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arXiv · 2605.15403

$\phi$-Balancing for Mixture-of-Experts Training

Lizhang Chen·Jonathan Li·Qi Wang·Runlong Liao·Shuozhe Li·Chen Liang·Ni Lao·Qiang Liu
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Abstract

Mixture-of-Experts (MoE) models rely on balanced expert utilization to fully realize their scalability. However, existing load-balancing methods are largely heuristic and operate on noisy mini-batch assignment statistics, introducing bias relative to population-level objectives. We propose $\phi$-balancing, a principled framework that directly targets population-level expert balance by minimizing a strictly convex, symmetric, and differentiable potential of the expected routing distribution. Using convex duality, we derive an equivalent min-max formulation and obtain a simple online algorithm via mirror descent, yielding an efficient EMA-based routing adjustment with negligible overhead. Across large-scale pretraining and downstream fine-tuning, $\phi$-balancing consistently outperforms prior Switch-style and loss-free baselines, demonstrating more stable and effective expert utilization.

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Lizhang Chen, Jonathan Li, Qi Wang, Runlong Liao, Shuozhe Li, Chen Liang, Ni Lao, Qiang Liu. 2026-05-14. $\phi$-Balancing for Mixture-of-Experts Training. https://arxiv.org/abs/2605.15403

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