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Deep neural networks have achieved tremendous success due to theirrepresentation power and adaptation to low-dimensional structures. Theirpotential for estimating structured regression functions has been recentlyestablished in the literature. However, most of the studies require the inputdimension to be fixed and consequently ignore the effect of dimension on therate of convergence and hamper their applications to modern big data with highdimensionality. In this paper, we bridge this gap by analyzing a $k^{th}$ ordernonparametric interaction model in both growing dimension scenarios ($d$ growswith $n$ but at a slower rate) and in high dimension ($d \gtrsim n$). In thelatter case, sparsity assumptions and associated regularization are required inorder to obtain optimal rates of convergence. A new challenge in divergingdimension setting is in calculation mean-square error, the covariance termsamong estimated additive components are an order of magnitude larger than thoseof the variances and they can deteriorate statistical properties without propercare. We introduce a critical debiasing technique to amend the problem. We showthat under certain standard assumptions, debiased deep neural networks achievea minimax optimal rate both in terms of $(n, d)$. Our proof techniques relycrucially on a novel debiasing technique that makes the covariances of additivecomponents negligible in the mean-square error calculation. In addition, weestablish the matching lower bounds.

Deep Neural Networks for Nonparametric Interaction Models with Diverging Dimension