2026-07-26T00:38:38Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17143 2024-03-06T10:59:04Z ddc:004 open_access

Learning Generalized Depth Three Arithmetic Circuits in the Non-Degenerate Case Bhargava, Vishwas Garg, Ankit Kayal, Neeraj Saha, Chandan Arithemtic Circuits Average-case Learning Depth 3 Arithmetic Circuits Learning Algorithms Learning Circuits Circuit Reconstruction Consider a homogeneous degree d polynomial f = T₁ + ⋯ + T_s, T_i = g_i(𝓁_{i,1}, …, 𝓁_{i, m}) where g_i’s are homogeneous m-variate degree d polynomials and 𝓁_{i,j}’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 T_i’s. The running time of the algorithm is poly(n, m, d, s) and the algorithm works under some non-degeneracy conditions on the linear forms and the g_i’s, and some additional technical assumptions n ≥ (md)², s ≤ n^{d/4}. The non-degeneracy conditions on 𝓁_{i,j}’s constitute non-membership in a variety, and hence are satisfied when the coefficients of 𝓁_{i,j}’s are chosen uniformly and randomly from a large enough set. The conditions on g_i’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 = Det_r(L^(1)) + … + Det_r(L^(s)), where L^(k) = (𝓁_{i,j}^(k))_{i,j} with 𝓁_{i,j}^(k)’s being linear forms in n variables chosen randomly, there is an algorithm which in time poly(n, r) outputs matrices (M^(k))_k of linear forms s.t. there exists a permutation π: [s] → [s] with Det_r(M^(k)) = Det_r(L^(π(k))). Our work follows the works [Neeraj Kayal and Chandan Saha, 2019; Garg et al., 2020] which use lower bound methods in arithmetic complexity to design average case learning algorithms. It also vastly generalizes the result in [Neeraj Kayal and Chandan Saha, 2019] about learning depth three circuits, which is a special case where each g_i 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 [Neeraj Kayal and Chandan Saha, 2019; Garg et al., 2020], 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Vishwas Bhargava and Ankit Garg and Neeraj Kayal and Chandan Saha 2022 Is Part Of LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2022.21 urn:nbn:de:0030-drops-171430 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.21 eng https://creativecommons.org/licenses/by/4.0/legalcode