# SGM: A Statistical Godel Machine for Risk-Controlled Recursive Self-Modification

arXiv: https://arxiv.org/abs/2510.10232 · HTML: https://mlsanity.com/paper/2510.10232/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 14 references that are in the map.
- First posted on arXiv 11 Oct 2025.

## Builds on (most cited first, 8 of 8)

- [Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization](https://mlsanity.com/paper/1603.06560/) (2016)
- [BOHB: Robust and Efficient Hyperparameter Optimization at Scale](https://mlsanity.com/paper/1807.01774/) (2018)
- [Population Based Training of Neural Networks](https://mlsanity.com/paper/1711.09846/) (2017)
- [Darwin Godel Machine: Open-Ended Evolution of Self-Improving Agents](https://mlsanity.com/paper/2505.22954/) (2025)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Approval policies for modifications to Machine Learning-Based Software as a Medical Device: A study of bio-creep](https://mlsanity.com/paper/1912.12413/) (2019)
- [Group Sequential Crossover Trial Designs with Strong Control of the Familywise Error Rate](https://mlsanity.com/paper/1710.03492/) (2017)

## Built on by (most cited first, 8 of 8)

- [Recursive self-improvement of AI research agents](https://mlsanity.com/paper/2609.26457/) (2026)
- [Self-Evolving Agents with Anytime-Valid Certificates](https://mlsanity.com/paper/2607.00871/) (2026)
- [SAHOO: Safeguarded Alignment for High-Order Optimization Objectives in Recursive Self-Improvement](https://mlsanity.com/paper/2603.06333/) (2026)
- [A Theory of Reliable Self-Evolution for Agent Harnesses](https://mlsanity.com/paper/2609.08175/) (2026)
- [When Does Pairing Seeds Reduce Variance? Evidence from a Multi-Agent Economic Simulation](https://mlsanity.com/paper/2512.24145/) (2025)
- [Generalized Agent Iteration: One Formal Framework for Iterative Policy Improvement and Recursive Self-Improvement](https://mlsanity.com/paper/2609.13406/) (2026)
- [Which Self-Improvements Should We Trust? Reliable Self-Improvement When Agents Reuse Their Benchmarks](https://mlsanity.com/paper/2609.33180/) (2026)
- [G"odel Forest: Balancing Search Depth and Breadth for Data-Centric Recursive Self-Improvement](https://mlsanity.com/paper/2609.36675/) (2026)

## Abstract (arXiv)

Recursive self-modification is increasingly central in AutoML, neural architecture search, and adaptive optimization, yet no existing framework ensures that such changes are made safely. Godel machines offer a principled safeguard by requiring formal proofs of improvement before rewriting code; however, such proofs are unattainable in stochastic, high-dimensional settings. We introduce the Statistical Godel Machine (SGM), the first statistical safety layer for recursive edits. SGM replaces proof-based requirements with statistical confidence tests (e-values, Hoeffding bounds), admitting a modification only when superiority is certified at a chosen confidence level, while allocating a global error budget to bound cumulative risk across rounds.We also propose Confirm-Triggered Harmonic Spending (CTHS), which indexes spending by confirmation events rather than rounds, concentrating the error budget on promising edits while preserving familywise validity.Experiments across supervised learning, reinforcement learning, and black-box optimization validate this role: SGM certifies genuine gains on CIFAR-100, rejects spurious improvement on ImageNet-100, and demonstrates robustness on RL and optimization benchmarks.Together, these results position SGM as foundational infrastructure for continual, risk-aware self-modification in learning systems.Code is available at: https://github.com/gravitywavelet/sgm-anon.