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This introduces a suggestion of mathematical notation protocol for machine learning.
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# Suggested Notation for Machine Learning ## Authors - **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)](mailto:[email protected]), [Tao Luo (罗涛)](mailto:[email protected]), [Zheng Ma (马征)](mailto:[email protected]), [Yaoyu Zhang (张耀宇)](mailto:[email protected])** - _Initial work_ ## Introduction > 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 - [Suggested Notation for Machine Learning](#suggested-notation-for-machine-learning) - [Authors](#authors) - [Introduction](#introduction) - [Tabel of Contents](#tabel-of-contents) - [Dataset](#dataset) - [Function](#function) - [Loss function](#loss-function) - [Activation function](#activation-function) - [Two-layer neural network](#two-layer-neural-network) - [General deep neural network](#general-deep-neural-network) - [Complexity](#complexity) - [Training](#training) - [Fourier Frequency](#fourier-frequency) - [Convolution](#convolution) - [Notation table](#notation-table) - [L-layer neural network](#l-layer-neural-network) - [Acknowledgements](#acknowledgements) ## Dataset Dataset $S=\\{ \mathbf{z}_i \\}^n\_{i=1}=\\{(\mathbf{x}_i, \mathbf{y}_i) \\}^n\_{i=1}$ is sampled from a distribution $\mathcal{D}$ over a domain $\mathcal{Z} = \mathcal{X} \times \mathcal{Y}$. * $\mathcal{X}$ is the instances domain (a set) * $\mathcal{Y}$ is the label domain (a set) * $\mathcal{Z}=\mathcal{X}\times\mathcal{Y}\$ is the example domain (a set) Usually, $\mathcal{X}$ is a subset of $\mathbb{R}^d$ and $\mathcal{Y}$ is a subset of $\mathbb{R}^{d_\text{o}}$, where $d$ is the input dimension, $d_\text{o}$ is the ouput dimension. $n=$#$S$ is the number of samples. Wihout specification, $S$ and $n$ are for the training set. ## Function A hypothesis space is denoted by $\mathcal{H}$. A hypothesis function is denoted by $f_{\mathbf{\theta}}(\mathbf{x})\in\mathcal{H}$ or $f(\mathbf{x};\mathbf{\theta})$ with $f_{\mathbf{\theta}}:\mathcal{X}\to\mathcal{Y}$. $\mathbf{\theta}$ denotes the set of parameters of $f_{\mathbf{\theta}}$. If there exists a target function, it is denoted by $f^\*$ or $f^\*:\mathcal{X}\to\mathcal{Y}$ satisfying $\mathbf{y}_i=f^*(\mathbf{x}_i)$ for $i=1,\dots,n$. ## Loss function A loss function, denoted by $\ell:\mathcal{H}\times\mathcal{Z}\to\mathbb{R}_{+}:=[0,+\infty)$ measures the difference between a predicted label and a true label, e.g., - $L^2$ loss: $\ell(f_{\mathbf{\theta}},\mathbf{z})=(f_{\mathbf{\theta}}(\mathbf{x})-\mathbf{y})^2$, where $\mathbf{z}=(\mathbf{x},\mathbf{y})$. $\ell(f_{\mathbf{\theta}},\mathbf{z})$ can also be written as $\ell(f_{\mathbf{\theta}},\mathbf{y}))$ for convenience. Empirical risk or training loss for a set $S=\\{(\mathbf{x}_i,\mathbf{y}_i)\\}^n\_{i=1}$ is denoted by $L_S(\mathbf{\theta})$ or $L_n(\mathbf{\theta})$ or $R_S(\mathbf{\theta})$ or $R_n(\mathbf{\theta})$, $$ L_S(\mathbf{\theta})=\frac{1}{n}\sum^n_{i=1}\ell(f_{\mathbf{\theta}}(\mathbf{x}_i),\mathbf{y}_i). $$ The population risk or expected loss is denoted by $L_{\mathcal{D}}$ or $R_{\mathcal{D}}$, $$ L_{\mathcal{D}}(\mathbf{\theta})=\mathbb{E}_{\mathcal{D}}\ell(f_{\mathbf{\theta}}(\mathbf{x}),\mathbf{y})), $$ where $\mathbf{z}=(\mathbf{x},\mathbf{y})$ follow
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matched fp:b17081f5563fb64d, llm:Repository introduces suggested mathematical notation protocol for machine learning (description). Language: TeX.