### Abstract

Motivated by generating personalized recommendations using ordinal (or preference) data, we study the question of learning a mixture of MultiNomial Logit (MNL) model, a parameterized class of distributions over permutations, from partial ordinal or preference data (e.g. pair-wise comparisons). Despite its long standing importance across disciplines including social choice, operations research and revenue management, little is known about this question. In case of single MNL models (no mixture), computationally and statistically tractable learning from pair-wise comparisons is feasible. However, even learning mixture with two MNL components is infeasible in general. Given this state of affairs, we seek conditions under which it is feasible to learn the mixture model in both computationally and statistically efficient manner. We present a sufficient condition as well as an efficient algorithm for learning mixed MNL models from partial preferences/comparisons data. In particular, a mixture of r MNL components over n objects can be learnt using samples whose size scales polynomially in n and r (concretely, r^{35}n^{3}(logn)^{4}, with r << n^{2}/^{7} when the model parameters are sufficiently incoherent). The algorithm has two phases: first, learn the pair-wise marginals for each component using tensor decomposition; second, learn the model parameters for each component using RANKCEN-TRALITY introduced by Negahban et al. In the process of proving these results, we obtain a generalization of existing analysis for tensor decomposition to a more realistic regime where only partial information about each sample is available.

Original language | English (US) |
---|---|

Pages (from-to) | 595-603 |

Number of pages | 9 |

Journal | Advances in Neural Information Processing Systems |

Volume | 1 |

Issue number | January |

State | Published - Jan 1 2014 |

Event | 28th Annual Conference on Neural Information Processing Systems 2014, NIPS 2014 - Montreal, Canada Duration: Dec 8 2014 → Dec 13 2014 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Networks and Communications
- Information Systems
- Signal Processing

### Cite this

*Advances in Neural Information Processing Systems*,

*1*(January), 595-603.

**Learning mixed multinomial logit model from ordinal data.** / Oh, Sewoong; Shah, Devavrat.

Research output: Contribution to journal › Conference article

*Advances in Neural Information Processing Systems*, vol. 1, no. January, pp. 595-603.

}

TY - JOUR

T1 - Learning mixed multinomial logit model from ordinal data

AU - Oh, Sewoong

AU - Shah, Devavrat

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Motivated by generating personalized recommendations using ordinal (or preference) data, we study the question of learning a mixture of MultiNomial Logit (MNL) model, a parameterized class of distributions over permutations, from partial ordinal or preference data (e.g. pair-wise comparisons). Despite its long standing importance across disciplines including social choice, operations research and revenue management, little is known about this question. In case of single MNL models (no mixture), computationally and statistically tractable learning from pair-wise comparisons is feasible. However, even learning mixture with two MNL components is infeasible in general. Given this state of affairs, we seek conditions under which it is feasible to learn the mixture model in both computationally and statistically efficient manner. We present a sufficient condition as well as an efficient algorithm for learning mixed MNL models from partial preferences/comparisons data. In particular, a mixture of r MNL components over n objects can be learnt using samples whose size scales polynomially in n and r (concretely, r35n3(logn)4, with r << n2/7 when the model parameters are sufficiently incoherent). The algorithm has two phases: first, learn the pair-wise marginals for each component using tensor decomposition; second, learn the model parameters for each component using RANKCEN-TRALITY introduced by Negahban et al. In the process of proving these results, we obtain a generalization of existing analysis for tensor decomposition to a more realistic regime where only partial information about each sample is available.

AB - Motivated by generating personalized recommendations using ordinal (or preference) data, we study the question of learning a mixture of MultiNomial Logit (MNL) model, a parameterized class of distributions over permutations, from partial ordinal or preference data (e.g. pair-wise comparisons). Despite its long standing importance across disciplines including social choice, operations research and revenue management, little is known about this question. In case of single MNL models (no mixture), computationally and statistically tractable learning from pair-wise comparisons is feasible. However, even learning mixture with two MNL components is infeasible in general. Given this state of affairs, we seek conditions under which it is feasible to learn the mixture model in both computationally and statistically efficient manner. We present a sufficient condition as well as an efficient algorithm for learning mixed MNL models from partial preferences/comparisons data. In particular, a mixture of r MNL components over n objects can be learnt using samples whose size scales polynomially in n and r (concretely, r35n3(logn)4, with r << n2/7 when the model parameters are sufficiently incoherent). The algorithm has two phases: first, learn the pair-wise marginals for each component using tensor decomposition; second, learn the model parameters for each component using RANKCEN-TRALITY introduced by Negahban et al. In the process of proving these results, we obtain a generalization of existing analysis for tensor decomposition to a more realistic regime where only partial information about each sample is available.

UR - http://www.scopus.com/inward/record.url?scp=84937904644&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937904644&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:84937904644

VL - 1

SP - 595

EP - 603

JO - Advances in Neural Information Processing Systems

JF - Advances in Neural Information Processing Systems

SN - 1049-5258

IS - January

ER -