The final ordering could have been unrelated to how many n-grams a head memorizes.
What this does not say
- Not a prediction. The types are fixed at the end of training and the curves are clustered after the fact. Whether early curves foretell a head’s later type is not tested.
- Not exact values. The estimator presumes a local minimum, which a checkpoint mid-training is unlikely to be; the authors trust only the ordering of estimates.
- Not a measured clustering quality. The cluster counts were chosen by eye, and the standard cluster metrics “did not appear to work well” with eight heads per layer.
- Not general. One architecture without MLPs, a 5,000-token vocabulary, one run in the main text.
- Not that the coefficient beats ablation. Ablation scores over training also separate the types, and their final ordering is close to the coefficient’s (appendix E).
Connections
- QUESTION-275. The reply’s “functional differentiation” is measured here, per component, over training. It is a baseline the question’s coupled model would have to beat on prediction.
- QUESTION-039. It times differentiation, not acquisition; it adds a per-component measure to the question’s list of what can be timed.
- THEORY-967. The same reading’s account of heads changing role.
- LIT-1534. Elhage et al.’s mathematical framework is where the two-layer attention-only setting and the QK/OV reading of heads used here come from; its path expansion is THEORY-988.