✗ Failed on benchmark 2026

Age-conditioned semi-Markov router

Implementation & benchmark of arXiv:2607.05173 — Semi-Markovian switching in a fluctuating harmonic trap: An age-structured formulation

Usefulness6/10

Difficulty5/10

Novelty7/10

Architecture Stability Accuracy Sample efficiency Moe routing Ssm Rnn Probability Stochastic processes Dynamical systems

Source paper: Semi-Markovian switching in a fluctuating harmonic trap: An age-structured formulation arXiv:2607.05173 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Augment a neural router with the age of its current expert or latent regime and use an age-dependent hazard to determine when switching is likely. Unlike ordinary token-wise softmax routing, the router can learn non-geometric residence times, suppressing unstable expert oscillations while still allowing rapid transitions when the current regime becomes inappropriate.

Formulas

$$S_i(a)=\int_a^{\infty} r_i(a')\,da',\qquad R_i(a)=\frac{r_i(a)}{S_i(a)}$$

$$S_i(n)=\prod_{m=0}^{n-1}\bigl(1-h_i(m)\bigr),\qquad p_i(n)=h_i(n)S_i(n)$$

$$h_i(a,z)=\sigma\!\left(w_i^{\top}z+u_i^{\top}e(a)+b_i\right),\qquad a_{t+1}=\begin{cases}a_t+1,&\text{with probability }1-h_i(a_t),\\0,&\text{with probability }h_i(a_t).\end{cases}$$

Mathematical statement

The paper defines a waiting-time density r_i(a) for regime i, where a is the elapsed time since entering that regime. Its survival probability is S_i(a)=integral from a to infinity of r_i(a') da', and its hazard rate is R_i(a)=r_i(a)/S_i(a), the conditional instantaneous switching rate given survival until age a. We discretize time into steps n and represent the hazard by h_i(n) in [0,1], with discrete survival S_i(n)=product from m=0 to n-1 of (1-h_i(m)) and discrete duration probability p_i(n)=h_i(n)S_i(n). Thus the augmented state consisting of hidden representation, current regime, and age is locally Markovian: remain in regime i with probability 1-h_i(a), or switch with probability h_i(a). A neural implementation estimates h_i(a,z) from the current representation z and an age embedding e(a), for example h_i(a,z)=sigmoid(w_i^T z+u_i^T e(a)+b_i), while retaining the expert output during a stay and recomputing expert selection only after a switch.

Implementation notes

(1) Integrate this into a causal MoE or recurrent/SSM block. At each sequence position t, maintain a discrete regime index i_t, an integer age a_t, and the hidden state z_t. Each regime corresponds to one expert MLP or one SSM parameter set. Do not compute a fresh unconstrained softmax route at every token. Retain the current expert while the age-conditioned hazard says to stay, and choose a new expert only after a switch.

(2) Pseudocode: initialize i=argmax_i router(z_0), a=0. For each token, compute y=Expert[i](z). Compute h=sigmoid(w_i^T z+u_i^T age_embedding[min(a,A_max)]+b_i). During training, use a straight-through Bernoulli sample or a differentiable two-way relaxation. If staying, return Expert[i](z) and set a=a+1. If switching, compute new logits q=Router(z), mask the current expert i, select i_new=argmax(q) or sample from softmax(q/temperature), set i=i_new and a=0. For a fully differentiable MVP, use the weighted output (1-h)Expert[i](z)+h sum_j softmax(q)_j Expert[j](z). Add an optional duration regularizer using the implied distribution p_i(n)=h_i(n)S_i(n), such as a penalty between its mean duration and a desired target.

(3) The paper supplies the survival and hazard construction; the network estimates h_i(a,z), expert-selection logits, and hidden representations by backpropagation. Estimate empirical duration histograms from route traces and compare them with p_i(n). Monitor switch rate, mean duration, expert-load entropy, and the fraction of consecutive tokens assigned to the same expert.

(4) First experiment: use a four-expert causal Transformer-MoE or GRU-MoE on WikiText-2, plus a synthetic sequence task whose labels follow alternating regimes with known geometric and heavy-tailed durations. Compare standard token-wise top-1 routing against age-conditioned routing at equal parameter count and FLOPs. Success means lower validation perplexity or synthetic sequence error, substantially fewer expert switches, and no expert-load collapse. On long-context data, also test whether retaining an expert for multiple tokens reduces router computation without hurting perplexity.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: No measurable effect

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 a readable age-conditioned semi-Markov router toy module with empirical duration-hazard estimation, age/evidence-dependent switching, and survival/duration verification. Across three seeds and noise levels, it reduced switching by about 27–35% and router calls by about 56–64%, while the survival identity error was at most 2.2e-16. However, accuracy was consistently lower than the token-wise baseline by roughly 1.5–6.2 percentage points, so this is a promising stability and routing-compute signal, not an accuracy improvement.

Agent confidence: 8/10

Baseline: At noise 0.65, mean over 3 seeds: accuracy 0.7794, switch reduction reference 0%, 20,000/20,000 router calls.
Idea: At noise 0.65, mean over 3 seeds: accuracy 0.7447, 34.6% fewer switches, 59.3% fewer router calls. Math survival identity error <=2.2e-16; implied and empirical mean durations differed by <5e-5.

Limitations: This is a scalar synthetic two-regime simulation, not a trained neural MoE, GRU, Transformer, WikiText-2 benchmark, or equal-FLOP learned comparison. The hazard is estimated from held-out synthetic duration traces and the experiment does not test end-to-end backpropagation, expert load balancing, or learned age embeddings.

How to run: python3 age_router_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.05173) Sep 4, 2026

No significant difference vs. the tuned baseline (+0.9%, p=0.3073, wins 2 / 8 paired seeds).

Benchmark: Sequence modeling
Model: transformer_tiny
Paired seeds: 8
Baseline mean: 1.0414
Idea mean: 1.0507
Effect (Δ): 0.0094 (+0.9%; negative = idea better)
Wins: 2 / 8 paired seeds
p-value: 0.3073 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±1.8%
Mechanism: Not confirmed ✗
Practical effect: No effect
Baseline tuning: swept over 6 configs

Limitations:

The experiment used a reduced local budget of 400 training samples, 200 test samples, 3 epochs, two experts, and hidden width 16 after the initial larger run exceeded 40 minutes. It did not test WikiText-2, a discrete stochastic router, actual FLOP/router-call savings, expert-load entropy, or longer training. The mechanism signature's observed trace-change proxy is not a direct empirical switch count, and its confirmed field is false.

How to run: python3 stage2_age_router_bench.py

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

Stage 2 — Benchmark Sep 4, 2026 latest Failed ✗

Agent confidence: 9/10

Implemented an end-to-end age-conditioned semi-Markov recurrent expert router on the structurally matched sequence forecasting track, with identical trained architecture and data for baseline and idea systems. The tuned baseline achieved test MSE 1.04138 versus 1.05074 for the idea; paired delta was +0.00935 with permutation p=0.3073, so there is no significant improvement. The trained-model signature also failed confirmation: mean predicted hazard was 0.5709, mean age 0.9239, and the measured trace-change proxy was 0.00972.

Baseline: sequence/transformer_tiny label: tuned lr=0.01, temperature=0.8; full 8-seed mean test MSE 1.0413843393, std 0.0757906863; per-seed [1.0304178, 0.8935724, 1.0985037, 1.0838863, 0.9647194, 1.0631779, 1.1534550, 1.0433422]
Delta mean: 0.0093522072
Idea: best hazard-conditioned setting lr=0.01, temperature=0.8, hazard_bias=0.0; full 8-seed mean test MSE 1.0507365465, std 0.0798404275; per-seed [1.0360348, 0.9041899, 1.1562473, 1.0952754, 0.9761891, 1.0409249, 1.1531434, 1.0438876]
Idea wins: 2
N pairs: 8
P value: 0.3073

Benchmark result