  $ $Let $F$ be a multivariate function from a product set $\Sigma^n$ to an
Abelian group $G$. A $k$-partition of $F$ with cost $\delta$ is a partition of
the set of variables $\mathbf{V}$ into $k$ non-empty subsets $(\mathbf{X}_1,
\dots, \mathbf{X}_k)$ such that $F(\mathbf{V})$ is $\delta$-close to
$F_1(\mathbf{X}_1)+\dots+F_k(\mathbf{X}_k)$ for some $F_1, \dots, F_k$ with
respect to a given error metric. We study algorithms for agnostically learning
$k$ partitions and testing $k$-partitionability over various groups and error
metrics given query access to $F$. In particular we show that
  $1.$ Given a function that has a $k$-partition of cost $\delta$, a partition
of cost $\mathcal{O}(k n^2)(\delta + \epsilon)$ can be learned in time
$\tilde{\mathcal{O}}(n^2 \mathrm{poly} (1/\epsilon))$ for any $\epsilon > 0$.
In contrast, for $k = 2$ and $n = 3$ learning a partition of cost $\delta +
\epsilon$ is NP-hard.
  $2.$ When $F$ is real-valued and the error metric is the 2-norm, a
2-partition of cost $\sqrt{\delta^2 + \epsilon}$ can be learned in time
$\tilde{\mathcal{O}}(n^5/\epsilon^2)$.
  $3.$ When $F$ is $\mathbb{Z}_q$-valued and the error metric is Hamming
weight, $k$-partitionability is testable with one-sided error and
$\mathcal{O}(kn^3/\epsilon)$ non-adaptive queries. We also show that even
two-sided testers require $\Omega(n)$ queries when $k = 2$.
  This work was motivated by reinforcement learning control tasks in which the
set of control variables can be partitioned. The partitioning reduces the task
into multiple lower-dimensional ones that are relatively easier to learn. Our
second algorithm empirically increases the scores attained over previous
heuristic partitioning methods applied in this context.
