Statistical Gödel Machine (SGM)

• Statistical Gödel Machine is a framework for risk-controlled recursive self-modification using rigorous statistical confidence tests.
• It employs techniques like harmonic allocation, Hoeffding and Empirical-Bernstein bounds to certify performance improvements.
• SGM ensures familywise error control, efficiently preventing spurious or harmful modifications in noisy, high-dimensional systems.

The Statistical Gödel Machine (SGM) is a framework for principled risk-controlled recursive self-modification in noisy and high-dimensional learning systems. In contrast to classical Gödel machines, which require formal proofs of improvement for a self-rewrite, SGM employs rigorous statistical confidence tests to certify that any code edit or hyperparameter change yields superior performance at a pre-specified global confidence level. By allocating a familywise error budget across an indefinite sequence of edits, SGM ensures that the cumulative probability of accepting any harmful modification does not exceed a set global tolerance.

1. Formal Specification and Workflow

At each outer round $t$, the SGM proceeds as follows:

1. Proposal Generation: A Proposer module $\Pi$ receives as input the transcript $\mathcal{T}_{t-1}$ of all previous proposals, outcomes, and decisions, as well as the current system $\theta_{t-1}$, and outputs a candidate modification $\theta_t'$.
2. Paired Evaluation: An evaluation harness $\mathcal{H}$ executes paired experiments comparing the performance of $\theta_t'$ against incumbent $\theta_{t-1}$, returning a set of differences $\{\Delta_{t,i}\}_{i=1}^{n_t}$.
3. Statistical Certification (Gate): The SGM Gate $\mathcal{G}$ applies a chosen statistical test at per-round level $\delta_t$ to produce a certificate $C_t(\delta_t)$ and a binary decision $D_t\in\{\mathsf{accept},\,\mathsf{reject}\}$.
4. Adoption: Only if the test certifies superiority (e.g., a lower confidence bound on $\mathbb{E}\Delta_{t,i}$ exceeds zero), the edit is adopted, with the new $\theta_t'$, and the registry $\mathcal{R}$ updated.

The SGM mandates two critical statistical guarantees:

• Per-edit safety: $\Pr(\text{harmful accept at }t)\le\delta_t$.
• Cumulative (familywise) error control: For any round horizon $T'$, the probability of one or more harmful edits is $\le \sum_{t=1}^{T'}\delta_t\le\delta$.

2. Statistical Safety Mechanisms and Error Allocation

SGM introduces several error budget allocation strategies and acceptance mechanisms tailored to recursive self-modification:

• Harmonic Allocation: A global familywise tolerance $\delta$ is split across rounds via a harmonic schedule:

$$\delta_t = \frac{\delta}{t H_T},\quad H_T = \sum_{i=1}^{T}\frac{1}{i}$$

This ensures $\sum_{t=1}^{T}\delta_t = \delta$.

• Hoeffding-Based Acceptance: Under bounded paired differences $\Delta_i \in [a, b]$ and sample mean $\hat\mu$, Hoeffding’s bound at level $\delta_t$ is:

$$\mathrm{LCB}_{1-\delta_t} = \hat\mu - (b-a)\sqrt{\frac{1}{2n}\ln\frac{1}{\delta_t}}$$

The proposal is accepted iff $\mathrm{LCB}_{1-\delta_t}>0$.

• Empirical-Bernstein (Variance-Adaptive):

$$\hat\mu - \sqrt{\frac{2\hat\sigma^2\ln(3/\delta)}{n}} - \frac{3(b-a)\ln(3/\delta)}{n}$$

is used as a tighter one-sided bound for acceptance.

• Anytime Testing with $e$-values: For normalized differences $X_{t,i}$ and per-sample $e$-values $e_{t,i}=1+\lambda_{t,i}X_{t,i}$, the wealth process $W_t=\prod_{s\le t}\prod_{i}e_{s,i}$ is a nonnegative supermartingale. Acceptance occurs upon $W_t\ge 1/\delta$.

These mechanisms enable SGM to achieve both per-edit and familywise risk control in the presence of stochasticity and indefinite iteration.

3. Confirm-Triggered Harmonic Spending (CTHS)

SGM introduces Confirm-Triggered Harmonic Spending (CTHS) as an event-based approach to error spending. Whereas standard harmonic allocation expends the risk budget every round regardless of outcome, CTHS only spends at confirmation events—i.e., at rounds where sufficiently promising proposals escalate for further verification. For $K$ total confirmed proposals and $H_K=\sum_{j=1}^K 1/j$, the spend at the $k$-th confirmation is:

$\delta_{(k)} = \frac{\delta}{k\,H_K}$

This design concentrates the risk budget on rounds with high decision-criticality, increasing overall sample efficiency while preserving familywise error rate (FWER) control, as $\sum_{k=1}^K \delta_{(k)}=\delta$.

4. Theoretical Guarantees

The SGM provides rigorous statistical safety guarantees:

• Union-bound risk control: If each modification is certified at level $\delta_t$, then the global probability across any horizon $T$ of accepting a harmful (non-improving) modification is $\leq \sum_{t=1}^T\delta_t\leq \delta$.
• Anytime validity: For $e$-value based tests with bounded and predictable stakes, Ville’s inequality guarantees that $W_t$ exceeding $1/\delta$ ever (over all time) occurs with probability at most $\delta$, preserving the safety property.
• Event-based spending correctness: CTHS achieves familywise validity, as the total cumulative risk remains exactly $\delta$ across all confirmed tests.

These guarantees hold under the condition of bounded, independent paired differences across rounds and any acceptance rule detailed above.

5. Comparison with Classical Gödel Machines

Classical Gödel machines, as introduced by Schmidhuber, require formal logical proofs that a code rewrite increases expected utility. In stochastic, high-dimensional contexts common to modern ML—such as deep neural networks or reinforcement learning—such proofs are intractable or impossible to obtain. SGM relaxes this requirement, leveraging PAC-style statistical confidence intervals or anytime-valid $e$-value tests as practical surrogates for certification. By structuring risk allocation via global error budgets (e.g., harmonic or CTHS), SGM maintains a conceptual safety bar equivalent to never admitting a non-improving rewrite with more than global probability $\delta$, thus providing a first statistically sound and tractable safety layer for recursive self-modification (Wu et al., 11 Oct 2025).

6. Empirical Evaluation

Extensive experiments validate the safety and selectivity properties of SGM:

Domain	Protocol/Notable Results	Risk Budget Behavior
CIFAR-100	Synthetic injection: +4.0 pp gain confirmed at first event ($k=1$) by CTHS (spend 0.0748 < 0.1); in 10 real runs, only 1 proposal certified (+5.51 pp)	Global $\delta=0.1$ not exceeded
ImageNet-100	No proposals achieved certified gain; e.g., mixup=0.1 proposal rejected ($\bar\Delta=-4.03$ pp, LCB$=-1.91$)	All spurious improvements filtered
RL: CartPole-v1	8 proposals, all rejected (mean improvement $-45.7$, LCB in $[-0.95, -0.79]$)	Cumulative risk $<$ global threshold
RL: LunarLander-v2	One warm-start proposal certified (+513.2, LCB=+0.04, $n=19$), all others rejected	Selectivity conserves error budget
Rastrigin20 (CMA-ES)	Only micro-gains with LCB crossing zero certified (e.g., LCB=+0.009), majority rejected	Prevents acceptance of spurious edits

On CIFAR-100, SGM certified only the genuine improvement (weight_decay=0.001, ema_decay=0.99; incumbent 56.05% to 61.56% accuracy) with an LCB of +0.31 under 30 seeds; all other proposals failed confirmation. On ImageNet-100, SGM rejected all proposals, including those with high screening gains but negative confirmation LCBs. In RL, SGM rejected all CartPole proposals and certified only a single LunarLander modification with robust statistical evidence. For black-box optimization, only micro-gains above the LCB acceptance threshold were admitted.

These results confirm that SGM reliably filters out spurious or noisy proposals, while certifying genuine gains, and that CTHS enables efficient error budget concentration while maintaining rigorous familywise risk control.

7. Significance and Prospective Applications

SGM extends the conceptual safety of classical Gödel machines into practical, high-dimensional, and stochastic machine learning contexts. Applicable to AutoML, neural architecture search, reinforcement learning, and continual adaptive systems, it provides a statistically principled gate for recursive code and hyperparameter edits, ensuring with quantifiable global risk bounds that harmful self-modifications are virtually never accepted. Confirm-Triggered Harmonic Spending further improves sample efficiency by allocating risk only to events necessitating confirmation. SGM thus constitutes foundational infrastructure for risk-aware continual self-modification in modern learning pipelines (Wu et al., 11 Oct 2025).