Learning Plackett-Luce Mixtures

from Partial Preferences

Ao Liu1, Zhibing Zhao1, Chao Liao2, Pinyan Lu3 and Lirong Xia1

01/31/2019

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Introduction: Learning from Partial Preferences

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Learn to Ratings Learn to Ranks (Preferences)

(Commonly used) (More robust)

{c1, c3}≻ c2 ≻ c4 ≻ c5

c1 ≻ c3 and c4 ≻ c5

c5 ≻ c4 ≻ others

c1 ≻ c2 ≻ c3 ≻ c4 ≻ c5

i.e., commented, “I prefer c5 to

c4, all others are worse.”

Full Ranking

Partial orders

Introduction: Learn to Rank

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Cluster Analysis from Partial Preference (e.g. Netflix data):

• Agents (a1 , …, an): users. Alternatives (c1 , …, cm): movies, etc..

• Data: agents’ partial preference (partial order) towards (a subset of) alternatives.

• Objectives: (1) Learn distribution of k clusters’ full rankings.

(2) Learn membership of each agent.

Application in Recommender Systems

Partial Preferences → Cluster Analysis → Recommendations/

Group Decisions : c1 ≻ cm-1 and c4 ≻ c1

: {c4, c1}≻ c3 ≻ cm-1

: c1 ≻ c2 and c3 ≻ c4

: c4 ≻ c5 ≻ c1 ≻ cm-1

Recommend alternatives

/

Make decision

c1 ≻ c4 ≻ … ≻ c2 ≻ c3

c4 ≻ c2 ≻ … ≻ c1 ≻ cm-1

Background: Plackett-Luce Model (PL)

A very classic and simple

model for social choice:

1. Alternatives with larger

parameters have higher

probability to be ranked

higher

2. Nicely fit real-world

ranking data. [Gormley and Murphy-2007]

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Background: Algorithm for full rankings

Linear Extensions:

Bridges partial orders to full rankings.

Linear extensions contains all full rankings

consistent with the given partial order.

For example:

partial order: c1 ≻ c2 and c1 ≻ c3

linear extensions:

c1 ≻ c2 ≻ c3

c1 ≻ c3 ≻ c2

Generating linear extensions is nontrivial

(Brute force search requires m! time)

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General Framework of Our Approach

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General Framework of Our Approach

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Advantages:

1. Work for arbitrary partial orders

2. Do not lose any information

General Framework of Our Approach

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2-step approach:

1. Generate linear extensions according to a distribution

similar to Plackett-Luce by generalized repeated inserting

method (GRIM). Complexity of GRIM is O(m2)

2. Using Markov Chain Monte-Carlo (MCMC) to tune the

distribution.

Our Algorithm for Generating Linear Extensions

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Generated distributions are similar (given c1 ≻ c3)

2-step approach:

1. Generate linear extensions according to

a distribution similar to Plackett-Luce

by generalized repeated inserting

method (GRIM)

2. Using Markov Chain Monte-Carlo

(Metropolis–Hastings algorithm) to

tune the distribution.

Our Algorithm for Generating Linear Extensions

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Theoretical Analysis to GRIM+MCMC

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A large subset of partial orders: layered-structure preferences:

1. Upper layers are strictly more preferred than lower layers.

2. No preference given inside any layers.

(i.e., c1 ≻ {cm-1, c4} ≻ c2).

Theorem. For any layered-structured preference, the mixing time of

GRIM-MCMC algorithm for Plackett-Luce model is,

O(η(m*) Poly(other inputs))

- η : maximum ratio between θi and θj .

- m*: number of alternatives associated with partial preferencs

- m* normally is much smaller than m

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Experiments

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Summary

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• Proposed the first algorithm to learn Plackett-Luce Mixture

from partial preference making full use of all information.

• Designed new MCMC algorithms to sample linear

extensions obeying Plackett-Luce model

• GRIM+MCMC: Theoretically guaranteed

• Gibbs Sampling: better performance on real experiments

References

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[Luce 1959] Robert Duncan Luce. Individual Choice Behavior: A Theoretical Analysis. Wiley, 1959. [Gormley and Murphy-2007] Gormley IC, Murphy TB. A latent space model for rank data. Statistical Network Analysis: Models, Issues, and New Directions 2007 (pp. 90-102). Springer, Berlin, Heidelberg. [Gormley 2008] Isobel Claire Gormley et al. Exploring voting blocs within the Irish electorate: A mixture modeling approach. Journal of the American Statistical Association, 103(483) 2008. [Lu 2014] Tyler Lu and Craig Boutilier. Effective sampling and learning for mallows models with pairwise-preference data. The Journal of Machine Learning Research, 15(1):3783–3829, 2014. [Liu 1996] Jun S. Liu. Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Statistics and Computing, 6(2):113–119, 1996.

Thanks for your time !

Poster ID: ML6217