Matching the Statistical Query Lower Bound for k-sparse Parity Problems with Stochastic Gradient Descent

• URL: http://arxiv.org/abs/2404.12376v1
• Date: Thu, 18 Apr 2024 17:57:53 GMT
• Title: Matching the Statistical Query Lower Bound for k-sparse Parity Problems with Stochastic Gradient Descent
• Authors: Yiwen Kou, Zixiang Chen, Quanquan Gu, Sham M. Kakade,
• Abstract summary: We show that gradient descent (SGD) can efficiently solve the $k$-parity problem on a $d$dimensional hypercube. We then demonstrate how a trained neural network with SGD, solving the $k$-parity problem with small statistical errors.
• Score: 83.85536329832722
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The $k$-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 $k$-parity problem with stochastic gradient descent (SGD) on two-layer fully-connected neural networks. We demonstrate that SGD 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, thus 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 SGD can effectively approximate this good network, solving the $k$-parity problem with small statistical errors. Our theoretical results and findings are supported by empirical evidence, showcasing the efficiency and efficacy of our approach.

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