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            <h1 id="seo-header">【迁移】菜鸟博主对神经网络的简单理解</h1>
            
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                    Last updated on March 19, 2024 pm
                  
                
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                <p>博主最近在看临床预测模型，里面涉及到一些神经网络的相关知识，这里记录一下博主的简单理解。</p>
<h2 id="基本概念">基本概念</h2>
<ul>
<li><code>X</code>：一个nx维的输入张量，可以是nx=0的标量，nx=1的向量，nx=2的矩阵，或更高维的张量</li>
<li><code>Y</code>：一个ny维的输出张量，可以是nx=0的标量，nx=1的向量，nx=2的矩阵，或更高维的张量</li>
<li><code>f</code>：一个特定结构的<a target="_blank" rel="noopener" href="https://zhuanlan.zhihu.com/p/159305118">神经网络</a>，如简单的BP神经网络、复杂的深度神经网络<strong>DNN</strong>等，用于将X映射到Y</li>
<li><code>W</code>：一个nw维的张量，用来代表f中的所有权重，nw与f、X、Y有关，具体多少不用管</li>
<li><code>L</code>：损失函数，可以是交叉熵、残差平方等，用来计算f(X)与实际Y的差距</li>
</ul>
<p>一个简单的例子：假设<code>f</code>是DNN，<code>X</code>是一个256*256图片的像素矩阵，<code>Y</code>是一个3维分类向量，那么<code>f</code>就可以用于将一张特定的256*256图片<code>X0</code>映射到一个特定的<code>Y0(P(🐱), P(🐕), P(🐖))</code>，那么我们输出<code>(🐱,🐕,🐖)[which.max(Y0)]</code>，就可以对图片<code>X0</code>进行<code>(🐱,🐕,🐖)</code>的分类。</p>
<h2 id="对训练的理解">对训练的理解</h2>
<p>假设我们有一个训练好的权重<code>W</code>，那么我们就可以得到任意<code>X</code>下对实际<code>Y</code>的良好估计<code>f(X,W)</code>,有<code>L(f(X,W),Y)</code>相对较小。我们训练的目的，就是根据已有的数据集<code>D&#123;(X,Y)&#125;</code>，通过适当的拟合来找到这样一个权重<code>W</code>，它可以实现前面假设中的效果。</p>
<p>那么如何进行适当的拟合呢？一个简单的思路是梯度下降法。根据定义，我们知道<code>L</code>是<code>f(X)</code>和<code>Y</code>的函数，那么变换视角，<code>L</code>就是关于<code>W</code>的函数。假设学习率为<code>η</code>，初始权重为<code>W0</code>，对于<code>D</code>中的任意<code>(X0,Y0)</code>，我们可以求<code>ΔW0=-η∇L(W0,(X0,Y0),f)</code>。根据梯度的特性，我们可以知道对于更新后的权重<code>W1=W0+ΔW0</code>，有<code>L(W1,(X0,Y0),f)&lt;L(W0,(X0,Y0),f)</code>。在适当的参数下，通过不断循环上述过程，就能得到最终的<code>W</code>。</p>
<p>那么<code>∇L(W,f)</code>如何求呢？作为医学生，我们不必了解具体的数学原理来编程，只要构建好我们想要的网络结构<code>f</code>，<a target="_blank" rel="noopener" href="https://pytorch.org/">PyTorch</a>的<a target="_blank" rel="noopener" href="https://pytorch.org/docs/stable/autograd.html">自动求导</a>功能就可以自动帮我们计算出<code>∇L(W,f)</code>啦。</p>

                
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