✗ Failed on benchmark 2026

Relevant-Noise RG Curriculum

Implementation & benchmark of arXiv:2608.02716 — Universal crossovers in weakly-monitored quantum critical states

Usefulness7/10

Difficulty4/10

Novelty6/10

Dynamics Stability Generalization Sample efficiency Training dynamics Regularization Transformer Rnn Renormalization Dynamical systems Probability Scaling

Source paper: Universal crossovers in weakly-monitored quantum critical states arXiv:2608.02716 ⓘ · analyzed Aug 31, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Inject weak, unpostselected stochastic perturbations into activations, attention links, or recurrent transitions, but scale their strength according to effective computational size. The schedule is designed so that noise is initially a weak perturbation and becomes dominant only beyond a controlled depth or sequence length, producing a measurable crossover rather than uncalibrated constant dropout.

Formulas

$$g' = b^{y_g} g, \qquad y_g > 0$$

$$x(N,g) = g N^{y_g}, \qquad N_* \asymp g^{-1/y_g}$$

$$Q(N,g) = N^{-\kappa} F\!\left(g N^{y_g}\right)$$

$$h_{\ell+1} = \phi\!\left(W_\ell h_\ell + b_\ell + \sqrt{g_\ell}\,\xi_\ell\odot s_\ell\right), \qquad g_\ell = g_0\left(\frac{\ell}{\ell_0}\right)^{y_g}$$

Mathematical statement

Let g >= 0 be the stochastic perturbation strength, b > 1 a coarse-graining factor, and y_g > 0 its relevant RG eigenvalue. Under coarse graining, g' = b^(y_g) g. For effective size N, the crossover variable is x = g N^(y_g), and the crossover scale is N_* approximately equal to g^(-1/y_g). For N much smaller than N_*, the network follows approximately clean dynamics; for N much larger, stochasticity controls long-range behavior. A scale observable Q(N,g) should obey Q(N,g) = N^(-kappa) F(g N^(y_g)), where kappa is the clean-system scaling exponent and F is a crossover function. In a network, N may be layer depth, recurrent horizon, token distance, or optimizer time. The exponent y_g is estimated from data collapse across several noise strengths.

Implementation notes

1. Integration point: use a residual MLP, transformer, or RNN and inject zero-mean Gaussian noise after normalization and before the residual addition. Let the noise variance at layer or recurrent step l be g_l. For a transformer, apply the perturbation to the value stream or residual stream; for an RNN, apply it to the hidden-state transition. Use ordinary unpostselected training: every noisy forward pass contributes to the loss. 2. Pseudocode: choose g_0, a trial exponent y_g, a reference depth l_0, and a maximum variance g_max. At each layer, compute g_l = min(g_max, g_0 (l/l_0)^(y_g)), sample xi from a standard normal distribution, add sqrt(g_l) xi elementwise to the chosen activation, and backpropagate the averaged loss over one or two noise samples. 3. The RG scaling law and crossover form come from the paper; y_g, kappa, and F are estimated empirically. 4. First experiment: train 2-, 4-, and 8-layer residual MLPs on MNIST or CIFAR-10, comparing clean training, constant Gaussian noise, dropout, and the RG schedule. Measure activation correlations, gradient variance, and validation loss as functions of depth and g_0. The quantitative prediction is that curves for different g_0 collapse when plotted against x = g_0 N^(y_g), and that the largest change occurs near N_* = g_0^(-1/y_g). Fit y_g on half the settings and predict held-out crossover locations; agreement within 20 percent is required.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Inconclusive

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 7/10

Built a runnable Relevant-Noise RG experiment with a toy crossover verification and matched residual-MLP comparison. The toy checks confirmed RG rescaling to numerical precision, recovered y=0.685 versus 0.700, and measured crossover scales within 14.1% of N*=g^(-1/y), satisfying the 20% criterion; data-collapse scatter was strongly reduced. In the MLP, the schedule improved validation loss over clean, constant-noise, and dropout at depth 4 (0.0731 versus 0.0797, 0.1215, and 0.1038), but gains were not consistent at depths 2 and 8, so the practical training win is preliminary rather than universal.

Agent confidence: 7/10

Baseline: Toy: y_true=0.700; crossover median measured/predicted=1.035; maximum relative crossover error=14.1%; MLP validation loss by depth: clean d2/d4/d8 = 0.0884/0.0797/0.0990; constant Gaussian = 0.0739/0.1215/0.1105; dropout = 0.0796/0.1038/0.0891.
Idea: Toy: fitted y=0.685, RG rescaling max absolute error=8.9e-16, collapse score=0.000975 versus 0.0285 at a poor exponent; MLP RG-scheduled validation loss d2/d4/d8 = 0.0755/0.0731/0.1002, with the strongest relative benefit at d4.

Limitations: The toy observable is an explicitly constructed stochastic crossover model, so it validates the proposed scaling mechanism rather than deriving it from a physical network. The neural experiment used a small synthetic binary dataset, one seed, one noise hyperparameter, three depths, and 180 optimization steps; MNIST/CIFAR, broader hyperparameter sweeps, gradient-variance curves, multiple seeds, and a true held-out crossover fit were not tested.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.02716) Aug 31, 2026

Ahead of the baseline (−2.0%) but not statistically significant (p=0.543, wins 4 / 8 paired seeds).

Benchmark: Sequence modeling
Model: transformer_tiny
Paired seeds: 8
Baseline mean: 0.5145
Idea mean: 0.5044
Effect (Δ): -0.0101 (−2.0%; negative = idea better)
Wins: 4 / 8 paired seeds
p-value: 0.543 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±7.0%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

Only the built-in sequence track, two transformer layers, 10 epochs, and the tested g0/y settings were evaluated; no vision, dynamics, longer horizons, larger models, or broader exponent fitting was tested.

How to run: python3 bench_rg_runner.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 9/10

Implemented Relevant-Noise RG activation perturbations in the structurally matched sequence track using a local transformer training loop. The RG identity check passed to numerical precision, and the trained-model signature confirmed predicted layer noise standard deviations within 0.23% relative error. The best idea setting reduced mean test MSE from 0.514515 to 0.504396, but the paired permutation test was nonsignificant (delta=-0.010118, p=0.543), so this round does not establish a real training win.

Baseline: sequence/transformer_tiny, clean, best lr=0.001: 8-seed MSE mean 0.514515, std 0.054153; baseline sweep means lr 0.001/0.003/0.006 = 0.514816/0.632799/0.892328.
Idea: Relevant-Noise RG, y=0.7, best g0=0.004, lr=0.001: 8-seed MSE mean 0.504396, std 0.052841; paired delta=-0.010118; 4/8 wins; permutation p=0.543; verdict no significant win.

Benchmark result