245 lines
9.4 KiB
TeX
245 lines
9.4 KiB
TeX
% Homework Template
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\documentclass[a4paper]{article}
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\usepackage{ctex}
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\usepackage{amsmath, amssymb, amsthm}
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\usepackage{moreenum}
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\usepackage{mathtools}
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\usepackage{url}
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\usepackage{bm}
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\usepackage{enumitem}
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\usepackage{graphicx}
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\usepackage{subcaption}
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\usepackage{booktabs} % toprule
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\usepackage[mathcal]{eucal}
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\usepackage[thehwcnt = 1]{iidef}
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\usepackage{listings}
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\usepackage[x11names]{xcolor}
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\usepackage{float}
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\usepackage[colorlinks, linkcolor=black, anchorcolor=green, citecolor=blue]{hyperref}
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\DeclareMathOperator{\arctanh}{arctanh}
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% \DeclareMathOperator{\diag}{diag}
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\setenumerate[1]{label=(\arabic{*})}
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\setenumerate[2]{label=\arabic{*})}
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\definecolor{codekeyword}{RGB}{171, 0, 216}
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\definecolor{codetypename}{RGB}{29, 37, 251}
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\definecolor{codevariable}{RGB}{10, 23, 126}
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\definecolor{codestring}{RGB}{157, 0, 25}
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\definecolor{codecomment}{RGB}{31, 129, 19}
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\newfontfamily\cascadia[Ligatures=ResetAll]{Cascadia Code}
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% \newfontfamily\codefont[Ligatures=ResetAll]{Cascadia Code}
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\newfontfamily\codefont[Ligatures=ResetAll]{Fira Code}[Contextuals={Alternate}]
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% To enable ligature in listing, go check lstfiracode's github page and copy firacodestyle's settings.
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\lstset{
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basicstyle = \small\codefont,
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% ---
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tabsize = 4,
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showstringspaces = false,
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numbers = left,
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numberstyle = \cascadia,
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% ---
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breaklines = true,
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captionpos = t,
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% ---
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frame = l,
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flexiblecolumns,
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columns = fixed,
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}
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\thecourseinstitute{清华大学电子工程系}
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\thecoursename{\textbf{媒体与认知} \space 课堂2}
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\theterm{2023-2024学年春季学期}
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\hwname{作业}
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\begin{document}
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\courseheader
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% 请在YOUR NAME处填写自己的姓名
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\name{高艺轩}
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\vspace{3mm}
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\centerline{\textbf{\Large{理论部分}}}
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\section{单选题（15分）}
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% 请在？处填写答案
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\subsection{\underline{B}}
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\subsection{\underline{A}}
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\subsection{\underline{B}}
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\subsection{\underline{A}}
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\subsection{\underline{B}}
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\section{计算题（15 分）}
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\subsection{设隐含层为$\mathbf{z}=\mathbf{W}^T\mathbf{x}+\mathbf{b}$，其中$\mathbf{x}\in R^{(m \times 1)}$，$\mathbf{z}\in R^{(n\times 1)}$，$\mathbf{W}\in R^{(m\times n)}$，$\mathbf{b} \in R^{(n\times 1)}$均为已知，其激活函数如下：
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$$\mathbf{y}=\delta(\mathbf{z})=tanh(\mathbf{z})$$
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tanh表示双曲正切函数。若训练过程中的目标函数为L，且已知L对$\mathbf{y}$的导数 $\frac{\partial L}{\partial \mathbf{y}}=[\frac{\partial L}{\partial y_1},\frac{\partial L}{\partial y_2},...,\frac{\partial L}{\partial y_n}]^T$和$\mathbf{y}=[y_1,y_2,...,y_n]^T$的值。
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}
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\subsubsection{请使用$\mathbf{y}$表示出$\frac{\partial \mathbf{y}^T}{\partial \mathbf{z}}$, 这里的$\mathbf{y}^T$ 为行向量。
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}
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\begin{proof}[解]
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首先，对$i \neq j$，$\dfrac{\partial y_i}{\partial z_j} = 0$。
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同时$y_i = \tanh(z_i) = \tanh(\arctanh(y_i))$，因此
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\[\frac{\partial y_i}{\partial z_i} = 1 - \tanh^2(z_i) = 1 - y_i^2\]
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因此
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\[\dfrac{\partial \boldsymbol{y}^\mathrm{T}}{\partial \boldsymbol{z}} = \diag\{1 - y_1^2, 1 - y_2^2, \dots, 1 - y_n^2\} \qedhere\]
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\end{proof}
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\subsubsection{请使用$\mathbf{y}$和$\frac{\partial L}{\partial \mathbf{y}}$表示$\frac{\partial L}{\partial \mathbf{x}}$，$\frac{\partial L}{\partial \mathbf{W}}$，$\frac{\partial L}{\partial \mathbf{b}}$。
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}
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提示:$\frac{\partial L}{\partial \mathbf{x}}$，$\frac{\partial L}{\partial \mathbf{W}}$，$\frac{\partial L}{\partial \mathbf{b}}$与x,W,b具有相同维度。
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\begin{proof}[解]
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由链式法则
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\[\frac{\partial L}{\partial \boldsymbol{x}} = \frac{\partial \boldsymbol{z}^\mathrm{T}}{\partial \boldsymbol{x}} \frac{\partial \boldsymbol{y}^\mathrm{T}}{\partial \boldsymbol{z}} \frac{\partial L}{\partial \boldsymbol{y}} = W \diag\{1 - y_1^2, 1 - y_2^2, \dots, 1 - y_n^2\} \frac{\partial L}{\partial \boldsymbol{y}}\]
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对于$\dfrac{\partial L}{\partial W}$，
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\[\frac{\partial \boldsymbol{z}^T}{\partial W} = \begin{bmatrix}
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\boldsymbol{x} & \boldsymbol{x} & \cdots & \boldsymbol{x}
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\end{bmatrix}_{m \times n}\]
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\begin{align*}
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\frac{\partial L}{\partial W} & = \frac{\partial \boldsymbol{z}^\mathrm{T}}{\partial W} \frac{\partial \boldsymbol{y}^\mathrm{T}}{\partial \boldsymbol{z}} \frac{\partial L}{\partial \boldsymbol{y}}\\
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& = \begin{bmatrix}
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\boldsymbol{x} & \boldsymbol{x} & \cdots & \boldsymbol{x}
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\end{bmatrix}_{m \times n} \diag\{1 - y_1^2, 1 - y_2^2, \dots, 1 - y_n^2\} \frac{\partial L}{\partial \boldsymbol{y}}
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\end{align*}
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对于$\dfrac{\partial L}{\partial \boldsymbol{b}}$，由链式法则
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\[\frac{\partial L}{\partial \boldsymbol{b}} = \frac{\partial \boldsymbol{z}^\mathrm{T}}{\partial \boldsymbol{b}} \frac{\partial \boldsymbol{y}^\mathrm{T}}{\partial \boldsymbol{z}} \frac{\partial L}{\partial \boldsymbol{y}} = I_n \frac{\partial \boldsymbol{y}^\mathrm{T}}{\partial \boldsymbol{z}} \frac{\partial L}{\partial \boldsymbol{y}} = \diag\{1 - y_1^2, 1 - y_2^2, \dots, 1 - y_n^2\} \frac{\partial L}{\partial \boldsymbol{y}} \qedhere\]
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\end{proof}
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\vspace{6mm}
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\centerline{\textbf{\Large{编程部分}}}
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\vspace{3mm}
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% 请根据是否选择自选课题的情况选择“编程作业报告”或“自选课题开题报告”中的一项完成
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\section{编程作业报告}
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% 请在此处完成编程作业报告
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完成后的代码也可以在 \href{https://git.unlockableworld.com/unlockable/MediaNCognition}{\url{https://git.unlockableworld.com/unlockable/MediaNCognition}}中找到。
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\begin{enumerate}
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\item 使用默认配置进行训练和测试。
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\begin{enumerate}
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\item 训练模型。
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输入：
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\lstinputlisting{codes/1.1.in.txt}
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输出：
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\lstinputlisting{codes/1.1.out.txt}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.9\linewidth]{img/1default_train.png}
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\end{figure}
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\item 测试模型。
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输入：
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\lstinputlisting{codes/1.2.in.txt}
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输出：
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\lstinputlisting{codes/1.2.out.txt}
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\end{enumerate}
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\item 调整参数、使用Adam优化器训练并测试。
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\begin{enumerate}
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\item 训练模型。
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输入：
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\lstinputlisting{codes/2.1.in.txt}
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输出：
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\lstinputlisting{codes/2.1.out.txt}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.9\linewidth]{img/2adam_optim.png}
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\end{figure}
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\item 测试性能。
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输入：
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\lstinputlisting{codes/2.2.in.txt}
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输出：
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\lstinputlisting{codes/2.2.out.txt}
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\end{enumerate}
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\item 使用效果最佳的模型测试。
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经过简单的尝试，发现使用
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\lstinputlisting{codes/self_train.in.txt}
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可以使测试集准确率达到88.8\%，有略微的提升。训练的loss曲线：
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\begin{figure}[H]
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\centering
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\includegraphics[width=.9\linewidth]{img/3found_best.png}
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\end{figure}
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使用它进行预测：
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\begin{figure}[H]
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\centering
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\begin{subfigure}[b]{.3\linewidth}
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\includegraphics[width=\linewidth]{img/predict/predict01.png}
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\subcaption{预测：A}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{.3\linewidth}
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\includegraphics[width=\linewidth]{img/predict/predict02.png}
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\subcaption{预测：B}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{.3\linewidth}
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\includegraphics[width=\linewidth]{img/predict/predict03.png}
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\subcaption{预测：M}
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\end{subfigure}
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\begin{subfigure}[b]{.3\linewidth}
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\includegraphics[width=\linewidth]{img/predict/predict04.png}
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\subcaption{预测：R}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{.3\linewidth}
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\includegraphics[width=\linewidth]{img/predict/predict05.png}
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\subcaption{预测：M}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{.3\linewidth}
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\includegraphics[width=\linewidth]{img/predict/predict06.png}
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\subcaption{预测：O}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{.3\linewidth}
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\includegraphics[width=\linewidth]{img/predict/predict07.png}
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\subcaption{预测：B}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{.3\linewidth}
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\includegraphics[width=\linewidth]{img/predict/predict08.png}
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\subcaption{预测：W}
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\end{subfigure}
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\hfill
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\end{figure}
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\item 遇到的问题及解决方法
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\begin{enumerate}
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\item 代码中对灰度图像的矩阵进行标准化时，\lstinline{numpy}显示不能对\lstinline{NumpyGenericArray}进行对\lstinline{float}的\lstinline{/}操作。改用\lstinline{np.div()}解决了这个问题。
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\item 在利用训练好的模型进行预测时，发现自己找到的大部分模型都预测错误；最后与训练集的图片进行了对比，发现主要问题是裁切字母时留下了过大的边距，导致模型不能正确理解输入。重新裁剪边框后，得到正确的结果。
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\end{enumerate}
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\item 建议：希望下次发布作业代码可以利用清华的git。
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\end{enumerate}
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% \section{自选课题开题报告}
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% 请在此处介绍自选课题
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\end{document}
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%%% Local Variables:
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%%% mode: late\rvx
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%%% TeX-master: t
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%%% End:
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