Modeling Choice via Self-Attention

Models of choice are a fundamental input to many now-canonical optimization problems in the field of Operations Management, including assortment, inventory, and price optimization. Naturally, accurate estimation of these models from data is a critical step in the application of these optimization problems in practice, and so it is perhaps surprising that such choice estimation has to now been accomplished almost exclusively, both in theory and in practice, (a) without the use of deep learning in any meaningful way, and (b) via evaluation on limited data with constantly-changing metrics. This is in stark contrast to the vast majority of similar learning applications, for which the practice of machine learning suggests that (a) neural network-based models are typically state-of-the-art, and (b) strict standardization on evaluation procedures (datasets, metrics, etc.) is crucial. Thus motivated, we first propose a choice model that is the first to successfully (both theoretically and practically) leverage a modern neural network architectural concept (self-attention). Theoretically, we show that our attention-based choice model is a low-rank generalization of the Halo Multinomial Logit model, a recent model that parsimoniously captures irrational choice effects and has seen empirical success. We prove that whereas the Halo-MNL requires $\Omega(m^2)$ data samples to estimate, where $m$ is the number of products, our model supports a natural nonconvex estimator (in particular, that which a standard neural network implementation would apply) which admits a near-optimal stationary point with $O(m)$ samples. We then establish the first realistic-scale benchmark for choice estimation on real data and use this benchmark to run the largest evaluation of existing choice models to date. We find that the model we propose is dominant over both short-term and long-term data periods.