We present a general method for discovering symbolic local learning rules that can replace backpropagation, applicable, in principle, to any deep learning architecture and task. The method automatically discovers local loss functions specific to each layer by (1) constructing a dictionary of monomials of local tensor variables, inspired by Penrose diagram notation, (2) regressing over coefficients weighting these monomials to build layerwise polynomial losses whose gradients together minimize a target global loss. We test the method on classification, reconstruction, and self-supervised learning tasks with small multilayer networks and minimal datasets. Across all tasks, the discovered rules induce non-trivial interactions between layers, which lead to significant reductions in the corresponding global losses. Furthermore, the method selects sparse combinations of monomials, finding local rules that are more interpretable than those generated by MLP-based alternatives. The potential implications of our method range from energy-efficient deep learning to the discovery of biologically plausible learning mechanisms.

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