✗ Failed on benchmark 2026

Steady-State First-Passage Sensitivity Regularizer

Implementation & benchmark of arXiv:2608.11202 — Exact First-Passage Time Response Theory from Steady-State Response

Usefulness7/10

Difficulty6/10

Novelty8/10

Dynamics Stability Accuracy Speedup Rnn Ssm Diffusion sampling Training dynamics Stochastic processes Probability Statistics Dynamical systems Control theory

Source paper: Exact First-Passage Time Response Theory from Steady-State Response arXiv:2608.11202 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Treat a neural hidden-state process as a finite or discretized continuous-time Markov chain and define a target event as first entry into a target state set. Instead of estimating the derivative of the mean hitting time by expensive long rollouts, build an auxiliary regenerative chain that resets to the source state after reaching the target and estimate the same response from its stationary distribution. Penalize disagreement between this response prediction and short empirical perturbation measurements, or directly optimize the perturbation direction that decreases target hitting time.

Formulas

$$T_{ij}(\theta)=\mathbb{E}_i[\tau_j]=\int_0^\infty \Pr_i(\tau_j>t)\,dt,$$

$$\widetilde{Q}^{ij}_{ab}(\theta)=Q_{ab}(\theta)\quad(a\neq j),\qquad \widetilde{Q}^{ij}_{ia}(\theta)\leftarrow \widetilde{Q}^{ij}_{ia}(\theta)+Q_{ja}(\theta),$$

$$\widetilde{\pi}^{ij}(\theta)\widetilde{Q}^{ij}(\theta)=0,\qquad \sum_a\widetilde{\pi}^{ij}_a(\theta)=1,$$

$$R_{ij}=\left.\frac{\partial T_{ij}(h)}{\partial h}\right|_{h=0},\qquad \widehat{R}_{ij}^{\mathrm{FD}}=\frac{\widehat{T}_{ij}(+h)-\widehat{T}_{ij}(-h)}{2h},\qquad \mathcal{L}_{\mathrm{FPT}}=\left|\widehat{R}_{ij}^{\mathrm{aux}}-\widehat{R}_{ij}^{\mathrm{FD}}\right|^2$$

Mathematical statement

Let $X_t$ be a continuous-time Markov chain with generator $Q(\theta)$, where $\theta$ is a neural-network parameter or controllable perturbation, and let $T_{ij}(\theta)=\mathbb{E}_i[\tau_j]$ with $\tau_j=\inf\{t\geq0:X_t=j\}$. Construct an auxiliary regenerative chain by retaining the original transitions until the process reaches target $j$ and then resetting it to source $i$; let $\widetilde{\pi}^{ij}(\theta)$ be its stationary distribution and $\widetilde{Q}^{ij}(\theta)$ its generator. The renewal-reward identity makes $T_{ij}$ the mean regenerative cycle duration, while the paper's exact correspondence expresses derivatives of $T_{ij}$ with respect to $\theta$ through steady-state response quantities of the auxiliary chain. For a scalar perturbation $h$, use the linear response $R_{ij}=\left.\partial_hT_{ij}\right|_{h=0}$ and estimate it from the stationary derivative of the auxiliary chain instead of long first-passage trajectories. The construction assumes finite $T_{ij}$, an ergodic auxiliary chain, and differentiable transition rates.

Implementation notes

1. Integration point: use a recurrent neural network or neural state-space model with hidden state $z_t$. Choose a source region $A$ and target region $B$, such as the initial hidden-state cluster and the set of states classified as a correct terminal prediction. Discretize hidden states into $K$ clusters, or define a differentiable local transition generator $Q_\phi(\theta)$ with nonnegative off-diagonal rates generated by softplus functions. Let $h$ perturb input noise, temperature, a control gain, or a selected parameter direction. 2. Pseudocode: collect hidden-state trajectories; estimate $Q_{ab}=N_{ab}/D_a$, where $N_{ab}$ is the number of transitions from state $a$ to $b$ and $D_a$ is total dwell time in $a$; copy $Q$; redirect every transition entering target $B$ to source $A$ to form $\widetilde Q^{AB}$; solve $\widetilde\pi\widetilde Q=0$ with normalization by an augmented linear system; repeat for $+h$ and $-h$; differentiate or finite-difference the stationary quantity; add $\lambda\mathcal L_{\mathrm{FPT}}$ to the network loss. 3. Computed from the paper's mechanism: the regenerative auxiliary construction and the equality between first-passage response and auxiliary steady-state response. Estimated empirically: transition rates, state clustering, stationary distributions, and finite-sample derivatives. 4. First cheap experiment: train a small GRU on sequential MNIST or a two-state switching task, defining the target as correct classification for three consecutive steps. Compare vanilla training, direct first-passage-loss training, and this auxiliary-response regularizer. Measure MFPT for perturbations $h\in[-0.1,0.1]$ using 10,000 rollouts and compare centered finite differences with the auxiliary estimate. The quantitative prediction is linear-response agreement as $h\to0$, with relative error below 20% after each auxiliary state has at least 100 observed transitions. The MFPT curve should have slope $R_{ij}$. A failure boundary is also predicted: when the auxiliary chain loses ergodicity or $T_{ij}$ becomes very large, the stationary solve becomes ill-conditioned and the response variance should increase sharply.

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 and numerically verified the regenerative fast-reset CTMC method in experiment.py. The exact paper response formula matched centered finite differences to 1.3e-10; auxiliary stationary-response error decreased from 11.8% at reset rate 2 to 0.031% at rate 1000, with observed log-log slope -0.996 versus the predicted -1 scaling. The nonlinear rational-response formula had maximum absolute error 0.050 across six finite perturbations, and the predicted symmetry/sign behavior was confirmed; the rollout baseline was much noisier.

Agent confidence: 9/10

Baseline: Gillespie finite-difference MFPT response: -0.789, standard error 0.335 using 2,500 rollouts per perturbation
Idea: Auxiliary stationary response at reset rate 1000: -0.63043 versus exact derivative -0.63063, absolute error 0.000196; exact formula error 1.27e-10
Predictions: Reset convergence predicted O(1/K): observed log-log slope -0.996; finite-perturbation formula maximum absolute error 0.0495; symmetry derivative 0.0; helpful 1->2 derivative -0.631 and harmful 2->1 derivative +1.150

Limitations: This is an exact four-state CTMC toy verification, not a neural RNN training experiment. It does not test learned state clustering, auxiliary regularizer optimization, GPU behavior, very large MFPT or ill-conditioned regimes, or the claimed 100-transition empirical threshold.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.11202) Sep 1, 2026

No significant difference vs. the tuned baseline (+7.7%, p=0.8951, wins 4 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0449
Idea mean: 0.0484
Effect (Δ): 0.0035 (+7.7%; negative = idea better)
Wins: 4 / 8 paired seeds
p-value: 0.8951 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±120.8%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

The complete registered-track run used a reduced equal budget of 4 epochs and 200 training/200 test examples after the initial full-size implementation exceeded the shared 40-minute limit. The intervention is a differentiable soft first-passage surrogate over the eight observed prefixes rather than a learned clustered continuous-time Markov generator with an explicitly solved stationary distribution; long rollouts, large-scale training, alternative target definitions, and ill-conditioned ergodic regimes were not tested.

How to run: python3 bench_fpt.py

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

Stage 2 — Benchmark Sep 1, 2026 latest Failed ✗

Agent confidence: 9/10

The steady-state first-passage sensitivity regularizer was implemented and evaluated on the registered dynamics/rnn_small benchmark with matched architecture and paired seeds. The best idea had higher test MSE than the tuned baseline (0.04840 vs 0.04493), with paired delta +0.00347 and permutation p=0.89505; therefore the idea did not win.

Baseline: mean test MSE 0.0449314, std 0.0383548, best lr=0.005
Idea: mean test MSE 0.0484015, std 0.0418776, best lr=0.005; paired delta +0.0034701, p=0.89505

Benchmark result