A Theoretical Framework for Auxiliary-Loss-Free Load Balancing of Sparse Mixture-of-Experts in Large-Scale AI Models

• URL: http://arxiv.org/abs/2512.03915v2
• Date: Thu, 04 Dec 2025 16:34:28 GMT
• Title: A Theoretical Framework for Auxiliary-Loss-Free Load Balancing of Sparse Mixture-of-Experts in Large-Scale AI Models
• Authors: X. Y. Han, Yuan Zhong,
• Abstract summary: In large-scale AI training, Sparse Mixture-of-Experts (s-MoE) layers enable scaling by activating only a small subset of experts per token.<n>We provide a theoretical framework for analyzing the Auxiliary-Loss-Free Load Balancing (ALF-LB) procedure.
• Score: 3.0247776995428945
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: In large-scale AI training, Sparse Mixture-of-Experts (s-MoE) layers enable scaling by activating only a small subset of experts per token. An operational challenge in this design is load balancing: routing tokens to minimize the number of idle experts, which is important for the efficient utilization of (costly) GPUs. We provide a theoretical framework for analyzing the Auxiliary-Loss-Free Load Balancing (ALF-LB) procedure -- proposed by DeepSeek's Wang et al. (2024) -- by casting it as a one-step-per-iteration primal-dual method for an assignment problem. First, in a stylized deterministic setting, our framework yields several insightful structural properties: (i) a monotonic improvement of a Lagrangian objective, (ii) a preference rule that moves tokens from overloaded to underloaded experts, and (iii) an approximate-balancing guarantee. Then, we incorporate the stochastic and dynamic nature of AI training using a generalized online optimization formulation. In the online setting, we derive a strong convexity property of the objective that leads to a logarithmic expected regret bound under certain step-size choices. Additionally, we present real experiments on 1B-parameter DeepSeekMoE models to complement our theoretical findings. Together, these results build a principled framework for analyzing the Auxiliary-Loss-Free Load Balancing of s-MoE in AI models.

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