✗ Failed on benchmark 2026

Entry-Exit Curvature Scheduler

Implementation & benchmark of arXiv:2608.04594 — Traveling fronts in a spatial epidemic model with slow loss of immunity

Usefulness7/10

Difficulty5/10

Novelty7/10

Dynamics Stability Speedup Optimizer Scheduler Training dynamics Dynamical systems Control theory Bifurcations Numerical analysis

Source paper: Traveling fronts in a spatial epidemic model with slow loss of immunity arXiv:2608.04594 ⓘ · analyzed Aug 31, 2026

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 Not peer-reviewed.

Idea description

Replace pointwise curvature-based learning-rate decisions with a slow-fast entry-exit scheduler. The optimizer maintains a slowly varying state representing effective curvature or gradient-noise level, accumulates the weak transverse growth rate along that slow trajectory, and changes learning regime only when the accumulated rate returns to zero. This permits controlled passage through locally unstable or poorly conditioned regions while preventing indefinite residence in a regime with net positive perturbation growth.

Formulas

$$\lambda_-(S)=c-\sqrt{c^2-4D\mathcal{R}_0S+4D}.$$

$$S(\zeta)=1+(S_\infty-1)e^{-\zeta/c},\qquad \lambda_-(\zeta)=c-\sqrt{a-Be^{-\zeta/c}},$$

$$\int_0^{T_E}\lambda_-(\zeta)\,d\zeta=0,\qquad a=c^2-4D(\mathcal{R}_0-1),\quad B=4D\mathcal{R}_0(S_\infty-1)<0.$$

$$G_{t+1}=G_t+\hat\lambda_t\Delta t,\qquad \eta_t=\begin{cases}\eta_{\mathrm{explore}},&G_t<0\ \text{and no exit has occurred},\\\eta_{\mathrm{exploit}},&G_t\geq 0\ \text{after the first crossing}.\end{cases}$$

Mathematical statement

In the paper's traveling-wave reduction, the weak eigenvalue transverse to the slow manifold is $\lambda_-(S)=c-\sqrt{c^2-4D\mathcal{R}_0S+4D}$, where $c$ is front speed, $D$ is infected diffusion, $\mathcal{R}_0$ is the reproduction parameter, and $S$ is the slow variable. Along the slow flow, $S(\zeta)=1+(S_\infty-1)e^{-\zeta/c}$. Defining $a=c^2-4D(\mathcal{R}_0-1)$ and $B=4D\mathcal{R}_0(S_\infty-1)<0$, the eigenvalue becomes $\lambda_-(\zeta)=c-\sqrt{a-Be^{-\zeta/c}}$. The entry-exit time $T_E$ is the nontrivial positive solution of $\int_0^{T_E}\lambda_-(\zeta)d\zeta=0$. The neural analogue uses a slow state $s_t$, an estimated transverse growth rate $\hat\lambda_t$, and an accumulated budget $G_{t+1}=G_t+\hat\lambda_t\Delta t$. The optimizer exits its exploratory regime when $G_t$ crosses zero from below after previously becoming negative. The transfer assumes that local optimizer perturbations admit a fast linear approximation whose dominant transverse exponent varies slowly through $s_t$.

Implementation notes

Integrate this as a wrapper around SGD or AdamW. Maintain a scalar slow state $s_t$, an accumulated budget $G_t$, and two learning rates $\eta_{\mathrm{explore}}>\eta_{\mathrm{exploit}}$. Every $K$ optimizer steps, estimate a dominant curvature direction with one or two Hessian-vector products. Draw a normalized vector $v_t$, compute $H_tv_t$ by automatic differentiation, set $q_t=v_t^TH_tv_t$, update $v_t\leftarrow H_tv_t/\|H_tv_t\|$, and smooth curvature using $s_{t+1}=(1-\epsilon)s_t+\epsilon q_t$, with small $\epsilon$. Define $\hat\lambda_t=\alpha(s_t-\mu)$, where $\mu$ is a target stability curvature and $\alpha$ is calibrated from short probe trajectories. Update $G\leftarrow G+K\hat\lambda_t$. Use the exploratory learning rate while $G<0$; after the first crossing from negative to nonnegative, permanently switch to the exploitative rate. The paper contributes the integral-crossing rule; curvature trajectories, calibration constants, and Hessian directions are estimated empirically. First test a two-layer MLP on MNIST and a small ResNet on CIFAR-10 against SGD, AdamW, cosine decay, and a pointwise curvature-sign scheduler with matched average learning-rate budget. Log $s_t$, $\hat\lambda_t$, $G_t$, loss, gradient norms, and nearby-trajectory separation. The prediction is that switching occurs at the first discrete root of $\sum_{j=1}^{m}K\hat\lambda_j=0$, after the first positive local $\hat\lambda_t$ whenever the later integrated rate is negative. Predicted switch times should match observed perturbation-contraction onset within 20 percent; pointwise sign switching should occur earlier.

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 8/10

Built an entry-exit curvature scheduler with slow curvature state, accumulated growth budget, permanent crossing-triggered regime switching, and a pointwise-sign comparator. The paper reduction confirmed three mechanism predictions: integral exit occurs after the pointwise zero (T_E/lambda-zero = 2.01–2.28), exit time decreases monotonically with S_inf, and T_E scales approximately linearly with c (T_E/c relative spread 1.6%); root residuals were below 5e-15. In the toy optimizer, entry-exit switched at step 145 versus pointwise step 50 and achieved final loss 7.86e-5 versus 1.47e-4 for pointwise, but it did not beat constant high-rate SGD (1.79e-9), so the demonstrated gain is specifically delayed integrated switching rather than general optimization superiority.

Agent confidence: 8/10

Baseline: Pointwise curvature-sign scheduler: final loss 1.4718e-4, switch step 50; constant SGD eta=0.025: 2.3186e-2; constant high eta=0.12: 1.7895e-9
Idea: Entry-exit scheduler: final loss 7.8579e-5, switch step 145, mean eta 0.0582, budget minimum -6.2091 and final budget 2.7894; math checks: maximum root residual 4.99e-15, T_E/lambda-zero 2.01–2.28, c-scaling spread 1.58%

Limitations: Only a deterministic nonstationary 2D quadratic optimizer toy was tested; no MNIST, CIFAR-10, AdamW, neural-network HVP implementation, FLOP matching, or nearby-trajectory separation experiment was run. The curvature signal in the optimizer demo is supplied analytically rather than estimated from automatic-differentiation HVPs.

How to run: python3 entry_exit_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.04594) Sep 2, 2026

Ahead of the baseline (−8.7%) but not statistically significant (p=0.7926, wins 4 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0035
Idea mean: 0
Effect (Δ): -0.0003 (−8.7%; negative = idea better)
Wins: 4 / 8 paired seeds
p-value: 0.7926 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±65.6%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

Only the registered dynamics track was tested; tabular, vision, and sequence tracks, longer training, larger models, and broader scheduler calibration sweeps were not evaluated.

How to run: python3 bench_entry_exit.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: 10/10

Implemented and evaluated the Entry-Exit Curvature Scheduler on the registered dynamics/rnn_small track using HVP curvature probes, slow EMA state, accumulated growth budget, and permanent zero-crossing switching. The best idea configuration had mean test MSE 0.00322614 versus the tuned Adam baseline mean 0.00353509, but paired delta was -0.00030895 with permutation p=0.7926, so the required significant-win criterion was not met. The budget did not cross zero in any paired run, and the mechanism signature was therefore not confirmed.

Baseline: Adam, dynamics/rnn_small, tuned lr=0.006; mean test MSE 0.0035350887 over 8 seeds
Idea: Entry-Exit Adam, lr=0.006; mean test MSE 0.0032261411 over 8 seeds; paired delta -0.0003089476; 4/8 wins; permutation p=0.7926

Benchmark result