{
 "artifacts": null,
 "category": "optimization",
 "description": "Replace point-estimate expert routing with a nominal allocation and a least-favourable allocation under uncertainty in expert quality. If both allocations agree, use that route confidently; if they disagree, profile or evaluate only the expert-input pairs responsible for the disagreement. The same mechanism can be used offline to assign workloads to LLMs or online to choose among heterogeneous experts under a latency or FLOP budget.",
 "formulas_latex": [
  "$$X^{\\star}[\\mu]=\\arg\\max_{x}\\sum_{j}\\sum_{k\\in\\mathcal{K}_{j}}w_{j}\\mu_{kj}x_{kj}\\quad\\text{s.t.}\\quad N\\sum_{j}\\sum_{k\\in\\mathcal{K}_{j}}w_{j}p_{kj}x_{kj}\\leq\\Gamma,\\quad \\sum_{k\\in\\mathcal{K}_{j}}x_{kj}=1\\ \\forall j.$$",
  "$$M_{kj}=\\mu^{\\mathrm{ref}}_{kj}+B_{kj}+C^{o}_{kj},\\qquad R_{kj}=\\mu^{\\mathrm{ref}}_{kj}+C^{r}_{kj}.$$",
  "$$\\mathcal U=\\{R-C\\mid C_{kj}\\in[0,\\eta_j]\\ \\text{for every }k\\in\\mathcal K_j\\text{ and }j\\}.$$",
  "$$\\mu^{\\mathrm{pess}}_{kj}=R_{kj}-\\eta_j,\\qquad x^{\\mathrm{nom}}=X^{\\star}[R],\\qquad x^{\\mathrm{pess}}=X^{\\star}[\\mu^{\\mathrm{pess}}].$$"
 ],
 "id": 2833,
 "implementation": "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.",
 "math_summary": "The paper defines binary allocation variables $x_{kj}\\in\\{0,1\\}$, where $k$ indexes models or experts and $j$ indexes workloads or input clusters, with workload weight $w_j$, quality estimate $\\mu_{kj}$, cost $p_{kj}$, and total budget $\\Gamma$. The nominal allocation is $X^{\\star}[\\mu]=\\arg\\max_x\\sum_j\\sum_{k\\in\\mathcal K_j}w_j\\mu_{kj}x_{kj}$ subject to $N\\sum_j\\sum_{k\\in\\mathcal K_j}w_jp_{kj}x_{kj}\\leq\\Gamma$ and $\\sum_{k\\in\\mathcal K_j}x_{kj}=1$ for every $j$. The observed randomized score is decomposed as $R_{kj}=\\mu^{\\mathrm{ref}}_{kj}+C^r_{kj}$, while observational measurements also contain selection and evaluator gaps: $M_{kj}=\\mu^{\\mathrm{ref}}_{kj}+B_{kj}+C^o_{kj}$. The uncertainty set is generated by evaluator-error tables $C$ satisfying $C_{kj}\\in[0,\\eta_j]$ for every expert-workload pair, giving $\\mathcal U=\\{R-C:C_{kj}\\in[0,\\eta_j]\\}$. Here $\\eta_j$ is the maximum evaluator error assumed for workload cluster $j$. Adapt the paper's certificate by computing the nominal optimum at $R$ and a least-favourable optimum over $\\mu\\in\\mathcal U$; agreement means the route is invariant over the assumed uncertainty set. For a fixed candidate allocation $x$, its pessimistic score is $\\min_{\\mu\\in\\mathcal U}\\sum_{j,k}w_j\\mu_{kj}x_{kj}=\\sum_{j,k}w_j(R_{kj}-\\eta_j)x_{kj}$ under the independent interval model, while the adversarial allocation used for certification should be selected to maximize the opportunity for an alternative allocation to beat the nominal one.",
 "math_tags": [
  "optimization",
  "statistics",
  "probability"
 ],
 "ml_areas": [
  "moe-routing",
  "inference-speedup",
  "training"
 ],
 "paper": {
  "arxiv_id": "2608.29560",
  "arxiv_url": "https://arxiv.org/abs/2608.29560",
  "summary_what_math_gives_to_ml": "The paper provides a decision-focused way to handle uncertain quality estimates: instead of requiring the entire model-quality table to be identified, test whether the same budget-feasible allocation remains optimal over an explicitly defined uncertainty set. This is transferable to mixture-of-experts routing, cascaded inference, and learned model selection, where expert accuracy, latency, and cost are uncertain and the routing decision is globally coupled by a compute budget. The most useful adaptation is a nominal-versus-adversarial allocation certificate, followed by targeted evaluation only on model-input pairs that can change the allocation. This can turn uncertainty estimates into a practical routing-freezing or active-profiling mechanism rather than merely adding a generic robustness penalty.",
  "title": "Which LLM for Which Work? Budgeted Model Allocation under Uncertain Evaluation",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "speedup",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Two-Solve Robust Expert Router",
 "url": "https://synthcore.org/idea/2833/two-solve-robust-expert-router",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}