The local learning coefficent is a measure of a model's "effective dimensionality" that captures its complexity (more background here). Lau et al recently described a sampling method (SGLD) using noisy gradients to find a stochastic estimate $^λ$ in a computationally tractable way (good explanation here):

Examples of points sampled by SGLD in a two-dimensional toy model (Lau et al, 2023)

I present results (Github repo) of the "Task Variability" project suggested on the DevInterp project list. To see how degeneracy scales with task difficulty and model class, I trained a fully-connected MLP and a CNN (both with ~120k parameters) on nine MNIST variants with different subsets of the labels (just the labels {0, 1}, then {0, 1, 2}, etc.). All models were trained to convergence using the same number of training data points. I implement Lau et al's algorithm on each of the trained models. The results below are averaged over five runs:

The results for the full MNIST dataset are comparable to Lau et al's results while using models ten times smaller trained for ten times fewer epochs.

The sampling method is finicky and sensitive to the hyperparameter choices of learning rate and noise factor. It will fail by producing negative or very high $^λ$ values if the noise $ϵ$ and the distance penalty $γ$ (see lines 6 and 7 in the pseudocode above) isn't calibrated to the model's level of convergence. The results show a linear scaling law relating number of labels to task complexity. The CNN typically has a lower $^λ$ than the MLP, which matches intuitions that some of the complexity is "stored" in the architecture because the convolutions apply a useful prior on functions good at solving image recognition tasks.