arXiv:2408.05451v1 Announce Type: new
Abstract: Superposition — when a neural network represents more “features” than it has dimensions — seems to pose a serious challenge to mechanistically interpreting current AI systems. Existing theory work studies emph{representational} superposition, where superposition is only used when passing information through bottlenecks. In this work, we present mathematical models of emph{computation} in superposition, where superposition is actively helpful for efficiently accomplishing the task.
We first construct a task of efficiently emulating a circuit that takes the AND of the $binom{m}{2}$ pairs of each of $m$ features. We construct a 1-layer MLP that uses superposition to perform this task up to $varepsilon$-error, where the network only requires $tilde{O}(m^{frac{2}{3}})$ neurons, even when the input features are emph{themselves in superposition}. We generalize this construction to arbitrary sparse boolean circuits of low depth, and then construct “error correction” layers that allow deep fully-connected networks of width $d$ to emulate circuits of width $tilde{O}(d^{1.5})$ and emph{any} polynomial depth. We conclude by providing some potential applications of our work for interpreting neural networks that implement computation in superposition.
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