Fast Heavy Inner Product Identification Between Weights and Inputs in Neural Network Training

• URL: http://arxiv.org/abs/2311.11429v1
• Date: Sun, 19 Nov 2023 21:40:16 GMT
• Title: Fast Heavy Inner Product Identification Between Weights and Inputs in Neural Network Training
• Authors: Lianke Qin, Saayan Mitra, Zhao Song, Yuanyuan Yang, Tianyi Zhou
• Abstract summary: We consider a heavy inner product identification problem, which generalizes the Light Bulb problem(citeprr89): Given two sets $A subset -1,+1d$ and $B subset -1,+1d$ with $|A|=|B| = n$, if there are exact $k$ pairs whose inner product passes a certain threshold. We provide an algorithm that runs in $O(n2 omega / 3+ o(1))$ time to find the $k$ inner product pairs that surpass $rho
• Score: 31.08452714165316
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we consider a heavy inner product identification problem, which generalizes the Light Bulb problem~(\cite{prr89}): Given two sets $A \subset \{-1,+1\}^d$ and $B \subset \{-1,+1\}^d$ with $|A|=|B| = n$, if there are exact $k$ pairs whose inner product passes a certain threshold, i.e., $\{(a_1, b_1), \cdots, (a_k, b_k)\} \subset A \times B$ such that $\forall i \in [k], \langle a_i,b_i \rangle \geq \rho \cdot d$, for a threshold $\rho \in (0,1)$, the goal is to identify those $k$ heavy inner products. We provide an algorithm that runs in $O(n^{2 \omega / 3+ o(1)})$ time to find the $k$ inner product pairs that surpass $\rho \cdot d$ threshold with high probability, where $\omega$ is the current matrix multiplication exponent. By solving this problem, our method speed up the training of neural networks with ReLU activation function.

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