Redesign Mixture-of-Experts Routers with Manifold Power Iteration
Authors: Songhao Wu, Ang Lv, Ruobing Xie, Yankai Lin
Abstract
Router is the cornerstone component to the Mixture-of-Experts models. Serving as expert proxies, the rows of the router matrix compute their similarity to the MoE inputs to determine which subset of experts is activated. Ideally, each router row is designed to encode the expert matrix into this representative vector, such that its dot-product with token can better reflect token-expert affinity. However, there exists no design principles to enforce this condensation. In this paper, we propose to align each router row with the principal singular direction of the associated expert, as this direction provides the most expressive mathematical description of a matrix. Based on this principle, we propose a router redesign with Manifold Power Iteration (MPI). Specifically, it introduces a "Power-then-Retract" paradigm, where a power iteration step is performed on the router weights, followed by a retraction to impose a norm constraint to ensure both efficiency and stability. Theoretically, we show that MPI drives router rows to converge toward the principal singular directions of associated experts. Empirically, we pretrain MoE model across scales from 1B to 11B parameters to confirm that this alignment facilitates more effective MoE models.
Mixture-of-experts (MoE) models enable scalable transformer architectures by activating only a subset of experts per token. Recent evidence suggests that performance improves with increasingly granular experts, i.e., many small experts instead of a few large ones. However, this regime substantially increases routing cost, which can dominate computation. We introduce adaptive inverted-index routing for MoE (AIR-MoE), an inverted-index-inspired routing architecture based on vector quantization (VQ). In a first stage, AIR-MoE performs coarse shortlisting by assigning tokens to VQ codewords to construct a candidate set of experts. In a second stage, fine scoring computes exact routing scores restricted to this shortlist. This two-stage procedure approximates true top-k routing while avoiding full expert scoring and, in contrast to prior work, imposing no structural constraints on expert parameters. AIR-MoE serves as a drop-in replacement for standard routers and requires no modifications to the model architecture or loss function. We further provide a lower bound on the mass recall achieved by AIR-MoE that yields insights into its inner workings. Empirically, we demonstrate that AIR-MoE achieves improved performance compared to existing routing approaches in granular MoE settings.
Klaus-Rudolf Kladny, Maximilian Mordig, Bernhard Schölkopf +1
Mixture-of-Experts (MoE) layers increase model capacity by activating only a small subset of experts per token, and typically rely on a learned router to map hidden states to expert assignments. In this work, we ask whether a dedicated learned router is strictly necessary for MoE routing. We propose Self-Routing, a parameter-free routing mechanism that uses a designated subspace of the token hidden state directly as expert logits, eliminating the router projection entirely while leaving the rest of the MoE layer unchanged. We evaluate Self-Routing on language modeling across different expert counts and model scales, and on ImageNet-1K classification by comparing it against a standard learned router, random-routing baselines, and dense non-MoE baselines. Our results show that Self-Routing remains competitive with the learned-router baseline while removing all dedicated routing parameters, and yields more balanced expert utilization, with about 17 % higher average normalized routing entropy and no explicit load-balancing loss. On ImageNet-1K with DeiT-S/16, Self-Routing also slightly improves over the corresponding learned-router MoE. These findings suggest that effective MoE routing can emerge from the hidden representation itself without requiring a separate learned router module.
Mixture-of-Experts (MoE) models decouple total parameters from per-token compute, but deployment still requires storing every expert. Recent theory shows that pruning experts with the smallest router-norm changes during fine-tuning can preserve accuracy, but assumes full fine-tuning. We test whether lightweight adaptation can recover this signal. We briefly fine-tune with a parameter-efficient adapter, rank experts by the induced ℓ2 router change, and prune the least-changed experts in one shot. On Mixtral-8×7B-Instruct (44.83% MMLU-Pro), router-only LoRA trains 0.002% of parameters and outperforms all-module LoRA at matched rank with half the experts removed (27.54% vs. 24.42%); signal quality declines as adaptation spreads to attention and expert weights. Accuracy improves monotonically with LoRA rank, reaching 28.76%. IA3, which leaves router weights frozen, matches direct router adaptation, whereas unconstrained additive adapters degrade the signal. Router-guided MMLU-Pro accuracy decays quasi-linearly rather than collapsing, remains nearly 1.8 times that of magnitude-based or random pruning at maximal compression, and reduces memory by 49% and per-token latency by 37%. At 25% compression, retention is competitive with methods using full activation statistics. The criterion also transfers to Qwen1.5-MoE fine-tuned for mathematics, retaining 49.7% mean accuracy over eleven benchmarks with half the experts removed while random pruning falls to single digits. Router sensitivity under lightweight fine-tuning therefore makes provably motivated expert pruning practical at scale.