HOPE: High-order Polynomial Expansion of Black-box Neural Networks

🚀Quickstart

Installation Guide

1. Download project

git clone https://github.com/HarryPotterXTX/HOPE.git
cd HOPE

2. Prepare the Conda enviroments

conda create -n HOPE python=3.10
conda activate HOPE
pip install -r requirements.txt

Install PyTorch corresponding to your device

Usage Guide

1. Expand ''black-box'' neural networks into explicit expressions

For the high-order Taylor expansion of neural networks, we provide two implementations, HOPE and Autograd. Users can conveniently compute high-order derivatives of neural networks and obtain Taylor series approximations for the network by calling hope.py or auto.py, while HOPE has significant advantages in terms of accuracy, speed, and memory cost compared with Autograd.

Expand 1-D and 2-D neural networks with HOPE

python demo/expansion/code/net1d.py
python hope.py -d demo/expansion/outputs/1D_MLP_Sine/net.pt -o 10 -p 0
python demo/expansion/code/plot.py -d demo/expansion/outputs/1D_MLP_Sine/npy/HOPE_[[0.0]]_10.npy -r 4.5

python demo/expansion/code/net2d.py
python hope.py -d demo/expansion/outputs/2D_Conv_AvePool/net.pt -o 8 -p 0,0
python demo/expansion/code/plot.py -d demo/expansion/outputs/2D_Conv_AvePool/npy/HOPE_[[0.0,0.0]]_8.npy -r 2

The results are saved in demo/expansion/outputs/{project_name}/figs/.

2. Interpretation of deep neural networks and function discovery

(a) Represent $y=\frac{x_1^2+x_2}{2}$ with implicit neural representation

python demo/discovery/code/train.py

Obtain local Taylor polynomials for (0,0), (0.5,0.5), (-0.5, -0.5)

python hope.py -d demo/discovery/outputs/discovery_{time}/model/best.pt -o 2 -s -p 0,0
python hope.py -d demo/discovery/outputs/discovery_{time}/model/best.pt -o 2 -s -p 0.5,0.5
python hope.py -d demo/discovery/outputs/discovery_{time}/model/best.pt -o 2 -s -p n0.5,n0.5

$$\begin{aligned} y&=0.50x_2+0.51x_1^2, \\ y&=0.38+0.50(x_1-0.5)+0.50(x_2-0.5)+0.50(x_1-0.5)^2\approx 0.50x_2+0.50x_1^2, \\ y&=-0.12-0.50(x_1+0.5)+0.50(x_2+0.5)+0.50(x_1+0.5)^2\approx 0.50x_2+0.50x_1^2. \end{aligned}$$

Expand the network on multiple reference inputs and find a global explanation

python global.py -d demo/discovery/outputs/discovery_{time}/model/best.pt -p 0,0 -o 4 -r 1 -n 30 -t 10

The local Taylor coefficients on reference point are marked in red, the average values of coefficients obtained from multiple reference points are marked in blue, and the coefficients that combine local and global properties are marked in black. One can easily find the function expressed by the neural network from the top coefficients. The final Taylor polynomial on reference point [0, 0] is $y=0.50x_2+0.51x_1^2$.

(b) Represent $y=0.5+x_1+0.6x_2x_3-0.8x_2x_4+x_1x_2x_3-x_3x_5$ with implicit neural representation

python demo/discovery/code/train1.py

Get the Taylor coefficients

python global.py -d demo/discovery/outputs/discovery1_3_{time}/model/best.pt -p 0,0,0,0,0 -o 3 -r 1 -n 5 -t 14

The final Taylor polynomial on reference point [0, 0, 0, 0, 0] is $y=0.50+1.00x_1+0.62x_2x_3-0.84x_2x_4+1.02x_1x_2x_3-1.04x_3x_5$.

3. Feature selection

(a) Train an MNIST digit classifier

python demo/heatmap/code/trainMNIST.py

(b) Separate it into ten equivalent single-output classifiers

python demo/heatmap/code/single_net.py -p demo/heatmap/outputs/MNIST_{time}

the results are saved in ''demo/heatmap/outputs/MNIST_{time}/model/SingleOutput''

(c) Heat maps by HOPE and perturbation-based method

python demo/heatmap/code/heatmaps.py -p demo/heatmap/outputs/MNIST_{time}

Citations

@article{xiao2024hope,
  title={HOPE: High-order Polynomial Expansion of Black-box Neural Networks},
  author={Xiao, Tingxiong and Zhang, Weihang and Cheng, Yuxiao and Suo, Jinli},
  journal={IEEE Transactions on Pattern Analysis and Machine Intelligence},
  year={2024},
  publisher={IEEE}
}

Contact

If you need any help or are looking for cooperation feel free to contact us. xtx22@mails.tsinghua.edu.cn