Suggested Notation for Machine Learning
This introduces a suggestion of mathematical notation protocol for machine learning.
The field of machine learning is evolving rapidly in recent years. Communication between different researchers and research groups becomes 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 notation are mentioned and more notation 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.
Tabel of Contents
Dataset
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. Wihout specification, 
and 
are for the training set.
Function
Hypothesis space is denoted by 
. 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 
.
Loss function
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 
,
Without ambiguity, 
is also used for 
.
The population risk or expected loss is denoted by
where follows the distribution .
Activation function
Activation function is denoted by 
.
Example 1. Some commonly used activation functions are
Two-layer neural network
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
General deep neural network
The counting of the layer number excludes the input layer. A -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,
Complexity
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
Training
The Gradient Descent is oftern denoted by GD. THe 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 
.
Fourier Frequency
The discretized frequency is denoted by 
, and the continuous frequency is denoted by 
.
Convolution
The convolution operation is denoted by 
.
Notation table
| 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{x}i |
|
convolution operation | * |
L-layer neural network
| 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 |
Acknowledgements
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).