Exponential Lower Bounds for Threshold Circuits of Sub-Linear Depth and Energy

Authors Kei Uchizawa, Haruki Abe



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Kei Uchizawa
  • Graduate School of Science and Engineering, Yamagata University, Yonezawa-shi Yamagata, Japan
Haruki Abe
  • Graduate School of Science and Engineering, Yamagata University, Yonezawa-shi Yamagata, Japan

Acknowledgements

We thank the anonymous reviewers for their careful reading and helpful comments.

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Kei Uchizawa and Haruki Abe. Exponential Lower Bounds for Threshold Circuits of Sub-Linear Depth and Energy. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.MFCS.2023.85

Abstract

In this paper, we investigate computational power of threshold circuits and other theoretical models of neural networks in terms of the following four complexity measures: size (the number of gates), depth, weight and energy. Here, the energy of a circuit measures sparsity of their computation, and is defined as the maximum number of gates outputting non-zero values taken over all the input assignments. As our main result, we prove that any threshold circuit C of size s, depth d, energy e and weight w satisfies log(rk(M_C)) ≤ ed (log s + log w + log n), where rk(M_C) is the rank of the communication matrix M_C of a 2n-variable Boolean function that C computes. Thus, such a threshold circuit C is able to compute only a Boolean function of which communication matrix has rank bounded by a product of logarithmic factors of s, w and linear factors of d, e. This implies an exponential lower bound on the size of even sublinear-depth and sublinear-energy threshold circuit. For example, we can obtain an exponential lower bound s = 2^Ω(n^{1/3}) for threshold circuits of depth n^{1/3}, energy n^{1/3} and weight 2^o(n^{1/3}). We also show that the inequality is tight up to a constant factor when the depth d and energy e satisfies ed = o(n/log n). For other models of neural networks such as a discretized ReLU circuits and descretized sigmoid circuits, we define energy as the maximum number of gates outputting non-zero values. We then prove that a similar inequality also holds for a discretized circuit C: rk(M_C) = O(ed(log s + log w + log n)³). Thus, if we consider the number gates outputting non-zero values as a measure for sparse activity of a neural network, our results suggest that larger depth linearly helps neural networks to acquire sparse activity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • Circuit complexity
  • disjointness function
  • equality function
  • neural networks
  • threshold circuits
  • ReLU cicuits
  • sigmoid circuits
  • sparse activity

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