Learning and Testing Variable Partitions
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Let F be a multivariate function from a product set Σ^n 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 (X_1, ..., X_k) such that F(V) is δ-close to F_1(X_1)+...+F_k(X_k) for some F_1, ..., 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 δ, a partition of cost O(k n^2)(δ + ϵ) can be learned in time Õ(n^2 poly (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 √(δ^2 + ϵ) can be learned in time Õ(n^5/ϵ^2). 3. When F is Z_q-valued and the error metric is Hamming weight, k-partitionability is testable with one-sided error and O(kn^3/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.
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