Learning a mixture of two multinomial logits

The classical Multinomial Logit (MNL) is a behavioral model for user choice. In this model, a user is offered a slate of choices (a subset of a finite universe of n items), and selects exactly one item from the slate, each with probability proportional to its (positive) weight. Given a set of observed slates and choices, the likelihood-maximizing item weights are easy to learn at scale, and easy to interpret. However, the model fails to represent common real-world behavior. As a result, researchers in user choice often turn to mixtures of MNLs, which are known to approximate a large class of models of rational user behavior. Unfortunately, the only known algorithms for this problem have been heuristic in nature. In this paper we give the first polynomial-time algorithms for exact learning of uniform mixtures of two MNLs. Interestingly, the parameters of the model can be learned for any n by sampling the behavior of random users only on slates of sizes 2 and 3; in contrast, we show that slates of size 2 are insufficient by themselves

Multinomial Logit Models with Implicit Variable Selection

Multinomial logit models which are most commonly used for the modeling of unordered multi-category responses are typically restricted to the use of few predictors. In the high-dimensional case maximum likelihood estimates frequently do not exist. In this paper we are developing a boosting technique called multinomBoost that performs variable selection and fits the multinomial logit model also when predictors are high-dimensional. Since in multicategory models the effect of one predictor variable is represented by several parameters one has to distinguish between variable selection and parameter selection. A special feature of the approach is that, in contrast to existing approaches, it selects variables not parameters. The method can distinguish between mandatory predictors and optional predictors. Moreover, it adapts to metric, binary, nominal and ordinal predictors. Regularization within the algorithm allows to include nominal and ordinal variables which have many categories. In the case of ordinal predictors the order information is used. The performance of the boosting technique with respect to mean squared error, prediction error and the identification of relevant variables is investigated in a simulation study. For two real life data sets the results are also compared with the Lasso approach which selects parameters