Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts

• URL: http://arxiv.org/abs/2402.03460v2
• Date: Sat, 25 May 2024 16:02:25 GMT
• Title: Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts
• Authors: Anastasis Kratsios, Haitz Sáez de Ocáriz Borde, Takashi Furuya, Marc T. Law,
• Abstract summary: Mixture-of-Experts (MoEs) can scale up beyond traditional deep learning models. We show that MoMLPs can generalize since the entire MoMLP model has a (finite) VC dimension of $tildeO(LmaxnL,JW)$.
• Score: 17.022107735675046
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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