Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts
<|reference_start|>Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts: 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.<|reference_end|>
@article{kratsios2024approximation,
    title={Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of
  Experts},
    author={Anastasis Kratsios and Haitz S'aez de Oc'ariz Borde and Takashi
  Furuya and Marc T. Law},
    journal={arXiv preprint arXiv:2402.03460},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03460},
    primaryClass={stat.ML cs.LG cs.NA cs.NE math.CO math.NA}
}