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The crossed random effects model is widely used, finding applications invarious fields such as longitudinal studies, e-commerce, and recommendersystems, among others. However, these models encounter scalability challenges,as the computational time for standard algorithms grows superlinearly with thenumber N of observations in the data set, commonly $\Omega(N^{3/2})$ or worse.Recent work has developed scalable methods for crossed random effects in linearmodels and some generalized linear models, but those works only allow forrandom intercepts. In this paper we devise scalable algorithms for models thatinclude random slopes. This problem brings a substantial difficulty inestimating the random effect covariance matrices in a scalable way. We addressthat issue by using a variational EM algorithm. In simulations, we see that theproposed method is faster than standard methods. It is also more efficient thanordinary least squares which also has a problem of greatly underestimating thesampling uncertainty in parameter estimates. We illustrate the new method on alarge dataset (five million observations) from the online retailer Stitch Fix.

Scalable solution to crossed random effects model with random slopes