Convergence Rates for Softmax Gating Mixture of Experts

• URL: http://arxiv.org/abs/2503.03213v1
• Date: Wed, 05 Mar 2025 06:11:24 GMT
• Title: Convergence Rates for Softmax Gating Mixture of Experts
• Authors: Huy Nguyen, Nhat Ho, Alessandro Rinaldo,
• Abstract summary: Mixture of experts (MoE) has emerged as an effective framework to advance the efficiency and scalability of machine learning models.<n>Central to the success of MoE is an adaptive softmax gating mechanism which takes responsibility for determining the relevance of each expert to a given input and then dynamically assigning experts their respective weights.<n>We perform a convergence analysis of parameter estimation and expert estimation under the MoE equipped with the standard softmax gating or its variants, including a dense-to-sparse gating and a hierarchical softmax gating.
• Score: 78.3687645289918
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Mixture of experts (MoE) has recently emerged as an effective framework to advance the efficiency and scalability of machine learning models by softly dividing complex tasks among multiple specialized sub-models termed experts. Central to the success of MoE is an adaptive softmax gating mechanism which takes responsibility for determining the relevance of each expert to a given input and then dynamically assigning experts their respective weights. Despite its widespread use in practice, a comprehensive study on the effects of the softmax gating on the MoE has been lacking in the literature. To bridge this gap in this paper, we perform a convergence analysis of parameter estimation and expert estimation under the MoE equipped with the standard softmax gating or its variants, including a dense-to-sparse gating and a hierarchical softmax gating, respectively. Furthermore, our theories also provide useful insights into the design of sample-efficient expert structures. In particular, we demonstrate that it requires polynomially many data points to estimate experts satisfying our proposed \emph{strong identifiability} condition, namely a commonly used two-layer feed-forward network. In stark contrast, estimating linear experts, which violate the strong identifiability condition, necessitates exponentially many data points as a result of intrinsic parameter interactions expressed in the language of partial differential equations. All the theoretical results are substantiated with a rigorous guarantee.

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