Looping a transformer is not free: an iso-depth sweep puts a number on what one recurrence is worth
AspireAI community · arXiv (Schwethelm et al.)
Most transformer scaling work varies parameters and data, which leaves weight-reuse schemes hard to compare. This paper holds training compute roughly fixed and sweeps recurrence counts instead, fitting a joint scaling law that separates one-time parameters from repeated ones. The measured recurrence-equivalence exponent is 0.46, meaning a looped block buys about half the value of a genuinely new unique block at matched compute: at four recurrences, a 410M looped model matches a 580M non-looped one but costs what a 1B model costs to train. The paper also shows the exponent is diagnostic rather than fixed, dropping to 0.38 under truncated backpropagation and rising with hyperconnections, which suggests looped models are often under-trained rather than fundamentally unable to compete. The caveat is that the sweep spans models up to about 1B parameters, so the numbers describe the small-model regime.
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Two sparse-autoencoder analyses have resisted scaling: matching semantically similar features across layers, and compressing large feature circuits into interpretable supernodes. This paper argues both are the same underlying problem, estimating semantic distances between SAE features that sit on different activation manifolds. Its representation treats each feature not as a single decoder vector but as an activation-weighted distribution over the hidden states expressing it, projects those distributions into a shared reference space, and compares them with Wasserstein distance. The representation is proved invariant to activation rescaling and stable under perturbation, and recovers true matches under finite-sample margin conditions. Empirically it outperformed decoder-vector and LLM-based baselines and captured subtle functional distinctions between related features, and it compressed large feature circuits into interpretable supernodes automatically. The limitation is that this is interpretability tooling: the paper demonstrates better feature matching and circuit compression, not that downstream interpretability claims become more accurate.