Imporve number of softmax-ce

5_nn/3-softmax_ce.ipynb

@@ -143,7 +143,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## 2. 推导过程\n",
+    "## 3. 推导过程\n",
     "\n",
     "首先，我们要明确一下我们要求什么，我们要求的是我们的$loss$对于神经元输出($z_i$)的梯度，即：\n",
     "\n",
@@ -158,14 +158,26 @@
     "$$\n",
     "\n",
     "有个人可能有疑问了，这里为什么是$a_j$而不是$a_i$，这里要看一下$softmax$的公式了，因为$softmax$公式的特性，它的分母包含了所有神经元的输出，所以，对于不等于$i$的其他输出里面，也包含着$z_i$，所有的$a$都要纳入到计算范围中，并且后面的计算可以看到需要分为$i = j$和$i \\ne j$两种情况求导。\n",
-    "\n",
-    "### 2.1 针对$a_j$的偏导\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 3.1 针对$a_j$的偏导\n",
     "\n",
     "$$\n",
     "\\frac{\\partial C}{\\partial a_j} = \\frac{(\\partial -\\sum_j y_j ln a_j)}{\\partial a_j} = -\\sum_j y_j \\frac{1}{a_j}\n",
     "$$\n",
-    "\n",
-    "### 2.2 针对$z_i$的偏导\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 3.2 针对$z_i$的偏导\n",
     "\n",
     "如果 $i=j$ :\n",
     "\n",
@@ -188,8 +200,14 @@
     "$$\n",
     "(\\frac{u}{v})' = \\frac{u'v - uv'}{v^2} \n",
     "$$\n",
-    "\n",
-    "### 2.3 整体的推导\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 3.3 整体的推导\n",
     "\n",
     "\\begin{eqnarray}\n",
     "\\frac{\\partial C}{\\partial z_i} & = & (-\\sum_j y_j \\frac{1}{a_j} ) \\frac{\\partial a_j}{\\partial z_i} \\\\\n",
@@ -234,7 +252,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## 3. 问题\n",
+    "## 4. 问题\n",
     "如何将本节所讲的softmax，交叉熵代价函数应用到上节所讲的BP方法中？"
    ]
   },