A Statistical Analysis for Supervised Deep Learning with Exponential Families for Intrinsically Low-dimensional Data

• URL: http://arxiv.org/abs/2412.09779v1
• Date: Fri, 13 Dec 2024 01:15:17 GMT
• Title: A Statistical Analysis for Supervised Deep Learning with Exponential Families for Intrinsically Low-dimensional Data
• Authors: Saptarshi Chakraborty, Peter L. Bartlett,
• Abstract summary: We consider supervised deep learning when the given explanatory variable is distributed according to an exponential family. Under the assumption of an upper-bounded density of the explanatory variables, we characterize the rate of convergence as $tildemathcalOleft( dfrac2lfloorbetarfloor(beta + d)2beta + dn-frac22beta + dn-frac22beta + dn-frac22beta + dn-
• Score: 32.98264375121064
• License:
• Abstract: Recent advances have revealed that the rate of convergence of the expected test error in deep supervised learning decays as a function of the intrinsic dimension and not the dimension $d$ of the input space. Existing literature defines this intrinsic dimension as the Minkowski dimension or the manifold dimension of the support of the underlying probability measures, which often results in sub-optimal rates and unrealistic assumptions. In this paper, we consider supervised deep learning when the response given the explanatory variable is distributed according to an exponential family with a $\beta$-H\"older smooth mean function. We consider an entropic notion of the intrinsic data-dimension and demonstrate that with $n$ independent and identically distributed samples, the test error scales as $\tilde{\mathcal{O}}\left(n^{-\frac{2\beta}{2\beta + \bar{d}_{2\beta}(\lambda)}}\right)$, where $\bar{d}_{2\beta}(\lambda)$ is the $2\beta$-entropic dimension of $\lambda$, the distribution of the explanatory variables. This improves on the best-known rates. Furthermore, under the assumption of an upper-bounded density of the explanatory variables, we characterize the rate of convergence as $\tilde{\mathcal{O}}\left( d^{\frac{2\lfloor\beta\rfloor(\beta + d)}{2\beta + d}}n^{-\frac{2\beta}{2\beta + d}}\right)$, establishing that the dependence on $d$ is not exponential but at most polynomial. We also demonstrate that when the explanatory variable has a lower bounded density, this rate in terms of the number of data samples, is nearly optimal for learning the dependence structure for exponential families.

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