  Empirical risk minimization over classes functions that are bounded for some
version of the variation norm has a long history, starting with Total Variation
Denoising (Rudin et al., 1992), and has been considered by several recent
articles, in particular Fang et al., 2019 and van der Laan, 2015. In this
article, 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 $\mathcal{F}_d$ allows
to bound the bracketing entropy of sieves of $\mathcal{F}_d$, and therefore
derive rates of convergence in nonparametric function estimation. Specifically,
for sieves whose growth is controlled by some rate $a_n$, we show that the
empirical risk minimizer has rate of convergence $O_P(n^{-1/3} (\log
n)^{2(d-1)/3} a_n)$. Remarkably, the dimension only affects the rate in $n$
through the logarithmic factor, making this method especially appropriate for
high dimensional problems.
  In particular, we show that in the case of nonparametric regression over
sieves of c\`adl\`ag functions with bounded sectional variation norm, this
upper bound on the rate of convergence holds for least-squares estimators,
under the random design, sub-exponential errors setting.
