Title:Score the Update, Not the Token: Descent-Aligned Routing for Combinatorial LoRA Experts

Abstract:Mixture-of-LoRA-experts methods raise the capacity of low-rank adaptation by routing each token to a few low-rank experts. Nearly all of them tie one input-side factor to one output-side factor per expert, and nearly all of them route by scoring the token: the router picks experts without seeing what any of them would write. We argue that the router should score the update. To first order, adding an expert's update to a layer output lowers the loss by the inner product between that update and the negative loss gradient at the output. This usefulness is quadratic in the token, so a router that is linear in the token sees only the part of it that runs through the token mean, and routers that rank experts by the norm of their own activations never see the output factor. If each expert is split into a reader (down-projection) and a writer (up-projection), the usefulness of every reader--writer pair becomes an inner product in the shared rank-$r$ space, and all $N_AN_B$ pairs can be scored from $N_A+N_B$ vectors. We build VANE on this identity. A low-rank compass predicts the descent direction of each token. VANE scores every pair by the alignment between its update and the compass without forming any update, activates the top-$k$ pairs with additive gates, and gives every pair its exact first-order router gradient. On single-domain commonsense reasoning and a four-domain multi-task mixture with Llama-3.2-3B and Llama-3.1-8B, VANE attains the best average among twelve PEFT and MoE-LoRA baselines, by 0.9--1.1 and 1.3--1.5 points respectively, with less than half the trainable parameters of an 8-expert MoE-LoRA. Its router scores also track the measured usefulness of experts far more closely than token routers do.

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