Downsampling for Testing and Learning in Product Distributions
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We study the domain reduction problem of eliminating dependence on n from the complexity of property testing and learning algorithms on domain [n]^d, and the related problem of establishing testing and learning results for product distributions over ℝ^d. Our method, which we call downsampling, gives conceptually simple proofs for several results: 1. A 1-page proof of the recent o(d)-query monotonicity tester for the hypergrid (Black, Chakrabarty Seshadhri, SODA 2020), and an improvement from O(d^7) to O(d^4) in the sample complexity of their distribution-free monotonicity tester for product distributions over ℝ^d; 2. An ( O(kd))-time agnostic learning algorithm for functions of k convex sets in product distributions; 3. A polynomial-time agnostic learning algorithm for functions of a constant number of halfspaces in product distributions; 4. A polynomial-time agnostic learning algorithm for constant-degree polynomial threshold functions in product distributions; 5. An ( O(k √(d)))-time agnostic learning algorithm for k-alternating functions in product distributions.
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