期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2021
卷号：21
语种：English
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：Consider a homogeneous degree d polynomial f=T1++Ts, Ti=gi(i1im) where gi's are homogeneous m-variate degree d polynomials and ij's are linear polynomials in n variables. We design a (randomized) learning algorithm that given black-box access to f, computes black-boxes for the Ti's. The running time of the algorithm is poly(nmds) and the algorithm works under some \emph{non-degeneracy} conditions on the linear forms and the gi's, and some additional technical assumptions n(md)2snd4 . The non-degeneracy conditions on ij's constitute non-membership in a variety, and hence are satisfied when the coefficients of ij's are chosen uniformly and randomly from a large enough set. The conditions on gi's are satisfied for random polynomials and also for natural polynomials common in the study of arithmetic complexity like determinant, permanent, elementary symmetric polynomial, iterated matrix multiplication. A particularly appealing algorithmic corollary is the following: Given black-box access to an f=Detr(L(1))++Detr(L(s)), where L(k)=(ij(k))ij with ij(k)'s being linear forms in n variables chosen randomly, there is an algorithm which in time \poly(nr) outputs matrices (M(k))k of linear forms \text{s.t.} there exists a permutation :[s][s] with Detr(M(k))=Detr(L((k))). Our work follows the works of Kayal-Saha(STOC'19) and Garg-Kayal-Saha(FOCS'20) which use lower bound methods in arithmetic complexity to design average case learning algorithms. It also vastly generalizes the result in \cite{Kayal-Saha'19} about learning depth-three circuits, which is a special case where each gi is just a monomial. At the core of our algorithm is the partial derivative method which can be used to prove lower bounds for generalized depth-three circuits. To apply the general framework in \cite{Kayal-Saha'19, Garg-Kayal-Saha'20}, we need to establish that the non-degeneracy conditions arising out of applying the framework with the partial derivative method are satisfied in the random case. We develop simple but general and powerful tools to establish this, which might be useful in designing average case learning algorithms for other arithmetic circuit models.
