# Hierarchical Copula-Gumbel-Top-K Routing: Two-Sided Dependence Control for Frozen Mixture-of-Experts at Fixed Per-Token Routing Laws

arXiv: https://arxiv.org/abs/2607.28670 · HTML: https://mlsanity.com/paper/2607.28670/

## 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.

## Placed in

- Mixture-of-experts / [Expert routing strategies](https://mlsanity.com/moe/expert-routing-strategies/)

## Builds on (most cited first, 12 of 12)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [GShard: Scaling Giant Models with Conditional Computation and Automatic Sharding](https://mlsanity.com/paper/2006.16668/) (2020)
- [Mixture-of-Experts with Expert Choice Routing](https://mlsanity.com/paper/2202.09368/) (2022)
- [BASE Layers: Simplifying Training of Large, Sparse Models](https://mlsanity.com/paper/2103.16716/) (2021)
- [Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacement](https://mlsanity.com/paper/1903.06059/) (2019)
- [A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learning](https://mlsanity.com/paper/2110.01515/) (2021)
- [Load Balancing Mixture of Experts with Similarity Preserving Routers](https://mlsanity.com/paper/2506.14038/) (2025)
- [Parameter-Efficient Routed Fine-Tuning: Mixture-of-Experts Demands Mixture of Adaptation Modules](https://mlsanity.com/paper/2508.02587/) (2025)
- [Route Experts by Sequence, not by Token](https://mlsanity.com/paper/2511.06494/) (2025)
- [Improving Routing in Sparse Mixture of Experts with Graph of Tokens](https://mlsanity.com/paper/2505.00792/) (2025)

## 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.