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<a class="post-title-link" href="/2018/02/27/batch-normalization/" itemprop="url">batch normalization</a></h1>
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<h1 id="batch-normalization-accelerating-deep-network-training-by-reducing-internal-covariate-shift-2015">Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift 2015</h1>
<h2 id="abstract">abstract</h2>
<p>把训练过程中,网络中间层输入数据分布的改变(因为前层网络参数随着反向传播变化,导致本层网络的输入数据的分布发生变化)称为:"Internal Covariate Shift"<br>
为解决这个问题,提出BN结构,并使其称为网络模型中的一部分,对每一个输入batch做标准化。 - 使用BN层,可以提高学习率 - 参数初始化不再那么重要 - 有规则化作用,从而可以省去dropout层 使用bach-normalized networks的集成,在ImageNet上得到了超过人的分类准确率。 <span class="math display">\[1+1=2\]</span></p>
<h2 id="introduction">introduction</h2>
<p>使用mini-batch gradient descent 与 stochastic gradient descent相比的优点<br>
- 随着batch_size的增大,梯度估计会更准确 - batch能够并行化,从而使计算更高效。</p>
<p>当一个学习系统的输入分布变化时,称为covariate shift,以往使用domain adaption解决。<br>
internal covariate shift就是对系统输入到子系统输入的推广,作者提出batch normalization来解决这个问题,它能够标准化输入从而称为均值为<span class="math inline">\(0\)</span>、方差为<span class="math inline">\(1\)</span>的输入,而且,BN对梯度反向传播也有作用,减少了梯度对参数大小及初始值的依赖?</p>
<h2 id="towards-reducing-internal-covariate-shift">towards reducing internal covariate shift</h2>
<p>通过修改网络或修改优化算法的参数。<strong>看的不太懂</strong></p>
<h2 id="normalization-via-mini-batch-statics">normalization via mini-batch statics</h2>
<p>两个简化 1. 使用标准化替代白化,假设一层网络的输入为<span class="math inline">\(d\)</span>维向量,<span class="math inline">\(x=(x^{(1)},x^{(2)}......x^{(d)})\)</span> > <span class="math display">\[\hat x_k = \frac{x^{(k)}-\bold{E}[x^{(k)}]}{\sqrt{Var[x^{(k)}]}} \quad \text{标准化}\]</span> 然而标准化操作会改变一层的表达能力,如sigmoid函数,如果它的输入被标准化,那么它的输出只分布在sigmoid线性部分,即要保证插入的层能够表示identity 变换,由此,添加变换 > <span class="math display">\[y^{(k)}=\gamma^{(k)} x^{(k)}+\beta ^{(k)}\]</span> 当<span class="math inline">\(\gamma^{k}=Var[x^{(k)}]\)</span>,<span class="math inline">\(\beta^{(k)}=\bold{E}[x^{(k)}]\)</span>时,相当于identity变换(没有变换)<br>
2. 在mini-batch内计算均值与标准差</p>
<p>算法1 > <span class="math display">\[\mu_B \leftarrow \frac{1}{m}\sum_i^m x_i\quad\text{mini-batch mean}\]</span> (1) > <span class="math display">\[\sigma_B^2 \leftarrow \frac{1}{m}\sum_i^m (x_i - \mu_B)^2\quad\text{mini-batch variance}\]</span>(2) > <span class="math display">\[\hat x_i \leftarrow \frac{x_i-\mu_B}{\sqrt{\sigma_B^2 +\epsilon}}\quad\text{normalize}\]</span>(3) > <span class="math display">\[y_i \leftarrow \gamma\hat x_i+\beta \quad \text{scale and shift}\]</span> (4) 求导部分 ### training and inference with batch-normalised networks 预测时,使用整个训练集的均值和方差 > <span class="math display">\[\hat x=\frac{x-\bold{E}(x)}{\sqrt{Var[x]+\epsilon}}\]</span> 在算法2中有<span class="math inline">\(\bold{E[x]}\)</span>,<span class="math inline">\(Var[x]\)</span>的计算公式 > <span class="math display">\[\bold{E}[x]\leftarrow \bold{E}_B[\mu_B]\]</span> > <span class="math display">\[Var[x]\leftarrow \frac{m}{m-1}\bold{E}_B[\sigma_B^2]\]</span> ### batch-normalized convolutional networks 在非线性激活函数之前加BN层,这样添加能够输出更稳定的响应。 对于<span class="math inline">\(Wx+b\)</span> BN层前面的偏置参数的作用,被BN层归一化时抵消掉了</p>
<h3 id="batch-normalization-enables-higher-leraning-rate">batch-normalization enables higher leraning rate</h3>
<h1 id="tips">tips</h1>
<p>没有找到<span class="math inline">\(\gamma,\beta\)</span>是如何初始化的<br>
有论文专门探讨dropout与bn层结合使用效果差的原因,并提出解决方法。</p>
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