Methods · Learning Dynamics and Mechanistic Interpretability
Refined Local Learning Coefficients (Localising Change to Components)
Survey sections: §10.2.1 Localising Change to Components
Wang et al. (2025b) introduced refined variants of the Local Learning Coefficient (Section 10.3.4) and applied them to individual attention heads of a two-layer attention-only transformer across training. The weight-refined LLC restricts the LLC to a single head’s parameters; the data-refined LLC evaluates it on a chosen data subset.
Assumptions and requirements
Once a change is localised in time, the components that changed in that window are candidates for the mechanism.
Intended applications
Heads of the same type cluster together in both their final values and the overall shape of the curve across training, revealing how heads differentiate into distinct functional roles.
The same analysis identifies a previously undocumented multigram circuit.
Unlike the proposed induction scores (Olsson et al., 2022) for identifying induction heads, the measure does not require specifying a head type in advance.
Limitations
However, estimating refined LLCs is computationally expensive, and the approach has so far been demonstrated only on toy models.
Localising a phase transition in a coarse-grained metric therefore does not help identify the responsible components. In contrast, component-level attribution may find only gradual change without being wrong.
Related methods
Key papers
Cited in §10.2.1.
- George Wang, Jesse Hoogland, Stan van Wingerden, Zach Furman, and Daniel Murfet. Differentiation and specialization of attention heads via the refined local learning coefficient. In International Conference on Learning Representations, 2025b.
- PreprintCatherine Olsson, Nelson Elhage, Neel Nanda, Nicholas Joseph, Nova DasSarma, Tom Henighan, Ben Mann, Amanda Askell, Yuntao Bai, Anna Chen, Tom Conerly, Dawn Drain, Deep Ganguli, Zac Hatfield-Dodds, Danny Hernandez, Scott Johnston, Andy Jones, Jackson Kernion, Liane Lovitt, Kamal Ndousse, Dario Amodei, Tom Brown, Jack Clark, Jared Kaplan, Sam McCandlish, and Chris Olah. In-context learning and induction heads. Transformer Circuits Thread, 2022. https://transformer-circuits.pub/2022/in-context-learning-and-induction-heads/index.html.
Also discussed in: §10.3.3 Attributing Transitions to Components
Other methods in Learning Dynamics and Mechanistic Interpretability
Trajectory-Based Attribution · Per-Token Learning Curves · Forked-Run Experiments and Linear Mode Connectivity · Loss-Curve Derivatives · Capability-Specific Metrics · Per-Example Decomposition (POLCA) · Latent State Models · Progress Measures · Local Learning Coefficient (LLC) · Forecasting from Smaller and Partial Runs · Forecasting Safety-Relevant Behaviours · Internal Precursors as Early Warning · Pretraining Data Filtering (Deep Ignorance) · Gradient Routing · Monitoring and Steering During Fine-Tuning (Persona Vectors) · Train-Time Intervention Loop