Hierarchical Copula-Gumbel-Top-K Routing: Two-Sided Dependence Control for Frozen Mixture-of-Experts at Fixed Per-Token Routing Laws
On the map: Mixture-of-experts → Expert routing strategies
Abstract (arXiv)
A stochastic Gumbel-Top-K router defines, for every token of a mixture-of-experts (MoE) model, a routing law: a distribution over ordered expert lists and mixture weights. We ask which joint distributions over the routing choices of different tokens are reachable while every individual token's complete routing law is held exactly fixed. We give a two-sided construction, Hierarchical Copula-Gumbel-Top-K (CGA). Within a group of related tokens, an exchangeable Gaussian copula positively correlates the Gumbel perturbations at each expert coordinate, which can increase within-group expert-set coherence. Across disjoint pairs of groups, a tunable antithetic construction introduces a selectable amount of negative dependence. We prove that both operations leave each token's ordered Top-K sample, mixture weights, and inclusion probabilities identical in distribution to independent routing at a routing layer conditioned on its pre-routing logits; conditional expected expert traffic is preserved as a consequence. We characterize the resulting trade-off: positive within-group coupling can only inflate the variance of realized expert loads relative to independent routing, while nonnegative cross-group opposition can only reduce it relative to flat coupling at the same within-group strength. Coherence and load dispersion are thus controlled by two complementary dependence dials on the invariance constraint surface. Because the base model is untouched, the dials can be driven by a small controller over frozen features, trainable with a score-function estimator: the frozen network is evaluated only in the forward direction, and gradients are confined to the controller. An initial small-scale pilot validates the mechanism and the training route, but does not establish task-level fine-tuning gains.
Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in Expert routing strategies, one of 12 directions in Mixture-of-experts. Builds on 12 of its 18 references that are in the map. First posted on arXiv 25 Jul 2026.
Builds on
12 of 18 references.
Closest ancestors · 6
Same direction, newest first
- 2025Route Experts by Sequence, not by Token9 Nov
- 2025Parameter-Efficient Routed Fine-Tuning: Mixture-of-Experts Demands Mixture of Adaptation Modules4 Aug
- 2025Load Balancing Mixture of Experts with Similarity Preserving Routers16 Jun
- 2025Improving Routing in Sparse Mixture of Experts with Graph of Tokens1 May
- 2022Mixture-of-Experts with Expert Choice Routing18 Feb
- 2021Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity11 Jan
Other references · 6
- 2021LoRA: Low-Rank Adaptation of Large Language Modelscited by 14,380
- 2017Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layercited by 3,509
- 2020GShard: Scaling Giant Models with Conditional Computation and Automatic Shardingcited by 1,631
- 2021BASE Layers: Simplifying Training of Large, Sparse Modelscited by 320
- 2019Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacementcited by 211
- 2021A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learningcited by 82
Built on by
Not cited yet.
Similar, not cited
Close in wording, no citation either way
- 2026A Persistent State for Auditable Mixture-of-Experts Routing
- 2026Share First, Route What Remains: A Unified Framework for Token-Adaptive MoE Computation
- 2026Path-Constrained Mixture-of-Experts
- 2026Routing-Free Mixture-of-Experts
- 2026Hierarchical Mixture-of-Experts with Two-Stage Optimization
- 2026Beyond Routing: Decoupling Expert Dispatch and Aggregation in Sparse Mixture-of-Experts
- 2026MoRE: Scaling mixture of experts with hardware-aware low-rank routing
- 2026ProbMoE: Differentiable Probabilistic Routing for Mixture-of-Experts