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In this paper, we propose a convex formulation for learning logisticregression model (logit) with latent heterogeneous effect on sub-population. Intransportation, logistic regression and its variants are often interpreted asdiscrete choice models under utility theory (McFadden, 2001). Two prominentapplications of logit models in the transportation domain are traffic accidentanalysis and choice modeling. In these applications, researchers often want tounderstand and capture the individual variation under the same accident orchoice scenario. The mixed effect logistic regression (mixed logit) is apopular model employed by transportation researchers. To estimate thedistribution of mixed logit parameters, a non-convex optimization problem withnested high-dimensional integrals needs to be solved. Simulation-basedoptimization is typically applied to solve the mixed logit parameter estimationproblem. Despite its popularity, the mixed logit approach for learningindividual heterogeneity has several downsides. First, the parametric form ofthe distribution requires domain knowledge and assumptions imposed by users,although this issue can be addressed to some extent by using a non-parametricapproach. Second, the optimization problems arise from parameter estimation formixed logit and the non-parametric extensions are non-convex, which leads tounstable model interpretation. Third, the simulation size insimulation-assisted estimation lacks finite-sample theoretical guarantees andis chosen somewhat arbitrarily in practice. To address these issues, we aremotivated to develop a formulation that models the latent individualheterogeneity while preserving convexity, and avoids the need forsimulation-based approximation. Our setup is based on decomposing theparameters into a sparse homogeneous component in the population and low-rankheterogeneous parts for each individual.

Convex Latent Effect Logit Model via Sparse and Low-rank Decomposition