# Statistical Gains from Looped Estimation under Parameter Budgets

arXiv: https://arxiv.org/abs/2609.25778 · HTML: https://mlsanity.com/paper/2609.25778/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 34 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Scaling Latent Reasoning via Looped Language Models](https://mlsanity.com/paper/2510.25741/) (2025)
- [SMELT: Scaling Laws for Compute-Matched MoE Looped Transformers](https://mlsanity.com/paper/2609.01343/) (2026)
- [Loop Neural Networks for Parameter Sharing](https://mlsanity.com/paper/2409.14199/) (2024)
- [Approximation Bounds for Recurrent Neural Networks with Application to Regression](https://mlsanity.com/paper/2409.05577/) (2024)
- [Recurrent neural networks approximate continuous functions](https://mlsanity.com/paper/2606.20325/) (2026)

## Abstract (arXiv)

Memory constraints in artificial intelligence motivate accurate function approximation with fewer parameters. We study looping, which repeatedly composes one update function with shared parameters; each output becomes the next input. A looped Transformer, for example, reuses one block, whereas its conventional untied counterpart uses separately parameterized blocks. We compare their parameter requirements for a given worst-case approximation accuracy, or equivalently, their approximation accuracy under a common budget limiting distinct trainable coefficients. We then ask whether this representational parsimony improves statistical accuracy. For general likelihood models, we establish an upper squared Hellinger risk bound for looped sieve maximum likelihood and a minimax lower bound for the jointly tuned untied family. Further loop iterations improve the approximation bound without adding parameters, while increasing computation and the fitted-class complexity bound. For targets of known H"older smoothness, looped residual feedforward networks and post-layer-normalized Transformers attain the minimax polynomial rate up to logarithmic factors with a fixed number of bounded real parameters. At sufficiently large fixed budgets, looped worst-case risk vanishes while optimal worst-case untied risk remains bounded away from zero. The loop-to-untied risk ratio also tends to zero under specified growing-budget conditions. Regression, binary response, and energy-based generative models illustrate the theory.