For language models, we only need to learn the token embeddings. And there they have such problems. I'm not sure if there is any bug in my code, but this is what I found. For trainable embedding tasks, like module addition and permutation symmetry, you need to set it to 64 digits. But for other tasks, like image classification or symbolic regression, there's no such problem because we don't have this representation problem. So maybe we can combine the BIMT penalty loss with some modularity loss people use in graph theory, something like that.

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