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This introduces a suggestion of mathematical notation protocol for machine learning.

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**Beijing Academy of Artificial Intelligence (北京智源人工智能研究院)****Peking University (北京大学)****Shanghai Jiao Tong University (上海交通大学)**-
**Zhi-qin John Xu (许志钦), Tao Luo (罗涛), Zheng Ma (马征), Yaoyu Zhang (张耀宇)**-*Initial work*

This introduces a suggestion of mathematical notation protocol for machine learning.

The field of machine learning has evolved rapidly in recent years. Communication between different researchers and research groups has become increasingly important. A key challenge for communication arises from inconsistent notation usages among different papers. This proposal suggests a standard for commonly used mathematical notation for machine learning. In this first version, only some notations are mentioned and more notations are left to be done. This proposal will be regularly updated based on the progress of the field. We look forward to more suggestions to improve this proposal in future versions.

Dataset is sampled from a distribution over a domain .

- is the instances domain (a set)
- is the label domain (a set)
- is the example domain (a set)

Usually, is a subset of and is a subset of , where is the input dimension, is the ouput dimension.

is the number of samples. Without specification, and are for the training set.

A hypothesis space is denoted by . A hypothesis function is denoted by or with .

denotes the set of parameters of .

If there exists a target function, it is denoted by or satisfying for .

A loss function, denoted by , measures the difference between a predicted label and a true label, e.g.,

- loss: , where . can also be written as for convenience.

Empirical risk or training loss for a set is denoted by or or or ,

The population risk or expected loss is denoted by or ,

where follows the distribution .

An activation function is denoted by .

**Example 1**. Some commonly used activation functions are

The neuron number of the hidden layer is denoted by , The two-layer neural network is

where is the activation function, is the input weight, is the output weight, is the bias term. We denote the set of parameters by

The counting of the layer number excludes the input layer. An -layer neural network is denoted by

where , , , , is a scalar function and "" means entry-wise operation. We denote the set of parameters by

This can also be defined recursively,

The VC-dimension of a hypothesis class is denoted as VCdim().

The Rademacher complexity of a hypothesis space on a sample set is denoted by or . The complexity is random because of the randomness of . The expectation of the empirical Rademacher complexity over all samples of size is denoted by

Gradient Descent is often denoted by GD. Stochastic Gradient Descent is often denoted by SGD.

A batch set is denoted by and the batch size is denoted by .

The learning rate is denoted by .

The discretized frequency is denoted by , and the continuous frequency is denoted by .

The convolution operation is denoted by .

| symbol | meaning | Latex | simplied | | ------------------------------------------------------------------------------------------------ | ----------------------------------------------------- | ------------------ | --------------------- | | | input |

\bm{x}|

\mathbf{x}| | | output, label |

\bm{y}|

\vy| | | input dimension |

d| | | | output dimension |

d_{\rm o}| | | | number of samples |

n| | | instances domain (a set) |

\mathcal{X}|

\fX| | | labels domain (a set) |

\mathcal{Y}|

\fY| | | example domain |

\mathcal{Z}|

\fZ| | | hypothesis space (a set) |

\mathcal{H}|

\mathcal{H}| | | a set of parameters |

\bm{\theta}|

\mathbf{\theta}| | | hypothesis function |

\f_{\bm{\theta}}|

f_{\mathbf{\theta}}| | or | target function |

f, f^*| | | loss function |

\ell| | | distribution of |

\mathcal{D}|

\fD| | | sample set | | , , , | empirical risk or training loss | | | population risk or expected loss | | | activation function |

\sigma| | | input weight |

\bm{w}_j|

\mathbf{w}_j| | | output weight |

a_j| | | bias term |

b_j| | or | neural network |

f_{\bm{\theta}}|

f_{\mathbf{\theta}}| | | two-layer neural network | | ) | VC-dimension of | | , | Rademacher complexity of on | | | Rademacher complexity over samples of size | | | gradient descent | | | stochastic gradient descent | | | a batch set |

B| | | | batch size |

b| | | | learning rate |

\eta| | | discretized frequency |

\bm{k}|

\mathbf{k}| | | continuous frequency |

\bm{\xi}|

\mathbf{xi}| | | | convolution operation |

*|

| symbol | meaning | Latex | simplied | | --------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------- | -------------- | ------------------ | | | input dimension |

d| | | | output dimension |

d_{\rm o}| | | | the number of -th layer neuron, , |

m_l| | | the -th layer weight |

\bm{W}^{[l]}|

\mathbf{W}^{[l]}| | | the -th layer bias term |

\bm{b}^{[l]}|

\mathbf{b}^{[l]}| | | entry-wise operation |

\circ| | | activation function |

\sigma| | | , parameters |

\bm{\theta}|

\mathbf{\theta}| | | | | | , -th layer output | | | , -layer NN |

Please cite this repository in your publications if it helps your research.

@misc{beijing2020Suggested, title = {Suggested Notation for Machine Learning}, author = {Beijing Academy of Artificial Intelligence}, howpublished = {\url{https://github.com/Mayuyu/suggested-notation-for-machine-learning}}, year=2020 }

Chenglong Bao (Tsinghua), Zhengdao Chen (NYU), Bin Dong (Peking), Weinan E (Princeton), Quanquan Gu (UCLA), Kaizhu Huang (XJTLU), Shi Jin (SJTU), Jian Li (Tsinghua), Lei Li (SJTU), Tiejun Li (Peking), Zhenguo Li (Huawei), Zhemin Li (NUDT), Shaobo Lin (XJTU), Ziqi Liu (CSRC), Zichao Long (Peking), Chao Ma (Princeton), Chao Ma (SJTU), Yuheng Ma (WHU), Dengyu Meng (XJTU), Wang Miao (Peking), Pingbing Ming (CAS), Zuoqiang Shi (Tsinghua), Jihong Wang (CSRC), Liwei Wang (Peking), Bican Xia (Peking), Zhouwang Yang (USTC), Haijun Yu (CAS), Yang Yuan (Tsinghua), Cheng Zhang (Peking), Lulu Zhang (SJTU), Jiwei Zhang (WHU), Pingwen Zhang (Peking), Xiaoqun Zhang (SJTU), Chengchao Zhao (CSRC), Zhanxing Zhu (Peking), Chuan Zhou (CAS), Xiang Zhou (cityU).