✗ Failed on benchmark 2026

Laplace-Heterogeneous MoE Routing

Implementation & benchmark of arXiv:2608.22122 — From Individual-Based Stochastic Epidemics to Heterogeneous SIR Equations

Usefulness6/10

Difficulty4/10

Novelty6/10

Architecture Scalability Stability Accuracy Moe routing Transformer Probability Statistics Dynamical systems

Source paper: From Individual-Based Stochastic Epidemics to Heterogeneous SIR Equations arXiv:2608.22122 ⓘ · analyzed Aug 29, 2026

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 Not peer-reviewed.

Idea description

Replace the usual hand-designed expert-load penalty with a heterogeneous survival penalty derived from a susceptibility distribution. Each expert receives an availability factor q_e=G(A_e), where A_e is its cumulative recent routing pressure and G_e is a learned or fixed mixture of exponentials; highly used experts are suppressed smoothly, while heterogeneous experts can have different resistance to pressure. The mixture produces adaptive curvature and long-tailed penalties that may reduce router collapse and expert overflow without adding auxiliary loss terms.

Formulas

$$G(a)=\int_{[0,\infty)}e^{-a\lambda}\,F(d\lambda)=\mathbb{E}[e^{-a\Lambda}],\qquad s(t)=s_0G(A(t)),\qquad A(t)=\tau\int_0^t\iota(u)\,du.$$

$$\overline{\lambda}(A)=-\frac{G'(A)}{G(A)}=\frac{\mathbb{E}[\Lambda e^{-A\Lambda}]}{\mathbb{E}[e^{-A\Lambda}]}.$$

$$G(A)=\sum_{k=1}^K w_k e^{-\lambda_kA},\qquad \pi_k(A)=\frac{w_ke^{-\lambda_kA}}{G(A)},\qquad m(A)=\sum_k\pi_k(A)\lambda_k.$$

$$m'(A)=-\operatorname{Var}_{A}(\Lambda)=-\left[\sum_k\pi_k(A)\lambda_k^2-m(A)^2\right]\le 0.$$

Mathematical statement

The paper defines a nonnegative susceptibility random variable Lambda with law F and Laplace transform G(a)=E[exp(-a Lambda)]=integral exp(-a lambda) dF(lambda). The susceptible fraction under cumulative pressure a is s(a)=s_0 G(a). Its effective susceptibility is the exponentially tilted mean m(a)=-G'(a)/G(a)=E[Lambda exp(-a Lambda)]/E[exp(-a Lambda)], where the tilted distribution weights susceptibility lambda proportionally to exp(-a lambda). For finite second moment, m'(a)=-Var_a(Lambda)<=0, so the pressure response weakens monotonically. In the neural adaptation, expert e has nonnegative pressure A_e, availability q_e=G(A_e), and routing logit z_e+log(q_e+epsilon). Use a finite mixture G(A_e)=sum_{k=1}^K w_k exp(-lambda_k A_e), with w_k>=0, sum_k w_k=1, and lambda_k>=0. The adaptive hazard is m(A_e)=sum_k pi_k(A_e) lambda_k, where pi_k(A_e)=w_k exp(-lambda_k A_e)/G(A_e).

Implementation notes

Integrate the mechanism into the router immediately before top-k expert selection. For a batch of tokens x_i, let r_ie=x_i^T v_e be the ordinary router logit and let p_e be an exponentially averaged expert-pressure state. First compute soft routing mass u_e=(1/B) sum_i softmax(r_i)_e, then update pressure with p_e <- beta p_e+(1-beta)u_e; use stop-gradient through p_e for the first experiment so the router cannot exploit state-update dynamics. Define nonnegative susceptibility values by lambda_k=softplus(a_k), mixture weights w_k=softmax(b)_k, and G(p_e)=sum_k w_k exp(-lambda_k p_e). Modify logits as r'_ie=r_ie+alpha log(G(p_e)+epsilon), then select top-k experts using r'. The pseudocode is: compute r; compute u from the unmodified softmax; update p; compute q_e=sum_k w_k exp(-lambda_k p_e); compute r'=r+alpha log(q_e+epsilon); select top-k from r'; dispatch and train normally. Initially fix lambda_k on a logarithmic grid, such as {0.25,1,4,16}, and use uniform w_k; optionally learn lambda_k and w_k with a weak entropy penalty on w and a constraint on the average lambda scale. The paper's mathematical guarantee is monotone decreasing effective hazard m'(p)<=0; verify this numerically during training. Monitor expert-load coefficient of variation, token overflow rate, router entropy, validation loss, and throughput. Start with a 6-layer Transformer on Wikitext-103 or a synthetic language task, using 8 experts and top-2 routing. Compare against standard top-2 routing with a Switch-style load-balancing loss at equal capacity and FLOPs. Success means lower overflow and utilization variance at matched perplexity, or improved perplexity after removing the auxiliary balancing loss. Include ablations for a single exponential, the mixture, and a fixed linear load penalty.

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 9/10

Built a reusable NumPy Laplace-heterogeneous MoE router with EMA pressure, finite-mixture availability, effective hazard, and top-k selection. The mechanism checks were confirmed: m'(A)=-Var_A(lambda) had max error 2.65e-8, q'/q=-m had error 4.12e-7, and the observed 1% asymptotic crossover pressure 7.65 closely matched the predicted 7.60. In the deterministic routing stress test, the mixture reduced load CV from 2.397 to 0.180 and overflow from 0.777 to 0.014 versus baseline, but this does not establish language-model quality improvement.

Agent confidence: 9/10

Baseline: Toy top-1 routing: mean load CV 2.3968, overflow 0.7773, entropy 1.4998; final loads [469,32,3,1,0,2,4,1].
Idea: Laplace mixture routing: mean load CV 0.1802, overflow 0.0141, entropy 1.9113; final loads [36,78,71,68,68,64,63,64]. Math errors: derivative identity 2.65e-8 and hazard identity 4.12e-7; crossover observed/predicted 7.65/7.60.

Limitations: Only a deterministic NumPy toy routing stress test was run; no trained Transformer, language-model loss, perplexity, CUDA benchmark, learned mixture parameters, top-2 dispatch capacity implementation, or equal-FLOP training comparison was tested.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.22122) Sep 2, 2026

No significant difference vs. the tuned baseline (+0.4%, p=0.8799, wins 3 / 8 paired seeds).

Benchmark: Heterogeneous_regime_moe
Model: custom_sparse_moe
Paired seeds: 8
Baseline mean: 0.303
Idea mean: 0.3042
Effect (Δ): 0.0012 (+0.4%; negative = idea better)
Wins: 3 / 8 paired seeds
p-value: 0.8799 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Not confirmed ✗
Practical effect: No effect
Baseline tuning: swept over 12 configs

Limitations:

The built-in tracks lacked MoE routing structure, so the custom track was officially promoted with bench.validate_and_promote. The reduced but complete protocol used 200 train samples, 200 test samples, 8 experts, top-2 routing, and 3 epochs; no large Transformer, learned mixture parameters, capacity overflow dispatch, or throughput study was tested.

How to run: python3 stage2_bench.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

The officially registered custom heterogeneous-regime MoE benchmark found no win for Laplace-Heterogeneous routing. The best idea MSE was 0.3042489160 versus tuned baseline MSE 0.3030446615, with paired delta +0.0012042541 and permutation p=0.8799. The trained-model mechanism signature was not confirmed: predicted raw pressure CV was 0.1783 while observed routed-load CV was 0.4250.

Baseline: MSE 0.3030446615, lr=0.006, aux=0.05, epochs=3, 8 seeds
Idea: MSE 0.3042489160, lr=0.006, alpha=2.0, beta=0.9, epochs=3, 8 seeds
Idea wins: 3
N pairs: 8
Paired delta mean: 0.0012042541056871414
Permutation pvalue: 0.8799

Benchmark result