  Modern recommendation systems rely on real-valued embeddings of categorical
features. Increasing the dimension of embedding vectors improves model accuracy
but comes at a high cost to model size. We introduce a multi-layer embedding
training (MLET) architecture that trains embeddings via a sequence of linear
layers to derive superior embedding accuracy vs. model size trade-off.
  Our approach is fundamentally based on the ability of factorized linear
layers to produce superior embeddings to that of a single linear layer. We
focus on the analysis and implementation of a two-layer scheme. Harnessing the
recent results in dynamics of backpropagation in linear neural networks, we
explain the ability to get superior multi-layer embeddings via their tendency
to have lower effective rank. We show that substantial advantages are obtained
in the regime where the width of the hidden layer is much larger than that of
the final embedding (d). Crucially, at conclusion of training, we convert the
two-layer solution into a single-layer one: as a result, the inference-time
model size scales as d.
  We prototype the MLET scheme within Facebook's PyTorch-based open-source Deep
Learning Recommendation Model. We show that it allows reducing d by 4-8X, with
a corresponding improvement in memory footprint, at given model accuracy. The
experiments are run on two publicly available click-through-rate prediction
benchmarks (Criteo-Kaggle and Avazu). The runtime cost of MLET is 25%, on
average.
