Softmax Routing Dynamics in Mixture-of-Experts Models: Collapse and Recovery

Mixture-of-experts (MoE) routers typically apply softmax to the router scores and keep the top-K experts, making every token use exactly K experts. Sparsity-inducing probability maps such as sparsemax, alpha-entmax and normmax can adaptively assign exact zeros to selected experts, and therefore appear to offer token-dependent expert participation, even when using the same top-K machinery. In this work, we study whether and how this sparsity survives training. We train matched 300M and 1B top-2 MoE language models with softmax, 1.5-entmax, sparsemax and 2-normmax, and find that the maps behave very differently once trained: at 1B, entmax discards 30% less probability mass than softmax while almost never dropping a selected expert, sparsemax retains the most mass, and normmax routes 21% of tokens to a single expert. These outcomes are not properties of the maps alone. Each map drops a selected expert only when the gap between the two largest scores reaches a fixed threshold, and the trained routers differ in the score distribution they learn: the entmax router learns scores with roughly half the spread of softmax's, which keeps its top-2 gaps below its threshold, while sparsemax and normmax, which share the same threshold, learn different gap distributions and hence different participation. Routers thus co-adapt their scores to the map, and a map's capacity to produce zeros does not by itself determine expert participation. While none of the sparse maps improves validation loss over softmax, they make the trained models far less sensitive to selecting more experts at inference: sparsemax trained with K=2 loses 0.02 nats when run with K=8, where softmax loses 0.58. Our results indicate that adaptive MoE routing has to be designed around the joint behavior of the probability map and the learned scores, rather than around the map alone.

Mixture-of-Experts (MoE) models scale by activating only a small subset of experts per token. However, training such models remains challenging because top-$k$ routing is discrete and non-differentiable, requiring gradient estimators for expert selection whose design remains a central open problem. We introduce ProbMoE, a probabilistic routing framework that models expert selection as a distribution over cardinality-constrained expert subsets and formulates routing as probabilistic inference in this discrete subset space. We first propose ProbMoE Exact-$k$ routing, which samples $k$-expert subsets in the forward pass, and the backward pass uses gradients through each expert's exact marginal probability as a tractable surrogate for the true gradient. ProbMoE naturally generalizes to a dynamic-$k$ routing setting, where both training and inference constrain the routing cardinality to the same predefined range, allowing adaptive expert allocation per token. Across benchmarks and model backbones, ProbMoE Exact-$k$ achieves strong performance compared to competitive baselines, with improved expert utilization and routing diversity; ProbMoE Dynamic-$k$ achieves comparable performance with fewer activated experts.

Softmax routing approaches hard top-1 routing as the temperature tends to zero, but the limiting passage is singular at router ties. This paper develops a boundary-layer calculus for this soft-to-hard limit in population squared-loss mixture-of-experts regression. For a router with logits $a_k(x;φ)$, the relevant local quantity is the top-two margin $Δ(x;φ)$, and the relevant global quantity is the boundary mass $\mathbb{P}(Δ(X;φ)\le w)$. Under smoothness and transversality assumptions, coarea and tubular-neighborhood estimates show how this mass scales with the slab width; in the binary case the leading coefficient is an explicit surface integral over the routing interface. These geometric estimates give quantitative bounds between the soft objective $L_τ$ and the hard objective $L_0$, including an $O(τ^α)$ uniform comparison under a margin-tail condition, and yield $Γ$-convergence of the soft objectives on compact parameter spaces. The main conclusion is that the zero-temperature approximation is controlled by the probability carried by an $O(τ)$ neighborhood of the routing interfaces, not by temperature alone. After isolating this boundary-layer part of the problem, we record a conditional landscape-transfer theorem from hard to small-temperature soft routing and a reduced two-expert Gaussian calculation illustrating local symmetry breaking. Synthetic diagnostics are included only as controlled checks of the boundary-layer predictions.