Let F be a multivariate function from a product set Σn to an
Abelian group G. A k-partition of F with cost δ is a partition of
the set of variables V into k non-empty subsets (X1,…,Xk) such that F(V) is δ-close to
F1(X1)+⋯+Fk(Xk) for some F1,…,Fk 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 δ, a partition
of cost O(kn2)(δ+ϵ) can be learned in time
O~(n2poly(1/ϵ)) for any ϵ>0.
In contrast, for k=2 and n=3 learning a partition of cost δ+ϵ is NP-hard.
2. When F is real-valued and the error metric is the 2-norm, a
2-partition of cost δ2+ϵ can be learned in time
O~(n5/ϵ2).
3. When F is Zq-valued and the error metric is Hamming
weight, k-partitionability is testable with one-sided error and
O(kn3/ϵ) non-adaptive queries. We also show that even
two-sided testers require Ω(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.Comment: Innovations in Theoretical Computer Science (ITCS) 202