Two-Solve Robust Expert Router

Integrate this at the offline router or model-profile-selection layer, rather than inside every token-level forward pass initially. Partition requests into $J$ workload clusters using embeddings, task IDs, sequence length, or predicted difficulty. For each cluster $j$ and candidate expert or model $k$, collect a quality estimate $R_{kj}$, cost $p_{kj}$, traffic weight $w_j$, and uncertainty radius $\eta_j$; quality can be validation accuracy, reward, pass rate, or negative loss, while cost can combine latency and FLOPs. First solve the paper's 0-1 multiple-choice knapsack with $\mu=R$ to obtain $x^{nom}$. Then solve it again with $\mu=R-\eta_j$ to obtain a pessimistic route. In a stronger certificate, search for an adversarial table in the interval box: for each candidate alternative allocation $y$, assign each uncertain entry to an endpoint that maximizes $\mathrm{score}(y,\mu)-\mathrm{score}(x^{nom},\mu)$, and retain the alternative with the largest margin. For large systems, approximate this by sampling endpoint tables and repeatedly solving the knapsack, or formulate the endpoint choices as a mixed-integer linear program. Freeze routing only when nominal and pessimistic allocations agree and the worst sampled alternative margin is non-positive. Otherwise, send only disputed pairs $(k,j)$ to an evaluation queue, update $R_{kj}$ and $\eta_j$ from new measurements, and repeat. The quantities computed from the paper's mathematics are the budget-constrained solves and interval adversary; quantities estimated empirically are $R$, $\eta$, costs, and cluster frequencies. The first experiment should use a 4-8 expert MoE or a cascade of small and large language models on GSM8K, MMLU subsets, or a synthetic mixture of easy and hard tasks. Compare nominal routing, always-largest-model routing, and uncertainty-aware routing at equal average FLOPs. Measure task accuracy, average cost, number of profiling evaluations, and the fraction of routes certified. Success is higher accuracy than nominal routing at the same cost, or equal accuracy with fewer expert evaluations; a useful secondary signal is that targeted evaluation resolves routing disagreements faster than uniform profiling.