Parameter Regrouping: Discovering Symmetries Within the Data

Many neural architectures gain efficiency by encoding known symmetries as fixed weight-sharing patterns. We study whether a useful sharing pattern can instead be learned from task gradients. We introduce parameter regrouping: connections share a small set of values, a group is split when its connection-level gradients disagree, and groups are merged when both their values and mean gradients are similar. We study the mechanism through Shared Architecture Inference (SHAI), a proof-of-concept model for same-dimensional next-state prediction. Under joint-invariance and full-rank assumptions, each layer's unique local least-squares optimum is equivariant, motivating the layer-local losses used during discovery. Across five dynamical benchmarks, one SHAI configuration obtains lower median test error than an MLP on four tasks while using $7$--$138\times$ fewer parameters, and lower error than a hand-designed convolution on four tasks. Controlled teachers provide evidence of value-faithful recovery when reusable sharing exists, while a dense negative control incurs substantially higher error. Additional experiments show partial recovery of heterogeneous structure, detection of planted coupling, and a ring-like kernel on Lenia. The results are promising but limited to same-dimensional dynamics; the current method over-splits, scales quadratically with state dimension, and recovers translation structure more reliably than more complex sharing patterns.