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Empirical risk minimization over classes functions that are bounded for someversion of the variation norm has a long history, starting with Total VariationDenoising (Rudin et al., 1992), and has been considered by several recentarticles, in particular Fang et al., 2019 and van der Laan, 2015. In thisarticle, we consider empirical risk minimization over the class $\mathcal{F}_d$of c\`adl\`ag functions over $[0,1]^d$ with bounded sectional variation norm(also called Hardy-Krause variation).  We show how a certain representation of functions in $\mathcal{F}_d$ allowsto bound the bracketing entropy of sieves of $\mathcal{F}_d$, and thereforederive rates of convergence in nonparametric function estimation. Specifically,for sieves whose growth is controlled by some rate $a_n$, we show that theempirical risk minimizer has rate of convergence $O_P(n^{-1/3} (\logn)^{2(d-1)/3} a_n)$. Remarkably, the dimension only affects the rate in $n$through the logarithmic factor, making this method especially appropriate forhigh dimensional problems.  In particular, we show that in the case of nonparametric regression oversieves of c\`adl\`ag functions with bounded sectional variation norm, thisupper bound on the rate of convergence holds for least-squares estimators,under the random design, sub-exponential errors setting.

Fast rates for empirical risk minimization over c\`adl\`ag functions with bounded sectional variation norm