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

Abstract

Softmax routing in mixture-of-experts models can concentrate probability on a subset of experts, suppressing access to the others. We study how this concentration forms, why it persists, and how suppressed probabilities recover. For fixed experts, a stationary-loss barrier identifies the face supporting the optimal routed output. In a common-target square-loss model with freely trainable expert outputs, slow adaptation tracks fixed-expert dynamics on finite horizons, while every fixed positive adaptation rate yields an interior routing limit. An exact normal-velocity decomposition separates complementary-expert covariance from relative gradient advantage. Under positive two-sided power bounds along recovery paths, uniform over initial masses, the resulting coefficient's order of vanishing gives logarithmic or power-law recovery times. Integrations of constructed fields reproduce these orders. An exact discrete log-odds identity relates neural updates to threshold crossing. In paired Transformer-MoE training, removing a class-conditional router-gradient contrast delays confirmed recovery and reduces accumulated log-odds on fixed tokens, both with and without capacity constraints.