Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: The Local Interaction Basis: Identifying Computationally-Relevant and Sparsely Interacting Features in Neural Networks, published by Marius Hobbhahn on May 20, 2024 on The AI Alignment Forum.This is a linkpost for our two recent papers, produced at Apollo Research in collaboration with Kaarel Hanni (Cadenza Labs), Avery Griffin, Joern Stoehler, Magdalena Wache and Cindy Wu:1. An exploration of using degeneracy in the loss landscape for interpretability https://arxiv.org/abs/2405.109272. An empirical test of an interpretability technique based on the loss landscape https://arxiv.org/abs/2405.10928Not to be confused with Apollo's recent Sparse Dictionary Learning paper.A key obstacle to mechanistic interpretability is finding the right representation of neural network internals. Optimally, we would like to derive our features from some high-level principle that holds across different architectures and use cases. At a minimum, we know two things:1. We know that the training loss goes down during training. Thus, the features learned during training must be determined by the loss landscape. We want to use the structure of the loss landscape to identify what the features are and how they are represented.2. We know that models generalize, i.e. that they learn features from the training data that allow them to accurately predict on the test set. Thus, we want our interpretation to explain this generalization behavior.Generalization has been linked to basin broadness in the loss landscape in several ways, most notably including singular learning theory, which introduces the learning coefficient as a measure of basin broadness that doubles as a measure of generalization error that replaces the parameter count in Occam's razor.Inspired by both of these ideas, the first paper explores using the structure of the loss landscape to find the most computationally natural representation of a network. We focus on identifying parts of the network that are not responsible for low loss (i.e. degeneracy), inspired by singular learning theory.These degeneracies are an obstacle for interpretability as they mean there exist parameters which do not affect the input-output behavior in the network (similar to the parameters of a Transformer WV and WO matrices that do not affect the product WOV).We explore 3 different ways neural network parameterisations can be degenerate:1. when activations are linearly dependent2. when gradient vectors are linearly dependent3. when ReLU neurons fire on the same inputs.This investigation leads to the interaction basis, and eventually the local interaction basis (LIB) that we test in the second paper. This basis removes computationally irrelevant features and interactions, and sparsifies the remaining interactions between layers.Finally, we analyse how modularity is connected to degeneracy in the loss landscape. We suggest a preliminary metric for finding the sorts of modules that the neural network prior is biased towards.The second paper tests how useful the LIB is in toy and language models. In this new basis we calculate integrated gradient based interactions between features, and analyse the graph of all features in a network. We interpret strongly-interacting features, and identify modules in this graph using the modularity metric of the first paper.To derive the LIB basis we coordinate-transform the activations of neural networks in two steps: Step 1 is a transformation into the PCA basis, removing activation space directions which don't explain any variance. Step 2 is a transformation of the activations to align the basis with the right singular vectors of the gradient vector dataset.The 2nd step is the key new ingredient which aims to make interactions between adjacent layers sparse, and removes directions which do not affect downstre...