Routing in Gradient Space: Balanced Usage Is Not Expert Specialization

Mixture-of-Experts models rely on learned routers to assign tokens to experts, yet standard softmax gating provides no principled mechanism to control the tradeoff between sparsity and utilization. We propose Grassmannian MoE (GrMoE), a routing framework that operates on the Grassmannian manifold of subspaces, where gating weights arise from the concentration parameters of Matrix Bingham distributions. This construction yields a single, interpretable knob -- the concentration matrix $Λ$ -- that continuously controls routing entropy, replacing discrete top-$k$ selection with a smooth, geometrically principled sparsity mechanism. We further develop an amortized variational inference procedure for posterior routing distributions, enabling uncertainty-aware expert assignment that naturally resists expert collapse. We formally prove tight bounds relating the Bingham concentration spectrum to routing entropy, expected top-$k$ mass, and an exponential bound on expert collapse, establishing the first formal theory of concentration-controlled sparsity. On synthetic routing tasks, a 350M-parameter MoE language model with 8 experts, a 1.3B-parameter model with 16 experts, and a 2.7B-parameter model with 32 experts, GrMoE achieves 0% routing collapse across all seeds, comparable or better perplexity with 15--30% improved load balance, and a smooth monotonic relationship between concentration and effective sparsity that enables post-hoc sparsity tuning without retraining. Token-level analysis reveals that experts learn heterogeneous concentration values that correlate with linguistic specialization, providing interpretable routing behavior.

Standard Mixture-of-Experts (MoE) models rely on centralized routing mechanisms that introduce rigid inductive biases. We propose Routing-Free MoE which eliminates any hard-coded centralized designs including external routers, Softmax, Top-K and load balancing, instead encapsulating all activation functionalities within individual experts and directly optimized through continuous gradient flow, enabling each expert to determine its activation entirely on its own. We introduce a unified adaptive load-balancing framework to simultaneously optimize both expert-balancing and token-balancing objectives through a configurable interpolation, allowing flexible and customizable resource allocation. Extensive experiments show that Routing-Free MoE can consistently outperform baselines with better scalability and robustness. We analyze its behavior in detail and offer insights that may facilitate future MoE design ad optimization.

Sparse mixture-of-experts (MoE) large language models scale model capacity by routing each token to a small subset of experts. Their routers are regularized with load balancing terms and learn affinity scores through the language-model objective. However, these objectives do not provide direct alignment between routing affinities and token-level error. We introduce token-error supervision for sparse routing in two forms. The first form predicts an error score per expert. The affinity-weighted aggregate of these scores is aligned to the next-token cross-entropy loss, while the individual scores attenuate affinity before top-$K$ selection. The second directly aligns the router's affinities to the model's objective without requiring an additional head or inference-time modification. Both formulations use the Itakura--Saito divergence or an exponential negative log-likelihood for aligning affinities and token errors. Across two sparse MoE backbones and four multiple-choice question-answering benchmarks, we evaluate both supervision mechanisms. On Granite, our method improves accuracy by approximately 2.3 percentage points on average over a parameter-matched routing baseline. With stronger supervision, the gain on ARC-Challenge reaches 2.94 points. Both mechanisms preserve the native sparse execution budget and aggregation policy. Our code is available in the supplementary materials.