A stochastic optimization approach to train non-linear neural networks with a higher-order variation regularization

• URL: http://arxiv.org/abs/2308.02293v2
• Date: Mon, 14 Aug 2023 05:06:09 GMT
• Title: A stochastic optimization approach to train non-linear neural networks with a higher-order variation regularization
• Authors: Akifumi Okuno
• Abstract summary: This study considers a $(k,q)$th order variation regularization ($(k,q)$-VR) $(k,q)$-VR is defined as the $q$th-powered integral of the absolute $k$th order derivative of the parametric models to be trained. Our numerical experiments demonstrate that the neural networks trained with the $(k,q)$-VR terms are more resilient'' than those with the conventional parameter regularization.
• Score: 3.0277213703725767
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: While highly expressive parametric models including deep neural networks have an advantage to model complicated concepts, training such highly non-linear models is known to yield a high risk of notorious overfitting. To address this issue, this study considers a $(k,q)$th order variation regularization ($(k,q)$-VR), which is defined as the $q$th-powered integral of the absolute $k$th order derivative of the parametric models to be trained; penalizing the $(k,q)$-VR is expected to yield a smoother function, which is expected to avoid overfitting. Particularly, $(k,q)$-VR encompasses the conventional (general-order) total variation with $q=1$. While the $(k,q)$-VR terms applied to general parametric models are computationally intractable due to the integration, this study provides a stochastic optimization algorithm, that can efficiently train general models with the $(k,q)$-VR without conducting explicit numerical integration. The proposed approach can be applied to the training of even deep neural networks whose structure is arbitrary, as it can be implemented by only a simple stochastic gradient descent algorithm and automatic differentiation. Our numerical experiments demonstrate that the neural networks trained with the $(k,q)$-VR terms are more ``resilient'' than those with the conventional parameter regularization. The proposed algorithm also can be extended to the physics-informed training of neural networks (PINNs).

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