{
 "artifacts": null,
 "category": "training",
 "description": "Replace a static MoE load-balancing penalty with a two-stage capacity allocator. First compute each expert's technically feasible token capacity from latency, memory, and overflow constraints; then redistribute capacity using cumulative proportional fairness so experts that were repeatedly under-served receive more capacity later. Constrain the redistribution by an explicit efficiency budget, so fairness cannot silently cause an uncontrolled increase in routing loss or expert compute.",
 "formulas_latex": [
  "$$J_t=\\frac{\\left(\\sum_{e=1}^{E}r_{e,t}\\right)^2}{E\\sum_{e=1}^{E}r_{e,t}^2},\\qquad r_{e,t}=\\frac{S_{e,t}}{D_{e,t}+\\epsilon},\\qquad S_{e,t}=\\sum_{\\tau\\le t}x_{e,\\tau}.$$",
  "$$0\\le q^{\\mathrm{fair}}_{e,t}\\le q^{\\mathrm{tech}}_{e,t},\\qquad \\sum_{e=1}^{E}q^{\\mathrm{fair}}_{e,t}\\le C_t.$$",
  "$$\\max_e r_{e,t}-\\min_e r_{e,t}\\le\\rho,\\qquad \\sum_{t,e}\\ell_{e,t}(q^{\\mathrm{fair}}_{e,t})\\le\\sum_{t,e}\\ell_{e,t}(q^{\\mathrm{tech}}_{e,t})+\\Delta.$$",
  "$$\\min_{q^{\\mathrm{fair}}}\\ \\sum_{t,e}w_{e,t}\\left(q^{\\mathrm{fair}}_{e,t}-q^{\\mathrm{tech}}_{e,t}\\right)^2+\\lambda\\sum_t(1-J_t).$$"
 ],
 "id": 2700,
 "implementation": "Integrate the allocator immediately before top-k MoE routing. At the start of every window of B minibatches, estimate unconstrained expert demand d[e] from router probabilities and compute a technical envelope q_tech[e] using current expert memory and latency limits: q_tech[e]=min(capacity_e, floor(C*available_compute_e/sum_e available_compute_e)). Maintain cumulative demand D[e] and accepted tokens S[e] across windows. Solve the small quadratic projection problem for q_fair with projected gradient descent or a differentiable bisection multiplier: initialize q=q_tech; repeatedly update q toward the Jain or disparity objective; clip q to [0,q_tech]; project onto sum(q)\u003c=C; and reject updates violating the efficiency budget Delta. During routing, assign at most q_fair[e] tokens to expert e; overflow tokens go to their next-best expert. After routing, update S using accepted assignments and measure routing loss, overflow, expert latency, and total FLOPs. The source-derived quantities are the Jain index, cumulative ratios, and two-stage feasibility/fairness construction. Estimate q_tech, the loss functions ell, and Delta empirically from expert latency, memory, overflow, and validation loss. First test a small Switch-Transformer or Mixtral-style model on C4, comparing standard top-k plus auxiliary load-balancing loss against this allocator at equal total FLOPs. Sweep rho and Delta. The quantitative prediction is that J_t rises toward one as rho is relaxed or Delta increases, while validation-loss or throughput degradation stays below Delta. When Delta is reduced below the minimum cost of equalization, the fairness constraint becomes active and produces a visible kink in J versus Delta. The allocator should make no change when the technical ratios already satisfy the disparity bound.",
 "math_summary": "For expert e in a set of E experts and training window t, let x_{e,t} be the number of tokens assigned to expert e, q^{tech}_{e,t} its technically feasible capacity, and d_{e,t} its unconstrained demand. Define cumulative served tokens S_{e,t}=\\sum_{\\tau\\le t}x_{e,\\tau} and cumulative demand D_{e,t}=\\sum_{\\tau\\le t}d_{e,\\tau}. The cumulative service ratio is r_{e,t}=S_{e,t}/(D_{e,t}+\\epsilon), and the Jain index is J_t=(\\sum_e r_{e,t})^2/[E\\sum_e r_{e,t}^2]. The first-stage envelope satisfies 0\\le q_{e,t}\\le q^{tech}_{e,t} and \\sum_e q_{e,t}\\le C_t, where C_t is the available token or compute budget. The second stage chooses q^{fair}_{e,t} by minimizing deviation from the technical envelope while enforcing a disparity limit and an efficiency budget \\Delta: \\max_e r_{e,t}-\\min_e r_{e,t}\\le\\rho and \\sum_t\\sum_e \\ell_{e,t}(q^{fair}_{e,t})\\le\\sum_t\\sum_e\\ell_{e,t}(q^{tech}_{e,t})+\\Delta. Here \\ell_{e,t} is routing loss or excess compute, and fairness cost can be \\Gamma_t=1-J_t.",
 "math_tags": [
  "optimization",
  "statistics",
  "control-theory",
  "dynamical-systems"
 ],
 "ml_areas": [
  "moe-routing",
  "training-dynamics",
  "scheduler"
 ],
 "paper": {
  "arxiv_id": "2608.23444",
  "arxiv_url": "https://arxiv.org/abs/2608.23444",
  "summary_what_math_gives_to_ml": "The paper offers a two-stage mechanism: first compute technically feasible time-varying capacity envelopes, then redistribute that capacity using cumulative proportional fairness subject to an explicit efficiency-loss budget. Its transferable asset is the separation between feasibility and fairness, together with a multi-period fairness state that prevents repeatedly disadvantaging the same participant. A neural-network analogue is a dynamic capacity allocator for MoE experts or training clients: derive technically feasible token or update budgets, then redistribute them according to cumulative service or loss exposure while bounding the increase in compute or task loss. The mechanism produces falsifiable signatures through the Jain fairness index, maximum cumulative disparity, and a sharp activation boundary when the efficiency budget becomes binding.",
  "title": "Fair Dynamic Operating Envelopes using Distributed Multi-Period Optimal Power Flow and Jain Index for Active Distribution Networks",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 5,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "scalability",
  "accuracy"
 ],
 "title": "Cumulative-Fair MoE Capacity Envelopes",
 "url": "https://synthcore.org/idea/2700/cumulative-fair-moe-capacity-envelopes",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}