Estimating Large-Scale Tree Logit Models via a Difference of Strictly Convex Functions

Abstract

We describe an efficient estimation method for large-scale tree logit models, using a novel change-of-variables transformation that allows us to express the negative log-likelihood as a difference of strictly convex functions. Exploiting this representation, we design a fast iterative method that computes a sequence of parameter estimates. At each iteration, parameters at leaf nodes are updated using a simple formula involving the Lambert-W function, while the parameters at non-leaf nodes are updated simultaneously by minimizing a strictly convex one-dimensional function over the unit interval. No step size or second-order derivative is required. The sequence of parameter estimates yields increasing likelihood values, and we show that every limit point is a stationary point. Numerical results show that our algorithm outperforms state-of-the-art optimization methods, especially for large-scale tree logit models with thousands of nodes.