Mixture-of-Experts (MoEs) can scale up beyond traditional deep learning models by employing a routing strategy in which each input is processed by a single "expert" deep learning model. This strategy allows us to scale up the number of parameters defining the MoE while maintaining sparse activation, i.e., MoEs only load a small number of their total parameters into GPU VRAM for the forward pass depending on the input. In this paper, we provide an approximation and learning-theoretic analysis of mixtures of expert MLPs with (P)ReLU activation functions. We first prove that for every error level $\varepsilon>0$ and every Lipschitz function $f:[0,1]^n\to \mathbb{ R}$, one can construct a MoMLP model (a Mixture-of-Experts comprising of (P)ReLU MLPs) which uniformly approximates $f$ to $\varepsilon$ accuracy over $[0,1]^n$, while only requiring networks of $\mathcal{ O}(\varepsilon^{ -1})$ parameters to be loaded in memory. Additionally, we show that MoMLPs can generalize since the entire MoMLP model has a (finite) VC dimension of $\tilde{ O}(L\max\{ nL,JW\})$, if there are $L$ experts and each expert has a depth and width of $J$ and $W$, respectively.
Categories: stat.ML cs.LG cs.NA cs.NE math.CO math.NA