A09 — MDL / Singular Learning Theory

This framework asks: what is the effective complexity of each circuit component — is it geometrically simple (specialized) or degenerate (polyfunctional)?

Singular Learning Theory (SLT) provides a geometric characterization of neural network parameters at convergence. The Local Learning Coefficient (LLC) — also called the real log canonical threshold (RLCT) — measures how many effective parameters a component uses relative to its nominal parameter count. A component with low LLC is geometrically simple: it sits near a low-dimensional singularity in parameter space, consistent with implementing a single clean function. A component with high LLC is geometrically complex: it occupies a high-dimensional region, consistent with polyfunctionality or redundancy.

The MDL connection: LLC directly controls the model’s Bayesian Information Criterion at the component level. Lower LLC means shorter description length — the component can be described with fewer bits. This maps onto the intuition that interpretable circuits should be compressible: a head that implements a single algorithmic role (e.g., “copy the previous token”) should have lower effective complexity than a head juggling multiple unrelated tasks.

Theoretical grounding

Source	Year	Key contribution
Watanabe, Algebraic Geometry and Statistical Learning Theory	2009	SLT: real log canonical threshold governs generalization in singular models
Lau et al., arXiv 2310.19470	2023	devinterp: practical LLC estimation for neural networks
Elhage et al., “A Mathematical Framework for Transformer Circuits”	2021	Circuit components as functional units amenable to complexity analysis
Olsson et al., “In-context Learning and Induction Heads”	2022	Phase transitions in learning; induction heads as discrete mechanistic structures

Core concept: the Local Learning Coefficient

For a model with parameter ( w ) near a critical point ( w_0 ), the LLC ( \lambda ) governs the asymptotic free energy:

[ F_n = nL_n(w_0) + \lambda \log n + O(\log \log n) ]

where ( L_n ) is the empirical loss. The LLC is estimated via the SGLD (Stochastic Gradient Langevin Dynamics) trace:

[ \hat{\lambda} = \frac{n}{m} \left( \hat{L}_n^{\text{SGLD}} - L_n(w_0) \right) ]

For a regular (non-singular) model, ( \lambda = d/2 ) (half the parameter count). For singular models — which neural networks always are — ( \lambda ) can be much smaller, reflecting the low effective dimensionality of the parameter region. Components that implement clean, specialized functions tend to sit near lower-dimensional singularities and thus have lower LLC.

Hyperparam sensitivity (varying learning rate, data subset, initialization) complements LLC by measuring functional stability: a component whose role changes drastically with small hyperparameter changes is likely sitting in a high-dimensional, degenerate region.

Instruments under A09

C10 — LLC Estimation (10_llc.py)

Estimates the local learning coefficient for each circuit component by running SGLD chains from the trained weights and measuring the gap between the SGLD average loss and the MAP loss:

[ \hat{\lambda}(c) = \frac{n}{m} \left( \frac{1}{m} \sum_{t=1}^m L_n(w_t) - L_n(w_0) \right) ]

where ( w_t ) are SGLD samples restricted to component ( c )‘s parameters.

What it establishes: Effective complexity/dimensionality of each component’s parameter region. Low LLC indicates a specialized, interpretable role; high LLC indicates polyfunctionality or redundancy.

What it does not establish: What function the component implements (only its geometric complexity). Must be paired with A05 or A01 for functional characterization.

Usage:

uv run python 10_llc.py --tasks ioi sva --n-prompts 40

C29 — Hyperparameter Sensitivity (29_hyperparam_sensitivity.py)

Measures how stable a component’s causal importance score is across hyperparameter variations (ablation method, prompt count, corruption type). High sensitivity indicates the component’s role is fragile or method-dependent:

[ \text{Sensitivity}(c) = \text{CV}\left[ AP(c; \theta_1), \ldots, AP(c; \theta_k) \right] ]

where ( \theta_i ) are different hyperparameter settings and CV is the coefficient of variation.

What it establishes: Robustness of causal conclusions to methodological choices.

What it does not establish: Which hyperparameter setting is “correct” — only the variance across them.

Usage:

uv run python 29_hyperparam_sensitivity.py --tasks ioi sva --n-prompts 40

Reading the scores

Pattern	What it means
Low LLC, low sensitivity	Clean specialized component; likely interpretable
High LLC, low sensitivity	Complex but stable; polyfunctional component reliably used
Low LLC, high sensitivity	Specialized but fragile; role depends on exact conditions
High LLC, high sensitivity	Degenerate and unstable; difficult to interpret reliably

Connection to other frameworks

A09 provides a complexity characterization that complements A05’s (MDC/Glennan) mechanistic claims: a component hypothesized to implement a simple logic gate should have low LLC. A04 (Woodward) measures robustness to intervention method; A09’s hyperparam sensitivity measures robustness to evaluation method — both address reliability of causal claims. A08 (PID) identifies redundancy between components; high redundancy predicts that the redundant components sit in a degenerate (high-LLC) parameter region where permuting them does not change the loss.