Mixture of Many Zero-Compute Experts: A High-Rate Quantization Theory Perspective

2025-10-03

AI Summary

This paper addresses the generalization performance of Mixture-of-Experts (MoE) models in regression tasks, systematically analyzing the trade-off between approximation error and estimation error through the lens of high-bit-rate quantization theory. Methodologically, it proposes a novel MoE architecture featuring “many small regions with zero-computation experts,” derives theoretically optimal partitioning and conditions for minimizing test error in one-dimensional input settings, establishes an upper bound on test error for multi-dimensional inputs, and integrates statistical learning theory to characterize the learnability of expert parameters under fixed partitions. Key contributions include: (i) the first application of high-bit-rate quantization theory to MoE regression analysis; (ii) a rigorous characterization of how the number of experts governs the error trade-off—increasing expert count reduces approximation error but amplifies estimation error, yielding an optimal expert scale; and (iii) empirical validation of the theoretically predicted generalization error “kink point.”

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper uses classical high-rate quantization theory to provide new insights into mixture-of-experts (MoE) models for regression tasks. Our MoE is defined by a segmentation of the input space to regions, each with a single-parameter expert that acts as a constant predictor with zero-compute at inference. Motivated by high-rate quantization theory assumptions, we assume that the number of experts is sufficiently large to make their input-space regions very small. This lets us to study the approximation error of our MoE model class: (i) for one-dimensional inputs, we formulate the test error and its minimizing segmentation and experts; (ii) for multidimensional inputs, we formulate an upper bound for the test error and study its minimization. Moreover, we consider the learning of the expert parameters from a training dataset, given an input-space segmentation, and formulate their statistical learning properties. This leads us to theoretically and empirically show how the tradeoff between approximation and estimation errors in MoE learning depends on the number of experts.

Problem

• Analyzing mixture-of-experts models using high-rate quantization theory
• Studying approximation error bounds for multidimensional input spaces
• Exploring tradeoffs between approximation and estimation errors

Innovation

• MoE uses zero-compute constant experts per region
• Large expert count minimizes input-space region sizes
• Quantization theory analyzes approximation-estimation error tradeoff