@inproceedings{
kou2024matching,
title={Matching the Statistical Query Lower Bound for \$k\$-Sparse Parity Problems with Sign Stochastic Gradient Descent},
author={Yiwen Kou and Zixiang Chen and Quanquan Gu and Sham M. Kakade},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=EbSSBvwUWw}
}
The $k$-sparse parity problem is a classical problem in computational complexity and algorithmic theory, serving as a key benchmark for understanding computational classes. In this paper, we solve the $k$-sparse parity problem with sign stochastic gradient descent, a variant of stochastic gradient descent (SGD) on two-layer fully-connected neural networks. We demonstrate that this approach can efficiently solve the $k$-sparse parity problem on a $d$-dimensional hypercube ($k\le O(\sqrt{d})$) with a sample complexity of $\tilde{O}(d^{k-1})$ using $2^{\Theta(k)}$ neurons, matching the established $\Omega(d^{k})$ lower bounds of Statistical Query (SQ) models. Our theoretical analysis begins by constructing a good neural network capable of correctly solving the $k$-parity problem. We then demonstrate how a trained neural network with sign SGD can effectively approximate this good network, solving the $k$-parity problem with small statistical errors. To the best of our knowledge, this is the first result that matches the SQ lower bound for solving $k$-sparse parity problem using gradient-based methods.
Matching the Statistical Query Lower Bound for k-Sparse Parity Problems with Sign Stochastic Gradient Descent
Yiwen Kou Zixiang Chen Quanquan Gu Sham M. Kakade