Task-dependent clustering in permutation-aligned neural network weights

Across biology, physics and computation, complex systems can occupy reproducible regions of state space despite different microscopic histories. Attractor concepts provide one framework for understanding how diverse trajectories may become confined to a limited set of outcomes. Whether stochastic neural-network training similarly organizes models into reproducible, task-dependent regions is difficult to test because functionally equivalent networks can differ through hidden-unit permutations and rectified linear unit (ReLU) rescalings.

Here, we followed independently initialized shallow networks trained on several image-classification problems and aligned their hidden units before comparing their weights.

Networks trained on the same dataset developed more similar weight structure than networks trained on different datasets. During the first few stochastic gradient descent (SGD) updates, accuracy improved sharply while displacement from initialization remained small. These small movements were not structurally neutral: dataset-dependent separation emerged almost immediately in both aligned weights and hidden representations. Changing the prediction rule while holding the input images fixed produced a corresponding task-dependent separation, showing that the organization reflected the learning objective rather than the image distribution alone. The principal finding persisted after removing ReLU scaling freedom. Git Re-Basin yielded stronger neuronal correspondence than our independently developed Hungarian procedure, although the latter still recovered functionally meaningful alignments.

Together, the results show that training can rapidly organize small parameter changes into dataset- and task-dependent clusters. The effectiveness of small early updates also motivates a target-rich landscape hypothesis in which useful solution regions are locally accessible despite non-unique parameterizations.