Scalable solution to crossed random effects model with random slopes

The crossed random-effects model is widely used in applied statistics, finding applications in various fields such as longitudinal studies, e-commerce, and recommender systems, among others. However, these models encounter scalability challenges, as the computational time grows disproportionately with the number of data points, typically following a cubic root relationship $(N^{(3/2)}$ or worse) with $N$. Our inspiration for addressing this issue comes from observing the recommender system employed by an online clothing retailer. Our dataset comprises over 700,000 clients, 5,000 items, and 5,000,000 measurements. When applying the maximum likelihood approach to fit crossed random effects, computational inefficiency becomes a significant concern, limiting the applicability of this approach in large-scale settings. To tackle the scalability issues, previous research by Ghosh et al. (2022a) and Ghosh et al. (2022b) has explored linear and logistic regression models utilizing fixed-effect features based on client and item variables, while incorporating random intercept terms for clients and items. In this study, we present a more generalized version of the problem, allowing random effect sizes/slopes. This extension enables us to capture the variability in effect size among both clients and items. Importantly, we have developed a scalable solution to address the aforementioned problem and have empirically demonstrated the consistency of our estimates. Specifically, as the number of data points increases, our estimates converge towards the true parameters. To validate our approach, we implement the proposed algorithm using Stitch Fix data.