  The Mixture-of-Experts (MoE) architecture is showing promising results in
improving parameter sharing in multi-task learning (MTL) and in scaling
high-capacity neural networks. State-of-the-art MoE models use a trainable
sparse gate to select a subset of the experts for each input example. While
conceptually appealing, existing sparse gates, such as Top-k, are not smooth.
The lack of smoothness can lead to convergence and statistical performance
issues when training with gradient-based methods. In this paper, we develop
DSelect-k: a continuously differentiable and sparse gate for MoE, based on a
novel binary encoding formulation. The gate can be trained using first-order
methods, such as stochastic gradient descent, and offers explicit control over
the number of experts to select. We demonstrate the effectiveness of DSelect-k
on both synthetic and real MTL datasets with up to $128$ tasks. Our experiments
indicate that DSelect-k can achieve statistically significant improvements in
prediction and expert selection over popular MoE gates. Notably, on a
real-world, large-scale recommender system, DSelect-k achieves over $22\%$
improvement in predictive performance compared to Top-k. We provide an
open-source implementation of DSelect-k.
