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Let $ f^{\star} $ be a function on $ \mathbb{R}^d $ with an assumption of aspectral norm $ v_{f^{\star}} $. For various noise settings, we show that $\mathbb{E}\|\hat{f} - f^{\star} \|^2 \leq \left(v^4_{f^{\star}}\frac{\logd}{n}\right)^{1/3} $, where $ n $ is the sample size and $ \hat{f} $ is eithera penalized least squares estimator or a greedily obtained version of suchusing linear combinations of sinusoidal, sigmoidal, ramp, ramp-squared or othersmooth ridge functions. The candidate fits may be chosen from a continuum offunctions, thus avoiding the rigidity of discretizations of the parameterspace. On the other hand, if the candidate fits are chosen from adiscretization, we show that $ \mathbb{E}\|\hat{f} - f^{\star} \|^2 \leq\left(v^3_{f^{\star}}\frac{\log d}{n}\right)^{2/5} $. This work bridgesnon-linear and non-parametric function estimation and includes single-hiddenlayer nets. Unlike past theory for such settings, our bound shows that the riskis small even when the input dimension $ d $ of an infinite-dimensionalparameterized dictionary is much larger than the available sample size. Whenthe dimension is larger than the cube root of the sample size, this quantity isseen to improve the more familiar risk bound of $v_{f^{\star}}\left(\frac{d\log (n/d)}{n}\right)^{1/2} $, also investigatedhere.

Risk Bounds for High-dimensional Ridge Function Combinations Including Neural Networks