Looped Transformers increase effective depth by re-executing shared layers, but three limitations hinder their adoption: memory grows with the loop count, compute can only be adjusted in whole-block steps, and the loop count is hard to change after training. We propose LooM (Loop-invariant Memory), which addresses each in turn.

In a causal encoder-decoder (CED), LooM loops only the decoder layers after the first, which read a global memory written once per token, and each of them publishes the key-value (KV) entry for a token only on its first visit, so the cache does not grow with the loop count. These layers loop independently, providing every budget from 3 to 21 layer executions per token, and one model trained on randomly sampled budgets covers all of them without a router.

With enough loops, LooM outperforms a single-pass baseline trained on the same data on language modeling, zero-shot accuracy and long-context perplexity, while using the same cache size. Our analysis further shows that the training budget distribution decides what extra depth buys.

These results suggest that a CED is a natural place to loop: memory is written once, and depth becomes a budget chosen at deployment.