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A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits

Authors Nikhil Gupta, Chandan Saha, Bhargav Thankey

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Nikhil Gupta
  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India
Chandan Saha
  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India
Bhargav Thankey
  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

Acknowledgements

We would like to thank Neeraj Kayal and Ankit Garg for sitting through a presentation of this work and giving us useful feedback. Thanks specially to Ankit for bringing the work of Chen and Tell [Lijie Chen and Roei Tell, 2019] to our notice. A part of this work is done at Microsoft Research India (MSRI), where CS is spending a sabbatical year. CS would like to thank MSRI for providing an excellent research environment and for the hospitality.

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Nikhil Gupta, Chandan Saha, and Bhargav Thankey. A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 23:1-23:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CCC.2020.23

Abstract

We show an Ω̃(n^2.5) lower bound for general depth four arithmetic circuits computing an explicit n-variate degree-Θ(n) multilinear polynomial over any field of characteristic zero. To our knowledge, and as stated in the survey [Amir Shpilka and Amir Yehudayoff, 2010], no super-quadratic lower bound was known for depth four circuits over fields of characteristic ≠ 2 before this work. The previous best lower bound is Ω̃(n^1.5) [Abhijat Sharma, 2017], which is a slight quantitative improvement over the roughly Ω(n^1.33) bound obtained by invoking the super-linear lower bound for constant depth circuits in [Ran Raz, 2010; Victor Shoup and Roman Smolensky, 1997].
Our lower bound proof follows the approach of the almost cubic lower bound for depth three circuits in [Neeraj Kayal et al., 2016] by replacing the shifted partials measure with a suitable variant of the projected shifted partials measure, but it differs from [Neeraj Kayal et al., 2016]’s proof at a crucial step - namely, the way "heavy" product gates are handled. Loosely speaking, a heavy product gate has a relatively high fan-in. Product gates of a depth three circuit compute products of affine forms, and so, it is easy to prune Θ(n) many heavy product gates by projecting the circuit to a low-dimensional affine subspace [Neeraj Kayal et al., 2016; Amir Shpilka and Avi Wigderson, 2001]. However, in a depth four circuit, the second (from the top) layer of product gates compute products of polynomials having arbitrary degree, and hence it was not clear how to prune such heavy product gates from the circuit. We show that heavy product gates can also be eliminated from a depth four circuit by projecting the circuit to a low-dimensional affine subspace, unless the heavy gates together account for Ω̃(n^2.5) size. This part of our argument is inspired by a well-known greedy approximation algorithm for the weighted set-cover problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • depth four arithmetic circuits
  • Projected Shifted Partials
  • super-quadratic lower bound

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