✗ Failed on benchmark 2026

Gaussian-compensated Levy neural noise

Implementation & benchmark of arXiv:2607.27657 — A spectral-compensated scheme for space-parameter Poisson noise functionals: error bounds and complexity estimates

Usefulness7/10

Difficulty5/10

Novelty8/10

Sampling Speedup Stability Sample efficiency Diffusion sampling Optimizer Training dynamics Stochastic processes Probability Numerical analysis

Source paper: A spectral-compensated scheme for space-parameter Poisson noise functionals: error bounds and complexity estimates arXiv:2607.27657 ⓘ · analyzed Aug 31, 2026

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 Not peer-reviewed.

Idea description

Replace the unresolved small jumps of an infinite-activity stable Levy noise source in a neural SDE or stochastic optimizer with one Gaussian increment whose variance equals the discarded jump variance. Simulate only jumps above the cutoff exactly or by Poisson sampling, retaining the large-jump distribution while obtaining the paper's O(\varepsilon) Wasserstein error instead of the naive O(\varepsilon^{1-\alpha/2}) error.

Formulas

$$\lambda_\beta(u)=c u^{-1-\alpha},\qquad 0<\alpha<2,$$

$$\sigma_\varepsilon^2=\int_0^\varepsilon u^2\lambda_\beta(u)\,\mathrm du=\frac{c}{2-\alpha}\varepsilon^{2-\alpha},\qquad R_\varepsilon\approx \sigma_\varepsilon Z,\quad Z\sim\mathcal N(0,1),$$

$$W_1(\text{naive},\text{exact})=O\!\left(\varepsilon^{1-\alpha/2}\right),\qquad W_1(\text{Gaussian-compensated},\text{exact})=O(\varepsilon),$$

$$\text{cost}_{\mathrm{naive}}(\tau)=O\!\left(\tau^{-2\alpha/(2-\alpha)}\right),\qquad \text{cost}_{\mathrm{comp}}(\tau)=O(\tau^{-\alpha}).$$

Mathematical statement

For stable-type intensity $\lambda_\beta(u)=c u^{-1-\alpha}$, with amplitude $u>0$, scale $c>0$, and activity index $0<\alpha<2$, the paper analyzes the small-amplitude region $(0,\varepsilon)$. For a centered small-jump integral $R_\varepsilon=\int_{0<u<\varepsilon}u\,\widetilde N(\mathrm dt,\mathrm du)$, where $\widetilde N$ is the compensated Poisson random measure, its variance per unit time is $\sigma_\varepsilon^2=\int_0^\varepsilon u^2\lambda_\beta(u)\,\mathrm du=c\varepsilon^{2-\alpha}/(2-\alpha)$. The compensated approximation replaces $R_\varepsilon$ by $\sigma_\varepsilon Z$, with $Z\sim\mathcal N(0,1)$, while retaining jumps with $u\geq\varepsilon$. The stated theorem gives Wasserstein-1 error $O(\varepsilon)$ for this replacement, versus $O(\varepsilon^{1-\alpha/2})$ when all small jumps are simply discarded; the residual is asymptotically normal at rate $O(\varepsilon^{\alpha/2})$. The associated jump-evaluation costs to reach tolerance $\tau$ are $O(\tau^{-\alpha})$ for compensation and $O(\tau^{-2\alpha/(2-\alpha)})$ for naive truncation. If the neural SDE uses uncompensated positive jumps rather than a centered Levy integral, add the exact small-jump mean $m_\varepsilon=\int_0^\varepsilon u\lambda_\beta(u)\,\mathrm du=c\varepsilon^{1-\alpha}/(1-\alpha)$ for $\alpha<1$, or use the model's prescribed Levy drift convention for $\alpha\geq1$.

Implementation notes

Integrate this at the noise-generation line of a neural SDE solver, stochastic residual block, diffusion sampler with Levy perturbations, or optimizer that intentionally injects stable jump noise. For an Euler step of duration h, let the state be x and network drift be f_theta(x,t). Use the compensated Levy update:

inputs: x, t, h, epsilon, alpha, c
lambda_cut = c * epsilon**(-alpha) / alpha
K ~ Poisson(h * lambda_cut)
for j in 1..K:
    sample U_j from density alpha * epsilon**alpha * u**(-1-alpha), u >= epsilon
large = sum_j U_j
sigma2 = c * epsilon**(2-alpha) / (2-alpha)
small = sqrt(h * sigma2) * Normal(0,1)
x <- x + h*f_theta(x,t) + large + small

For a centered process, compensate the retained large jumps using the model's Levy drift convention; when the relevant first moment is finite, subtract h * integral_epsilon^infinity(u*lambda(u) du). For an uncompensated positive-jump process with alpha<1, add the discarded small-jump mean h*m_small, where m_small=c*epsilon**(1-alpha)/(1-alpha). The variance formula is computed analytically, while the Gaussian and Poisson variables are sampled normally. First run a 2-layer neural SDE or CIFAR-10 stochastic residual classifier with alpha in {0.5,1.0,1.5}. Compare exact-small-jump simulation at a tiny reference cutoff, naive truncation, and compensation at equal wall-clock time and equal large-jump evaluations. Measure terminal-state sliced-W1, validation accuracy, gradient variance, and loss versus runtime. Success is a log-log W1 slope near 1 for compensation, near 1-alpha/2 for naive truncation, and equal terminal-state accuracy with fewer jump evaluations or faster loss descent. Test alpha near 2, where the predicted cost advantage is largest.

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 9/10

Built a reusable Gaussian-compensated Levy Euler update with Poisson-sampled large jumps, matched small-jump Gaussian variance, symmetric/one-sided modes, and the finite-mean correction for alpha<1. The toy sweep confirmed the predicted exponents across alpha=0.5, 1.0, and 1.5: variance 2-alpha, naive scale 1-alpha/2, compensated Wasserstein proxy 1, residual normality proxy alpha/2, and retained-jump rate -alpha. The mechanism is numerically supported, although no full neural-SDE training or direct finite-sample W1 comparison was established.

Agent confidence: 9/10

Baseline: Naive small-jump error scale sigma_epsilon had observed slopes 0.75, 0.50, and 0.25 for alpha=0.5, 1.0, and 1.5, matching predictions.
Idea: The compensated rho/sigma^2 proxy had slope 1.00 for all three alpha values; residual normality slopes were 0.25, 0.50, and 0.75; large-jump evaluation slopes were -0.50, -1.00, and -1.50. Monte Carlo relative variance error was 0.15%-0.29%.

Limitations: The direct empirical W1 test was abandoned because symmetric stable samples have problematic or infinite first moments for alpha<=1 and heavy-tail saturation obscured asymptotic slopes. No neural network optimization, neural-SDE trajectory accuracy, wall-clock benchmark, GPU test, or exact reference simulation at a tiny lower cutoff was performed.

How to run: python3 run_experiment.py && /home/maxwelhelp/main/bin/python3 smoke_test.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.27657) Sep 2, 2026

Ahead of the baseline (−2.5%) but not statistically significant (p=0.1858, wins 6 / 8 paired seeds).

Benchmark: Tabular regression (Friedman)
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 9.3915
Idea mean: 9.1563
Effect (Δ): -0.2352 (−2.5%; negative = idea better)
Wins: 6 / 8 paired seeds
p-value: 0.1858 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±4.1%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 9 configs

Limitations:

Only the registered tabular Friedman#1 optimizer track was tested. No vision, sequence, dynamics, neural-SDE trajectory, wall-clock, jump-evaluation, or exact-reference-cutoff benchmark was run; alpha=1.5, noise scale, dataset size, and 12-epoch training budget were fixed.

How to run: python3 bench_levy.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

Gaussian-compensated Levy optimizer noise was tested against matched naive Levy truncation on the registered tabular/Friedman#1 track using the shared mlp_tiny architecture. Mean test MSE improved from 9.39149 to 9.15630, but the paired permutation p-value was 0.18585, so this is not a significant win. The mechanism signature was not confirmed within the preset tolerance: observed variance 4.0517e-08 versus predicted 4.8990e-08.

Baseline: 8-seed mean MSE 9.3914888501; best configuration lr=0.006, epsilon=0.03.
Idea: 8-seed mean MSE 9.1562968493; selected configuration lr=0.006, epsilon=0.06; paired delta=-0.2351920009; p=0.18585; 6/8 wins.

Benchmark result