Sparse Mixture-of-Experts for Compositional Generalization: Empirical Evidence and Theoretical Foundations of Optimal Sparsity

Sparse Mixture-of-Experts (SMoE) architectures have gained prominence for their ability to scale neural networks, particularly transformers, without a proportional increase in computational cost. Despite their success, their role in compositional generalization, i.e., adapting to novel combinations of known components, remains under-explored. This study challenges the assumption that minimal expert activation suffices for task generalization and investigates the relationship between task complexity and optimal sparsity in SMoE models. Through empirical evaluations on the SRAVEN symbolic reasoning task and the SKILL-MIX benchmark, we demonstrate that (i) the number of activated experts consistently increases with the perceived task difficulty to maintain performance; and (ii) the optimal number of activated experts scales proportionally with task complexity. Our theoretical analysis derives a scaling law for optimal sparsity by balancing approximation and estimation errors, revealing alignment with empirical observations. We formally show that the optimal sparsity lies between minimal activation (1-2 experts) and full activation, with the exact number scaling proportionally to task complexity and further influenced by the size of the training data and the complexity of the model. These findings offer practical insights for designing SMoE models that achieve computational efficiency while enabling robust compositional generalization.