Implicit Bias and Invariance: How Hopfield Networks Efficiently Learn Graph Orbits

Abstract:Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the corresponding minimum-norm hard-margin memorizer. Our main result shows that, for independent uniform samples from any finite permutation orbit, the exact sample hard-margin support vector machine (HSVM) concentrates exponentially around the invariant full-orbit HSVM. Consequently, an orbit-size-independent polynomial number of samples suffices both for approximate parameter invariance and for simultaneous memorization of every orbit element; directional convergence transfers this conclusion asymptotically to MEF gradient descent. For graph-isomorphism orbits, we characterize the invariant parameters as a three-dimensional subspace and show that every such orbit is memorizable. For cliques of fixed linear density, additional symmetry sharpens the uniform memorization bound to $O(v^4\log(1/\delta))$, exponentially smaller than the orbit size. Together with experiments across several learning rules, these results give a finite-sample account of how optimization bias can recover symmetry from partial group-structured data.

Machine Learning (cs.LG)

arXiv:2512.14338 [cs.LG]

(or arXiv:2512.14338v4 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2512.14338 Focus to learn more arXiv-issued DOI via DataCite

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