Learning Generalized Depth Three Arithmetic Circuits in the Non-Degenerate Case

Bhargava, V and Garg, A and Kayal, N and Saha, C (2022) Learning Generalized Depth Three Arithmetic Circuits in the Non-Degenerate Case. In: 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022, 19 - 21 September 2022, Virtual, Urbana-Champaign.

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Abstract

Consider a homogeneous degree d polynomial f = T1 + · · · + Ts, Ti = gi(ℓi, 1, ..., ℓi, m) where gi's are homogeneous m-variate degree d polynomials and ℓi, 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 Ti'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 gi's, and some additional technical assumptions n ≥ (md)2, s ≤ nd/4. The non-degeneracy conditions on ℓi, j's constitute non-membership in a variety, and hence are satisfied when the coefficients of ℓi, j'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) = (ℓ(k)i, j)i, j with ℓ(k)i, j'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 Detr(M(k)) = Detr(L(π(k))). Our work follows the works [22, 7] which use lower bound methods in arithmetic complexity to design average case learning algorithms. It also vastly generalizes the result in [22] 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 [22, 7], 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.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to the Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. .
Keywords:	Image reconstruction; Matrix algebra; Polynomials; Timing circuits, Arithemtic circuit; Arithmetic circuit; Average-case; Average-case learning; Case learning; Circuit reconstruction; Condition; Depth 3; Depth 3 arithmetic circuit; Learning circuit, Learning algorithms
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	04 Nov 2022 06:43
Last Modified:	04 Nov 2022 06:43
URI:	https://eprints.iisc.ac.in/id/eprint/77649

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