Almost Optimal Proper Learning and Testing Polynomials
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We give the first almost optimal polynomial-time proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. For s-sparse polynomial over n variables and ϵ=1/s^β, β>1, our algorithm makes q_U=(s/ϵ)^logβ/β+O(1/β)+ Õ(s)(log1/ϵ)log n queries. Notice that our query complexity is sublinear in 1/ϵ and almost linear in s. All previous algorithms have query complexity at least quadratic in s and linear in 1/ϵ. We then prove the almost tight lower bound q_L=(s/ϵ)^logβ/β+Ω(1/β)+ Ω(s)(log1/ϵ)log n, Applying the reduction in <cit.> with the above algorithm, we give the first almost optimal polynomial-time tester for s-sparse polynomial. Our tester, for β>3.404, makes Õ(s/ϵ) queries.
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