Understanding the difficulty of training deep feedforward neural networks (Xavier)

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转自：http://blog.csdn.net/shuzfan/article/details/51338178
“Xavier”初始化方法是一种很有效的神经网络初始化方法，方法来源于2010年的一篇论文

《Understanding the difficulty of training deep feedforward neural networks》

可惜直到近两年，这个方法才逐渐得到更多人的应用和认可。

为了使得网络中信息更好的流动，每一层输出的方差应该尽量相等。

基于这个目标，现在我们就去推导一下：每一层的权重应该满足哪种条件。

文章先假设的是线性激活函数，而且满足0点处导数为1，即 

现在我们先来分析一层卷积： 
 
其中ni表示输入个数。

根据概率统计知识我们有下面的方差公式： 

特别的，当我们假设输入和权重都是0均值时（目前有了BN之后，这一点也较容易满足），上式可以简化为： 

进一步假设输入x和权重w独立同分布，则有： 

于是，为了保证输入与输出方差一致，则应该有： 

对于一个多层的网络，某一层的方差可以用累积的形式表达： 

特别的，反向传播计算梯度时同样具有类似的形式： 

综上，为了保证前向传播和反向传播时每一层的方差一致，应满足：

但是，实际当中输入与输出的个数往往不相等，于是为了均衡考量，最终我们的权重方差应满足：

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学过概率统计的都知道 [a,b] 间的均匀分布的方差为： 

因此，Xavier初始化的实现就是下面的均匀分布：

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下面，我们来看一下caffe中具体是怎样实现的，代码位于include/caffe/filler.hpp文件中。

<code class="language-C++ hljs haskell has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: "Source Code Pro", monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-title" style="box-sizing: border-box;">template</span> <typename <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype</span>>
<span class="hljs-class" style="box-sizing: border-box;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">class</span> <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">XavierFiller</span> : public <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Filler</span><<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype</span>> {public:explicit <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">XavierFiller</span><span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">const</span> <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">FillerParameter</span>& <span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">param</span>)</span>: <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Filler</span><<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype</span>><span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">param</span>)</span> {}virtual void <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Fill</span><span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Blob</span><<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype</span>>* <span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">blob</span>)</span> {<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">CHECK</span><span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">blob</span>-><span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">count</span>()</span>);int fan_in = blob->count<span class="hljs-container" style="box-sizing: border-box;">()</span> / blob->num<span class="hljs-container" style="box-sizing: border-box;">()</span>;int fan_out = blob->count<span class="hljs-container" style="box-sizing: border-box;">()</span> / blob->channels<span class="hljs-container" style="box-sizing: border-box;">()</span>;<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype</span> n = fan_in;  // default to fan_inif <span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">this</span>-><span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">filler_param_</span>.<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">variance_norm</span>()</span> ==<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">FillerParameter_VarianceNorm_AVERAGE</span>) {n = <span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">fan_in</span> + <span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">fan_out</span>)</span> / <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype</span><span class="hljs-container" style="box-sizing: border-box;">(2)</span>;} else if <span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">this</span>-><span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">filler_param_</span>.<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">variance_norm</span>()</span> ==<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">FillerParameter_VarianceNorm_FAN_OUT</span>) {n = fan_out;}<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype</span> scale = sqrt<span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype(3)</span> / <span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">n</span>)</span>;caffe_rng_uniform<<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Dtype</span>><span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">blob</span>-><span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">count</span>()</span>, -scale, scale,blob->mutable_cpu_data<span class="hljs-container" style="box-sizing: border-box;">()</span>);<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">CHECK_EQ</span><span class="hljs-container" style="box-sizing: border-box;">(<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">this</span>-><span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">filler_param_</span>.<span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">sparse</span>()</span>, -1)<< "<span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Sparsity</span> not supported by this <span class="hljs-type" style="box-sizing: border-box; color: rgb(102, 0, 102);">Filler</span>.";}
};</span></code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li></ul>

由上面可以看出，caffe的Xavier实现有三种选择

（1） 默认情况，方差只考虑输入个数： 

（2） FillerParameter_VarianceNorm_FAN_OUT，方差只考虑输出个数： 

（3） FillerParameter_VarianceNorm_AVERAGE，方差同时考虑输入和输出个数： 

之所以默认只考虑输入，我个人觉得是因为前向信息的传播更重要一些

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