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On Sinkhorn's Algorithm and Choice Modeling

Zhaonan Qu, Alfred Galichon, Wenzhi Gao, Johan Ugander

TL;DR

This work establishes a formal equivalence between maximum likelihood estimation for Luce-type choice models and the canonical matrix balancing problem solved by Sinkhorn's algorithm. By casting choice data into a participation matrix $A$ with marginals $(p,q)$, the authors show that MLE solutions correspond to finite scalings $D^0,D^1$ that balance $A$ to match $p$ and $q$, thereby unifying a broad class of choice-modeling algorithms under Sinkhorn's framework. They then derive a global linear convergence result for Sinkhorn on general nonnegative matrices, with a rate governed by the algebraic connectivity of the associated bipartite graph, and provide a sharp asymptotic rate in terms of the second eigenvalue of a scaled Gram-like operator. The paper also clarifies the distinctions between strong and weak existence, introduces regularized Sinkhorn variants with guaranteed convergence, and connects these insights to a wide spectrum of optimization, economics, and network-optimization problems. Overall, the results illuminate the central role of connectivity and orthogonality in matrix balancing and open cross-disciplinary pathways between choice modeling and matrix balancing with practical implications for large-scale inference and optimization.

Abstract

For a broad class of models widely used in practice for choice and ranking data based on Luce's choice axiom, including the Bradley--Terry--Luce and Plackett--Luce models, we show that the associated maximum likelihood estimation problems are equivalent to a classic matrix balancing problem with target row and column sums. This perspective opens doors between two seemingly unrelated research areas, and allows us to unify existing algorithms in the choice modeling literature as special instances or analogs of Sinkhorn's celebrated algorithm for matrix balancing. We draw inspirations from these connections and resolve some open problems on the study of Sinkhorn's algorithm. We establish the global linear convergence of Sinkhorn's algorithm for non-negative matrices whenever finite scaling matrices exist, and characterize its linear convergence rate in terms of the algebraic connectivity of a weighted bipartite graph. We further derive the sharp asymptotic rate of linear convergence, which generalizes a classic result of Knight (2008). To our knowledge, these are the first quantitative linear convergence results for Sinkhorn's algorithm for general non-negative matrices and positive marginals. Our results highlight the importance of connectivity and orthogonality structures in matrix balancing and Sinkhorn's algorithm, which could be of independent interest. More broadly, the connections we establish in this paper between matrix balancing and choice modeling could also help motivate further transmission of ideas and lead to interesting results in both disciplines.

On Sinkhorn's Algorithm and Choice Modeling

TL;DR

This work establishes a formal equivalence between maximum likelihood estimation for Luce-type choice models and the canonical matrix balancing problem solved by Sinkhorn's algorithm. By casting choice data into a participation matrix with marginals , the authors show that MLE solutions correspond to finite scalings that balance to match and , thereby unifying a broad class of choice-modeling algorithms under Sinkhorn's framework. They then derive a global linear convergence result for Sinkhorn on general nonnegative matrices, with a rate governed by the algebraic connectivity of the associated bipartite graph, and provide a sharp asymptotic rate in terms of the second eigenvalue of a scaled Gram-like operator. The paper also clarifies the distinctions between strong and weak existence, introduces regularized Sinkhorn variants with guaranteed convergence, and connects these insights to a wide spectrum of optimization, economics, and network-optimization problems. Overall, the results illuminate the central role of connectivity and orthogonality in matrix balancing and open cross-disciplinary pathways between choice modeling and matrix balancing with practical implications for large-scale inference and optimization.

Abstract

For a broad class of models widely used in practice for choice and ranking data based on Luce's choice axiom, including the Bradley--Terry--Luce and Plackett--Luce models, we show that the associated maximum likelihood estimation problems are equivalent to a classic matrix balancing problem with target row and column sums. This perspective opens doors between two seemingly unrelated research areas, and allows us to unify existing algorithms in the choice modeling literature as special instances or analogs of Sinkhorn's celebrated algorithm for matrix balancing. We draw inspirations from these connections and resolve some open problems on the study of Sinkhorn's algorithm. We establish the global linear convergence of Sinkhorn's algorithm for non-negative matrices whenever finite scaling matrices exist, and characterize its linear convergence rate in terms of the algebraic connectivity of a weighted bipartite graph. We further derive the sharp asymptotic rate of linear convergence, which generalizes a classic result of Knight (2008). To our knowledge, these are the first quantitative linear convergence results for Sinkhorn's algorithm for general non-negative matrices and positive marginals. Our results highlight the importance of connectivity and orthogonality structures in matrix balancing and Sinkhorn's algorithm, which could be of independent interest. More broadly, the connections we establish in this paper between matrix balancing and choice modeling could also help motivate further transmission of ideas and lead to interesting results in both disciplines.
Paper Structure (44 sections, 13 theorems, 160 equations, 2 figures, 2 tables, 3 algorithms)

This paper contains 44 sections, 13 theorems, 160 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Choice Modeling
  4. Matrix Balancing
  5. Preliminaries on Choice Modeling and Matrix Balancing
  6. Maximum Likelihood Estimation of Luce Choice Models
  7. The Canonical Matrix Balancing Problem
  8. Connecting Choice Modeling and Matrix Balancing
  9. Maximum Likelihood Estimation of Luce Choice Models as Matrix Balancing
  10. Algorithmic Connections between Matrix Balancing and Choice Modeling
  11. Linear Convergence of Sinkhorn's Algorithm for Non-negative Matrices
  12. Preliminaries
  13. Global Linear Convergence
  14. Strong vs. Weak Convergence of Sinkhorn's Algorithm
  15. Sharp Asymptotic Rate
  16. ...and 29 more sections

Key Result

Theorem 1

Let $(A,p,q)$ be constructed from a choice dataset as follows: $p\in\mathbb{R}^{n}$ with $p_{i}=R_i$, the multiplicity of choice set $S_i$; $q\in\mathbb{R}^{m}$ with $q_{j}=W_j$, the total number of times item $j$ is chosen in the choice dataset; $A\in\mathbb{R}^{n\times m}$ with $A_{ij}=1\{j\in S_{ if and only if $s \in \Delta_m$ with $s_j=D^0_j$ satisfies the optimality conditions eq:optimality

Figures (2)

  • Figure 1: Representation of Luce choice data with participation matrix $A$, $R_i$ the frequency of appearances of choice set $i$, and $W_j$ the frequency of choices of item $j$ as a matrix balancing problem $(A,p,q)$ with target marginals $p_i=R_i$ and $q_j=W_j$.
  • Figure 2: Left: folded Gaussian distribution. Right: uniform distribution

Theorems & Definitions (20)

  • Remark 1: Importance of Graph Connectivity
  • Remark 2: Reduction to Unique Choice Sets
  • Theorem 1: Equivalence of Problems
  • Proposition 1: Equivalence of Assumptions
  • Remark 3: Aggregate Data as Sufficient Statistics
  • Corollary 1
  • Theorem 2: Equivalence of Algorithms
  • Remark 4: Optimization Connections
  • Theorem 3: Global Linear Convergence
  • Remark 5: Importance of Assumptions for Linear Convergence
  • ...and 10 more