For instance, we illustrate how different choices of $\alpha$ assign embedding dimensions in Fig \[fig:fixed\_budget\_alpha\_assignments\].

Matrix Factorization
--------------------

MF is a canonical algorithm for CF. From the perspective of this work, pure MF model for CF tasks is important because it offers the most distilled, simplest setting for which we can investigate the effects of mixed dimension embeddings. In our experiments we use the MovieLens dataset [@Harper2015]. We train at learning rate $10^{-2}$ using Amsgrad optimizer [@Reddi2018] for 100 epochs, taking model that scored the lowest validation loss. We always run 3 replicates per experiment.

In Fig. \[fig:mf\_train\_loss\_sample\], we report learning curves for MF with three different embedding layers, plotting the number of training epochs against the validation loss for each epoch. We plot the learning curve for $d = 16$ uniform dimension embeddings, which has the lowest validation loss (under early termination). We also plot the learning curve for $d = 4$ as a baseline. Third, we plot the learning curve for mixed dimension embedding layers at $\alpha=0.6$, which uses a number of parameters equivalent to uniform dimension embedding layers at $d = 4$.

[0.49]{} ![Loss under different parameters budgets[]{data-label="fig:mf_performance"}](images/learning_curve_mf.png "fig:"){width="\textwidth"}

[0.49]{} ![Loss under different parameters budgets[]{data-label="fig:mf_performance"}](images/mf_rate_performance_9_19.png "fig:"){width="\textwidth"}

Notice 