Beyond the Previous Layer: Residual Predictive Structure in Sparse MoE Routing

Sparse mixture-of-experts (MoE) language models route each token to multiple experts, suggesting a geometric account of their benefit: co-selected experts should contribute distinct representation directions. Existing evidence often conflates route coherence, candidate quality, and candidate-by-context interaction. We distinguish these quantities using an Expert Subspace Separation Index (ESSI), matched-route residuals, and a prefix-controlled $2\times2$ factorial; frozen-route interventions and a controlled Top-$k$ study assess functional value. Three paired contrasts organize the findings. First, across six MoE architectures, expert subspaces overlap substantially, yet actual routes explain token representations better than matched alternatives. Second, across the 39 factorial cells in OLMoE, Mixtral, and DeepSeek, the selected candidate explains more of the residual representation than the strongest unselected rival in every cell, yet the actual prefix narrows this advantage throughout: all interactions are negative, and every 95% confidence interval lies below zero. Third, this geometric narrowing does not imply functional redundancy: adding later experts improves next-token prediction in 24 of 39 frozen-route comparisons, while the other 15 estimates are inconclusive; a controlled training study also favors Top-2 over Top-1 in all three seeds. We call this joint pattern coherent overlap: routing selects token-relevant experts from a shared geometric neighborhood, while useful multi-expert computation persists without disjoint linear coverage. Separating these quantities clarifies why geometric similarity alone cannot determine redundancy or pruning value.

Sparse Mixture-of-Experts (MoE) routers commonly use the same scores both to select experts and to weight their already-computed outputs. We study whether these two roles, dispatch and aggregation, should be coupled. On pretrained OLMoE-1B-7B, we keep selected Top-8 expert IDs, expert computation, and total selected router mass fixed and change only within-set aggregation. A structured oracle improves full-horizon cross-entropy by 0.0160 +/- 0.0039 across three seeds; the router's top-scored expert is the counterfactual-best vertex only 17.2% of the time, with router-utility Spearman 0.030. We therefore train Fixed-Dispatch Adaptive Aggregation (FDAA), a 301K-parameter post-compute head optimized directly with the language-modeling objective while freezing the backbone, router, and experts. On OLMoE, FDAA improves fresh WikiText-103 test by Delta CE = -0.1523 +/- 0.0031 across three seeds, and mixed-domain training gives robust gains on WikiText-103, C4, and held-out Penn Treebank under frozen confirmatory evaluation. We also replicate the fixed-dispatch audit on DeepSeek-V2-Lite, which uses Top-6 routed experts plus shared experts. Best-vertex headroom remains significant on WikiText and C4, while router Top1 identifies the best selected expert in only 12.5% and 16.7% of audited examples. In a one-seed mixed-domain replication, FDAA improves locked WikiText and PTB, while C4 is statistically neutral. These results support a cross-architecture distinction between expert selection and expert commitment.

Mixture-of-experts (MoE) models have recently moved beyond routing a fixed number of complete experts. Shared-expert designs preserve reusable knowledge, fine-grained methods vary computation within experts, and dynamic routers adapt the number of active experts. Yet these decisions are usually made independently, overlooking a basic dependency: extracting reusable computation changes both what remains and how much expert capacity the remainder needs. We study this dependency by decomposing sparsely upcycled feed-forward experts into key-value channels. Co-activated experts align at a subset of value positions; removing these positions changes expert preference; and greater shared coverage is associated with lower residual expert demand. These observations lead to one principle: share first, then route what remains. We instantiate it in UniF-MoE, a unified framework for token-adaptive MoE computation. Each expert is partitioned into aligned blocks. A shared-demand score sets the shared block count and pathway weight, key prototypes select the shared content, and the complementary demand determines the residual expert count through cumulative routing mass. A Gram regularizer separates and normalizes router embeddings, promoting diverse routing directions, sparse expert overlap, and a simple routing geometry. Experiments on DomainBed and GLUE show that this unified design improves predictive performance over representative static and dynamic MoEs while reducing activated computation, inference latency, and memory. Code is available at https://github.com/existence0420/UniF-MoE.