Modular addition without black-boxes: Compressing explanations of MLPs that compute numerical integration

• URL: http://arxiv.org/abs/2412.03773v1
• Date: Wed, 04 Dec 2024 23:29:07 GMT
• Title: Modular addition without black-boxes: Compressing explanations of MLPs that compute numerical integration
• Authors: Chun Hei Yip, Rajashree Agrawal, Lawrence Chan, Jason Gross,
• Abstract summary: We present the first case study in rigorously compressing nonlinear feature-maps. We target a non-vacuous bound on the behaviour of the ReLU in time linear in the parameter-count of the circuit.
• Score: 1.7679702431368263
• License:
• Abstract: The goal of mechanistic interpretability is discovering simpler, low-rank algorithms implemented by models. While we can compress activations into features, compressing nonlinear feature-maps -- like MLP layers -- is an open problem. In this work, we present the first case study in rigorously compressing nonlinear feature-maps, which are the leading asymptotic bottleneck to compressing small transformer models. We work in the classic setting of the modular addition models, and target a non-vacuous bound on the behaviour of the ReLU MLP in time linear in the parameter-count of the circuit. To study the ReLU MLP analytically, we use the infinite-width lens, which turns post-activation matrix multiplications into approximate integrals. We discover a novel interpretation of} the MLP layer in one-layer transformers implementing the ``pizza'' algorithm: the MLP can be understood as evaluating a quadrature scheme, where each neuron computes the area of a rectangle under the curve of a trigonometric integral identity. Our code is available at https://tinyurl.com/mod-add-integration.

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