ClawLab
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MathLab


			
				
					
						
						
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    <title>6 - 计算图</title>
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    <h1>👾 Day 6: 计算图</h1>
    
    <div class="module">
        <div class="module-title">1️⃣【技术债与演进动机】The Technical Debt & Evolution</div>
        手动推导梯度复杂易错，深度学习模型嵌套多层。计算图自动记录前向过程，反向自动求导。
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        <div class="module-title">2️⃣【直觉建立】Visual Intuition</div>
        想象一个流程图：数据从左到右经过各种运算节点，每个节点记录输入输出，反向时从右到左传递梯度。
        <div class="youtube">🎬 B 站搜索：<code>计算图 自动求导 原理</code></div>
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        <div class="module-title">3️⃣【符号解码字典】The Symbol Decoder</div>
        
        <div class="symbol-map"><strong>$x$</strong> → <code>input_tensor</code> (输入张量)</div>
        <div class="symbol-map"><strong>$y$</strong> → <code>output_tensor</code> (输出张量)</div>
        <div class="symbol-map"><strong>$\frac{\partial y}{\partial x}$</strong> → <code>gradient</code> (梯度)</div>
        <div class="symbol-map"><strong>$\mathcal{G}$</strong> → <code>computation_graph</code> (计算图结构)</div>
        
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        <div class="module-title">4️⃣【核心推导】The Math</div>
        
### 计算图结构

$$y = f(x) = f_L(f_{L-1}(\cdots f_1(x)\cdots))$$

### 链式法则

$$\frac{\partial y}{\partial x} = \frac{\partial f_L}{\partial u_{L-1}} \cdot \frac{\partial f_{L-1}}{\partial u_{L-2}} \cdots \frac{\partial f_1}{\partial x}$$

### 反向传播

前向：计算每一层的输出 $u_l = f_l(u_{l-1})$

反向：从后往前计算梯度

$$\frac{\partial L}{\partial u_l} = \frac{\partial L}{\partial u_{l+1}} \cdot \frac{\partial u_{l+1}}{\partial u_l}$$

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        <div class="module-title">5️⃣【工程优化点】The Optimization Bottleneck</div>
        计算图占用内存存储中间结果，大模型下显存爆炸。使用梯度检查点（checkpoint）节省内存。
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        <div class="module-title">6️⃣【今日靶机】The OJ Mission</div>
        <div class="warning">🎯 任务：<code>cd exercises/ && python3 day6_task.py</code></div>
        实现简易计算图的 forward 和 backward，验证链式法则的正确性。
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