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

Yehuda Dar

TL;DR

We study a zero-compute, 1-sparse Mixture of Experts (ZC-1SMoE) for regression by partitioning the input space into many small regions and assigning a constant predictor to each region. By leveraging high-rate quantization theory, we derive both 1D and multidimensional approximation-error characterizations, including optimal segment densities and upper bounds that connect region geometry to prediction error. We also analyze learning the region constants via least squares, proving unbiasedness and establishing a tradeoff between approximation and estimation errors as the number of experts $m$ grows, with empirical validation in 1D. The framework links MoE design to quantization-density optimization, offers guidance on choosing $m$ given data, and lays groundwork for extending to more complex sparsity patterns and segmentation-learning strategies.

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.

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

TL;DR

We study a zero-compute, 1-sparse Mixture of Experts (ZC-1SMoE) for regression by partitioning the input space into many small regions and assigning a constant predictor to each region. By leveraging high-rate quantization theory, we derive both 1D and multidimensional approximation-error characterizations, including optimal segment densities and upper bounds that connect region geometry to prediction error. We also analyze learning the region constants via least squares, proving unbiasedness and establishing a tradeoff between approximation and estimation errors as the number of experts grows, with empirical validation in 1D. The framework links MoE design to quantization-density optimization, offers guidance on choosing given data, and lays groundwork for extending to more complex sparsity patterns and segmentation-learning strategies.

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.

Paper Structure

This paper contains 40 sections, 13 theorems, 128 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Relate Work
  3. Theory for Mixture of Experts
  4. High-Rate Quantization Theory
  5. High-Rate Scalar Quantization Theory
  6. High-Rate Vector Quantization
  7. Problem and Model Definition
  8. Approximation Error of MoE for One-Dimensional Inputs
  9. Small Subintervals Assumption and Locally-Linear Approximations
  10. Optimizing the Routing Segmentation using Expert Segment Density
  11. Differences Between the 1D-MoE Learning and Scalar Quantization
  12. Empirical Results: Approximation Error, Optimal Segmentation and Experts
  13. Approximation Error of MoE for Multidimensional Inputs
  14. Learning the Constant Experts: A Tradeoff in the Number of Experts
  15. Learning Expert Constants via Least Squares
  16. ...and 25 more sections

Key Result

Theorem 1

Given a segmentation of $[0,1]$ by $m$ subintervals $\{[a_{i-1},a_i)\}_{i=1}^m$, the optimal expert constants are The corresponding test error can be formulated as where $\Delta_{\sf max}= \underset{i\in\{1,\dots,m\}}{\max} \Delta_i$ is the largest subinterval length.

Figures (4)

  • Figure 1: Examples for optimal segmentation, best predictor in $\mathcal{H}_{m,1}$, and approximation error curves for a cosine$\beta$ and truncated Gaussian $p_{\mathrm{x}}$. In (e)-(g), the red markers on the $x$-axis denote the optimal segmentation points $\{a_i\}_{i=1}^m$ that were formed from the optimal segment density in (d).
  • Figure 2: Examples for optimal segmentation, best predictor in $\mathcal{H}_{m,1}$, and approximation error curves for a cosine with a constant segment$\beta$ and truncated Gaussian $p_{\mathrm{x}}$. In (e)-(g), the red markers on the $x$-axis denote the optimal segmentation points $\{a_i\}_{i=1}^m$ that were formed from the optimal segment density in (d).
  • Figure 3: Examples for learning experts for uniform segmentation. This experiment is for a cosine$\beta$ and truncated Gaussian $p_{\mathrm{x}}$ from Figs. \ref{['fig:example 1 - beta']}, \ref{['fig:example 1 - pdf x']}, respectively. Here, in (a)-(b), the red markers on the $x$-axis denote the given uniform segmentation points $\{a_i=\frac{i}{m}\}_{i=1}^m$; the dotted magenta lines show the best predictor in $\mathcal{H}_{m,1}^{c}\left(\{a_i=\frac{i}{m}\}_{i=1}^m\right)$; the black lines show the predictor learned from a training data of 200 examples. (c) shows the empirical and theoretical approximation error curves of the best predictors in $\mathcal{H}_{m,1}^{c}\left(\{a_i=\frac{i}{m}\}_{i=1}^m\right)$. (d) shows the empirical test error curves for the learned predictors for three sizes of training dataset.
  • Figure 4: Examples for learning experts for uniform segmentation. This experiment is for a cosine with a constant segment$\beta$ and truncated Gaussian $p_{\mathrm{x}}$ from Figs. \ref{['fig:example 2 - beta']}, \ref{['fig:example 2 - pdf x']}, respectively. Here, in (a)-(b), the red markers on the $x$-axis denote the given uniform segmentation points $\{a_i=\frac{i}{m}\}_{i=1}^m$; the dotted magenta lines show the best predictor in $\mathcal{H}_{m,1}^{c}\left(\{a_i=\frac{i}{m}\}_{i=1}^m\right)$; the black lines show the predictor learned from a training data of 200 examples. (c) shows the empirical and theoretical approximation error curves of the best predictors in $\mathcal{H}_{m,1}^{c}\left(\{a_i=\frac{i}{m}\}_{i=1}^m\right)$. (d) shows the empirical test error curves for the learned predictors for three sizes of training dataset.

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Theorem 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 3 more