[Paper Review] Learning Mixtures of Ranking Models

Pranjal Awasthi, Avrim Blum+2 more|arXiv (Cornell University)|Oct 31, 2014

Machine Learning and Algorithms27 references31 citations

TL;DR

This paper presents the first polynomial-time algorithm with theoretical guarantees for learning the parameters of a mixture of two Mallows models, using tensor decomposition to identify top-k position patterns in rankings. It achieves provable identifiability and exact recovery of central permutations and model parameters under arbitrary mixing, without requiring separation between base rankings.

ABSTRACT

This work concerns learning probabilistic models for ranking data in a heterogeneous population. The specific problem we study is learning the parameters of a Mallows Mixture Model. Despite being widely studied, current heuristics for this problem do not have theoretical guarantees and can get stuck in bad local optima. We present the first polynomial time algorithm which provably learns the parameters of a mixture of two Mallows models. A key component of our algorithm is a novel use of tensor decomposition techniques to learn the top-k prefix in both the rankings. Before this work, even the question of identifiability in the case of a mixture of two Mallows models was unresolved.

Motivation & Objective

• To address the lack of provably correct algorithms for learning mixtures of Mallows models, especially in heterogeneous populations.
• To resolve the long-standing open question of identifiability for a mixture of two Mallows models.
• To develop a method that learns parameters (central permutations, weights, dispersion) without requiring separation between the underlying rankings.
• To extend the use of tensor decomposition techniques to ranking models, which lack natural higher-order moments.

Proposed method

• Leverages tensor decomposition on third-order moment tensors constructed from pairwise and triplewise ranking probabilities.
• Defines novel moment vectors for Mallows models based on the probability of elements appearing at specific positions, especially the first position.
• Uses dynamic programming to compute exact probabilities f(i→j) for element i at position j in a Mallows model.
• Applies a tensor decomposition algorithm to recover the underlying structure of the mixture, even in the presence of noise from finite samples.
• Employs a clustering-like strategy based on position frequency distributions to initialize the decomposition, followed by refinement via parameter estimation.
• Uses Chernoff bounds to determine sample complexity for estimating position probabilities within required accuracy.

Experimental results

Research questions

• RQ1 Is a mixture of two Mallows models uniquely identifiable from i.i.d. samples?
• RQ2 Can we learn the parameters of a mixture of two Mallows models in polynomial time with provable guarantees?
• RQ3 Can tensor decomposition techniques be adapted to ranking models that lack natural higher-order moments?
• RQ4 Does the algorithm work without requiring strong separation between the central rankings of the two components?

Key findings

• The paper establishes that a mixture of two Mallows models is uniquely identifiable from polynomially many i.i.d. samples.
• The algorithm learns the central permutations π₁ and π₂ exactly, and the parameters φ₁, φ₂, w₁, w₂ up to ǫ-accuracy in time polynomial in n, (min{w₁,w₂})⁻¹, 1/φ₁(1−φ₁), 1/φ₂(1−φ₂), and 1/ǫ.
• The method achieves exact recovery of the full rankings π₁ and π₂ by first recovering the top-k elements using moment estimation and then extending via Lemma 10.2.
• The sample complexity is O(1/γ²w_min² log(n/δ)) where γ = min(gain(φ₁), gain(φ₂)) and w_min = min(w₁,w₂).
• The algorithm works without any separation condition between the two underlying permutations, making it applicable to arbitrary mixtures.
• Theoretical analysis shows that the total variation distance between the true and estimated mixture distributions is at most ǫn²/φ_min, ensuring robustness to parameter estimation errors.