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Cartman0/逆誤差伝播.ipynb

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逆誤差伝播
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逆誤差伝播.ipynb
{
"cells": [
{
"metadata": {},
"cell_type": "markdown",
"source": "# 逆誤差伝播"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## 復習"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### ベクトルの内積"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$\\vec{x} = \\begin{pmatrix}\nx_1 \\\\\nx_2 \\\\\nx_3\n\\end{pmatrix}$, $\\vec{w} = \\begin{pmatrix}\nw_1 \\\\\nw_2 \\\\\nw_3\n\\end{pmatrix}$\n\n\\begin{eqnarray}\n\\vec{w} \\cdot \\vec{x}\n&=& w_1 x_1 + w_2 x_2 + w_3 x_3 \\text{ (scalar)}\\\\\n&=& \\sum_i w_i x_i\\\\\n\\end{eqnarray}\n\n$\\vec{w}$が行ベクトルであることを意識すると，\n\n\\begin{eqnarray}\n\\vec{w}^T \\cdot \\vec{x}\n&=& \n\\begin{pmatrix}\nw_1 & w_2 & w_3\n\\end{pmatrix}\n\\begin{pmatrix}\nx_1 \\\\\nx_2 \\\\\nx_3\n\\end{pmatrix} \\\\\n&=& w_1 x_1 + w_2 x_2 + w_3 x_3 \n\\end{eqnarray}\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 行列積"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "\\begin{eqnarray}\nW_{2 \\times 3} \n&=& \\begin{pmatrix}\nw_{11} & w_{12} & w_{13} \\\\\nw_{21} & w_{22} & w_{23} \\\\\n\\end{pmatrix}\n&=&\n\\begin{pmatrix}\n\\vec{w_{1・}} \\\\\n\\vec{w_{2・}} \\\\\n\\end{pmatrix}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nX_{3 \\times 2} \n&=& \\begin{pmatrix}\nx_{11} & x_{12} \\\\\nx_{21} & x_{22} \\\\\nx_{31} & x_{32}\n\\end{pmatrix}\n&=&\n\\begin{pmatrix}\n\\vec{x_{\\cdot 1}} & \\vec{x_{\\cdot 2}} \\\\\n\\end{pmatrix}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{2 \\times 3} X_{3 \\times 2} \n&=& \n\\begin{pmatrix}\nw_{11} & w_{12} & w_{13} \\\\\nw_{21} & w_{22} & w_{23} \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nx_{11} & x_{12} \\\\\nx_{21} & x_{22} \\\\\nx_{31} & x_{32}\n\\end{pmatrix} \\\\\n&=&\n\\begin{pmatrix}\nw_{11}x_{11} + w_{12}x_{21} + w_{13}x_{31} & w_{11}x_{12} + w_{12}x_{22} + w_{13} x_{32}\\\\\nw_{21}x_{11}+w_{22}x_{21} + w_{23}x_{31} & w_{21}x_{12}+w_{22}x_{22} + w_{23}x_{32}\\\\\n\\end{pmatrix}\\\\\n&=&\n\\begin{pmatrix}\n\\sum_{i} w_{1i}x_{i1} & \\sum_{i} w_{1i}x_{i2} \\\\\n\\sum_{i} w_{2i}x_{i1} & \\sum_{i} w_{2i}x_{i2}\\\\\n\\end{pmatrix}\\\\\n&=& \n\\begin{pmatrix}\n\\vec{w_{1・}} \\\\\n\\vec{w_{2・}} \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n\\vec{x_{\\cdot 1}} & \\vec{x_{\\cdot 2}} \\\\\n\\end{pmatrix}\\\\\n&=& \n\\begin{pmatrix}\n\\vec{w_{1・}} \\cdot \\vec{x_{\\cdot 1}} & \\vec{w_{1・}} \\cdot \\vec{x_{\\cdot 2}} \\\\\n\\vec{w_{2・}} \\cdot \\vec{x_{\\cdot 1}} & \\vec{w_{2・}} \\cdot \\vec{x_{\\cdot 2}} \\\\\n\\end{pmatrix}\\\\\n\\end{eqnarray}\n"
},
{
"metadata": {
"trusted": false
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"cell_type": "code",
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},
{
"metadata": {},
"cell_type": "markdown",
"source": "## ニューラルネット"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "入力値に対して重みをかけバイアスを足して出力を表現"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 1入力バイアスあり1出力"
},
{
"metadata": {
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"trusted": false
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"cell_type": "code",
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{
"metadata": {},
"cell_type": "markdown",
"source": "### biasは定数項を使って表せる"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$$\nw_b \\cdot 1 = b\n$$\n"
},
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},
{
"metadata": {},
"cell_type": "markdown",
"source": "## 順伝播（forward）の計算"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 1入力1出力（バイアスなし）のネットワーク"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$$\no = wx \n$$\n\n"
},
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"outputs": [
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"source": "### 3入力1出力に拡張"
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"metadata": {},
"cell_type": "markdown",
"source": "ここで，活性化関数（非線形関数）をシグモイド関数にすると，\n\n$$\ng(x) = \\frac{1}{1 + \\exp{(-x)}}\n$$\n\n$$\no_2 \n= g(o)\n= \\frac{1}{1 + \\exp{(-o)}}\\\\\n= \\frac{1}{1 + \\exp{\\left\\{-(w_1x_1 + w_2 x_2 + w_3 x_3)\\right\\}}}\n$$"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 1入力3出力"
},
{
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"outputs": [
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"source": "### 3入力3出力"
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"source": "$$\n\\mathbf{W} = \\begin{pmatrix}\nw_{11} & w_{12} & w_{13} \\\\\nw_{21} & w_{22} & w_{23} \\\\\nw_{31} & w_{32} & w_{33} \\\\\n\\end{pmatrix}\n= \\begin{pmatrix}\n\\vec{w_{1\\cdot}}\\\\\n\\vec{w_{2 \\cdot}}\\\\\n\\vec{w_{3 \\cdot}} \\\\\n\\end{pmatrix}\n$$\n\n$$\n\\vec{x} = \n\\begin{pmatrix}\nx_{1} \\\\\nx_{2} \\\\\nx_{3} \\\\\n\\end{pmatrix}\n$$\n\n$$\n\\vec{o}\n=\n\\mathbf{W} \\vec{x}\n= \\begin{pmatrix}\n\\vec{w_{1\\cdot}}\\\\\n\\vec{w_{2 \\cdot}}\\\\\n\\vec{w_{3 \\cdot}} \\\\\n\\end{pmatrix}\n\\vec{x}\n= \\begin{pmatrix}\n\\vec{w_{1\\cdot}} \\cdot \\vec{x} \\\\\n\\vec{w_{2 \\cdot}} \\cdot \\vec{x} \\\\\n\\vec{w_{3 \\cdot}} \\cdot \\vec{x} \\\\\n\\end{pmatrix}\n$$\n\n展開した出力の式は，\n\n$$\n\\begin{pmatrix}\no_1 \\\\\no_2 \\\\\no_3 \\\\\n\\end{pmatrix}\n=\n\\begin{pmatrix}\nw_{11}x_{1} + w_{12}x_{2} + w_{13}x_{3} \\\\\nw_{21}x_{1} + w_{22}x_{2} + w_{23}x_{3} \\\\\nw_{31}x_{1} + w_{32}x_{2} + w_{33}x_{3} \\\\\n\\end{pmatrix}\n$$\n"
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"cell_type": "markdown",
"source": "### 3入力隠れ1層(2次元)1出力"
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"source": "\\begin{eqnarray}\n\\vec{h} \n&=& \\mathbf{W} \\vec{x}\\\\\n\\begin{pmatrix}\nh_1\\\\\nh_2\n\\end{pmatrix}\n&=& \n\\begin{pmatrix}\nw_{11} & w_{12} & w_{13}\\\\\nw_{21} & w_{22} & w_{23}\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nx_1\\\\\nx_2 \\\\\nx_3\n\\end{pmatrix} \\\\\n&=& \n\\begin{pmatrix}\nw_{11}x_1 + w_{12}x_2 + w_{13}x_3 \\\\\nw_{21}x_1 + w_{22}x_2 + w_{23}x_3\n\\end{pmatrix}\\\\\n\\end{eqnarray}\n\n\\begin{eqnarray}\no \n&=& \\mathbf{W_h} \\vec{h}\\\\\n&=& \\vec{w_h} \\cdot \\vec{h}\\\\\n&=& w_{h_1} h_1 + w_{h_2} h_2\n\\end{eqnarray}"
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]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "#### 非線形関数が必要な理由"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "\\begin{eqnarray}\no \n&=& \\mathbf{W_h} \\vec{h} \\\\\n&=& \\mathbf{W_h} \\mathbf{W} \\vec{x} \\\\\n&=& \\vec{w_h}\\mathbf{W} \\vec{x} \\\\\n&=& \n\\begin{pmatrix}\nw_{h_1} & w_{h_2} \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nw_{11} & w_{12} & w_{13}\\\\\nw_{21} & w_{22} & w_{23}\\\\\n\\end{pmatrix}\n\\vec{x} \\\\\n&=& \n\\begin{pmatrix}\nw_{h_1}w_{11}+ w_{h_2}w_{21} & w_{h_1}w_{12}+ w_{h_2}w_{22} & w_{h_1}w_{13}+ w_{h_2}w_{23}\\\\\n\\end{pmatrix}\n\\vec{x} \\\\\n&=&\n\\begin{pmatrix}\nw_{1}^{*} & w_{2}^{*} & w_{3}^{*} \\\\\n\\end{pmatrix}\n\\vec{x} \\\\\n\\end{eqnarray}\n\n1層分の重みにまとめられるので，隠れ層の出力には非線形関数を入れないと意味のない層にになってしまう．\n\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## 重みの更新"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "n次元入力の線形和を表す関数（バイアスなし，活性化関数なし）を考える．\n\n$$\nf(\\vec{x}) = w_1 x_1 + w_2 x_2 + \\cdots + w_n x_n\n$$\n\n入力値$\\vec{x^{*}}$に対して得たい出力値$y^{*}$として，\n最初はその適切な重みはわからないので，その二乗誤差関数を考えると，\n\n$$\nE(\\vec{w}) = \\frac{1}{2}(y^{*} - f(\\vec{x^{*}}))^2\n$$\n(今，入力値$\\vec{x^{*}}$と得たい出力値$y^{*}$は定数となっている．)\n\nこの関数の最小化を考えると，重みを以下の式で更新\n\n$$\nw_{i}^{\\text{new}} = w_{i}^{\\text{old}} - l \\frac{\\partial E(\\vec{w})}{\\partial w_i } \n$$\n\n勾配の逆方向にズラしていけば，誤差の小さい真の重みが決まる．\n\nl:学習率$0.1 \\sim 0.0001$くらいとなるのが多い．\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "この誤差関数の微分は，\n\n$$\n\\frac{\\partial E(\\vec{w})}{\\partial w_i}\n= (y^{*} - f(\\vec{x^{*}}))(y^{*} - f(\\vec{x^{*}}))'\n= (y^{*} - f(\\vec{x^{*}}))(-x_i^{*})\n= - x_i^{*}(y^{*} - f(\\vec{x^{*}}))\n$$"
},
{
"metadata": {
"scrolled": false,
"trusted": false
},
"cell_type": "code",
"source": "import scipy as sp\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nw = sp.linspace(-2, 4, 100)\n\ny_target = 1\nx1 = 1\nx2 = 0\ndef f(w1):\n w2 = 1\n return w1 * x1 + w2 * x2\n\nE = 1/2 * (y_target - f(w))**2\n\nplt.plot(w, E)\nplt.scatter([3], [1/2 * (y_target - f(3))**2])\nplt.quiver(3, 1/2 * (y_target - f(3))**2, 1, -x1 * (y_target - f(3)), linestyles=\"--\")\nplt.quiver(3, 1/2 * (y_target - f(3))**2, -1, x1 * (y_target - f(3)), color=\"r\", linestyle=\"dashed\")\nplt.scatter(2, 1/2 * (y_target - f(2))**2, color=\"red\")\nplt.vlines([3, 2], [0, 0], [1/2 * (y_target - f(3))**2, 1/2 * (y_target - f(2))**2], linestyles=\"--\")\nplt.text(2.8, -.4, \"$w^{old}$\", fontdict={'fontsize':16})\nplt.text(2.5, 2.7, \"$\\\\frac{\\partial E(w_1)}{\\partial w_1}$\", fontdict={'fontsize' :20})\nplt.text(1.5, 2., \"$-\\\\frac{\\partial E(w_1)}{\\partial w_1}$\", fontdict={'fontsize' :20}, color=\"r\")\nplt.text(2.1, 0.4, \"$-l \\\\frac{\\partial E(w_1)}{\\partial w_1}$\", fontdict={'fontsize':14}, color=\"r\")\nplt.quiver(3, 0, -1, 0, units='x', scale=1, color=\"r\")\nplt.text(1.8, -.4, \"$w^{new}$\", fontdict={'fontsize':16}, color=\"r\")\nplt.xlabel(\"$w_1$\")\nplt.ylabel(\"$E(w_1) = \\\\frac{1}{2}(y^{*} - f(w_1))^2$\")\nplt.ylim(-0.5, 5)\nplt.grid()\nplt.show()",
"execution_count": 117,
"outputs": [
{
"data": {
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\n",
"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 多層（3入力隠れ1層(2次元)1出力"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "さらに多層を考える．\n\n- 入力3次元\n- 隠れ1層2次元\n- 出力1次元\n\n"
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
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"cell_type": "markdown",
"source": "\\begin{eqnarray}\n\\vec{h} \n&=& \\mathbf{W} \\vec{x}\\\\\n\\begin{pmatrix}\nh_1\\\\\nh_2\n\\end{pmatrix}\n&=& \n\\begin{pmatrix}\n\\vec{w_{1}} \\cdot \\vec{x} \\\\\n\\vec{w_{2}} \\cdot \\vec{x}\n\\end{pmatrix}\\\\\n&=&\n\\begin{pmatrix}\nw_{11} & w_{12} & w_{13}\\\\\nw_{21} & w_{22} & w_{23}\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nx_1\\\\\nx_2 \\\\\nx_3\n\\end{pmatrix} \\\\\n&=& \n\\begin{pmatrix}\nw_{11}x_1 + w_{12}x_2 + w_{13}x_3 \\\\\nw_{21}x_1 + w_{22}x_2 + w_{23}x_3\n\\end{pmatrix}\\\\\n\\end{eqnarray}\n\n$$\n\\vec{h_f}\n=\n\\begin{pmatrix}\nh_{f_1} \\\\\nh_{f_2}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\nf(h_1) \\\\\nf(h_2)\n\\end{pmatrix}\n= \n\\begin{pmatrix}\n\\frac{1}{1 + \\exp{(- h_1)}} \\\\\n\\frac{1}{1 + \\exp{(- h_2)}}\n\\end{pmatrix}\n$$\n\n\n\\begin{eqnarray}\no \n&=& \\mathbf{W_h} \\vec{h_f}\\\\\n&=& \\vec{w_h} \\cdot \\vec{h_f}\\\\\n&=& w_{h_1} h_{f_1} + w_{h_2} h_{f_2}\\\\\n&=& w_{h_1} \\frac{1}{1 + \\exp{(- h_1)}} + w_{h_2} \\frac{1}{1 + \\exp{(- h_2)}}\\\\\n&=& w_{h_1} \\frac{1}{1 + \\exp{\\left\\{ - (w_{11}x_1 + w_{12}x_2 + w_{13}x_3) \\right\\}}} + w_{h_2} \\frac{1}{1 + \\exp{\\left\\{ - (w_{21}x_1 + w_{22}x_2 + w_{23}x_3) \\right\\} } }\\\\\n\\end{eqnarray}\n\n\\begin{eqnarray}\no_f \n= f(o)\n&=& \\frac{1}{1+\\exp{(-o)}}\\\\\n&=& \\frac{1}{1+\\exp{\\left\\{ -(w_{h_1} \\frac{1}{1 + \\exp{\\left\\{ - (w_{11}x_1 + w_{12}x_2 + w_{13}x_3) \\right\\}}} + w_{h_2} \\frac{1}{1 + \\exp{\\left\\{ - (w_{21}x_1 + w_{22}x_2 + w_{23}x_3) \\right\\} } }) \\right\\} }}\\\\\n\\end{eqnarray}\n"
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"metadata": {},
"cell_type": "markdown",
"source": "これを$w_{11}$についてそのまま微分することは困難のため，\n逆誤差伝播を使う．"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## 合成関数の微分（連鎖律）"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "逆誤差伝播には微分の連鎖律を利用している．\n\n（高校数学では合成関数の微分として紹介されている．）\n\n\n重み変数$w$から出力関数$o$が決まり，\n出力関数$o$から活性化関数（非線形関数）$f$が決まるとすると，\n合成関数の微分より，以下の式で導出できる．\n\n\\begin{eqnarray}\n\\frac{\\partial f}{\\partial w}\n&=& \\frac{\\partial f}{\\partial o}\n\\frac{\\partial o}{\\partial w}\n\\end{eqnarray}\n\nなので，以下2つの微分がわかれば計算できる．\n- （出力に対する）活性化関数の微分\n- （重みに対する）出力関数の微分\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "参考：https://mathtrain.jp/rensaritsu\n\n\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 例：1入力1出力、活性化関数なしのニューラル"
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"source": "活性化関数なしは，\n$$\nf(x) = x\n$$\nと同義になる．微分すると$1$.\n\n$$\no = wx+b = A + b\n$$\n\n中間出力を$A = wx$ とおく．\n\nバイアスの勾配：\n\n$$\n\\frac{\\partial o}{\\partial b} = 1\n$$\n\n重みの勾配：\n\n$$\n\\frac{\\partial o}{\\partial w}\n=\\frac{\\partial o}{\\partial A}\\frac{\\partial A}{\\partial w} \n= 1 \\cdot x \n= 3\n$$\n\n入力値xの勾配：\n\n$$\n\\frac{\\partial o}{\\partial x}\n=\\frac{\\partial o}{\\partial A}\\frac{\\partial A}{\\partial x} \n= 1 \\cdot w\n= 0.1\n$$\n\nこの例では，微分は簡単なので逆誤差伝播は必要ないが，\nポイントとして，\n- 重みの勾配は,出力値側の微分$\\frac{\\partial o}{\\partial A}$と入力値$x$との積\n- 入力値$x$の勾配は，出力値側の微分$\\frac{\\partial o}{\\partial A}$と重み$w$との積で計算される．\n"
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"execution_count": 3,
"outputs": [
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"text/plain": "<IPython.core.display.HTML object>"
},
"metadata": {},
"output_type": "display_data"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## sigmoidを例に関数の微分を逆誤差伝播で計算"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 通常にsigmoid関数の微分導出"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$$\nf(x) = \\frac{1}{1 + \\exp{(- x)}}\n$$"
},
{
"metadata": {
"trusted": false,
"scrolled": true
},
"cell_type": "code",
"source": "import scipy as sp\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\nx = sp.linspace(-5, 5, 100)\nf = 1 / (1 + sp.exp(- x))\n\nplt.plot(x, f)\nplt.grid()\nplt.show()",
"execution_count": 21,
"outputs": [
{
"data": {
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\n",
"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$$\nf(x) = \\frac{1}{1 + \\exp{(- x)}}\n$$\n\n分数関数の微分\n\n\\begin{eqnarray}\nf'(x) \n&=& \\frac{-(1 + \\exp{(- x)})'}{(1 + \\exp{(- x)})^2}\\\\\n&=& \\frac{-(-\\exp{(- x)})}{(1 + \\exp{(- x)})^2}\\\\\n&=& \\frac{\\exp{(- x)}}{(1 + \\exp{(- x)})^2} \\\\\n&=& \\frac{1 + \\exp{(- x)}}{(1 + \\exp{(- x)})^2} - \\frac{1}{(1 + \\exp{(- x)})^2}\\\\\n&=& \\frac{1}{1 + \\exp{(- x)}} - \\frac{1}{(1 + \\exp{(- x)})^2}\\\\\n&=& \\frac{1}{1 + \\exp{(- x)}} \\left(1 - \\frac{1}{1 + \\exp{(- x)}} \\right)\\\\\n&=& f(x)(1-f(x))\\\\\n\\end{eqnarray}"
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "%matplotlib inline\n\nx = sp.linspace(-5, 5, 100)\nf = 1 / (1 + sp.exp(- x))\ndiff = f * (1 - f)\nplt.plot(x, diff)\nplt.title(\"diff sigmoid\")\nplt.grid()\nplt.show()",
"execution_count": 22,
"outputs": [
{
"data": {
"image/png": 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"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {
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"output_type": "display_data"
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]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### sigmoid関数の微分をforwardとbackwardで求める"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$$\nf(x) = \\frac{1}{1+\\exp{(-x)}}\n$$\n\nforward:\n\n```\nx=1 -> (-1)x -> exp(x) -> x+1 -> 1/x -> o=\n```\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "出力側から考える．\n\n- C -> D:\n\nCの値はforwardにより事前に求まっている．\n\n$$\n\\frac{\\partial D}{\\partial C}\n= (\\frac{1}{C})' \n= - \\frac{1}{C^2}\n= - \\frac{1}{1.37^2}\n= -0.533\n$$\n\n出力側には，さらにDを入力してDが出力されるユニットがあると拡張して考えると，\nその連鎖律は，\n\n$$\n\\frac{\\partial D}{\\partial D} \\frac{\\partial D}{\\partial C}　= \n1 \\cdot \\frac{\\partial D}{\\partial C}　= -0.533\n$$\n\n- B -> C:\n\n偏微分\n\n$$\n\\frac{\\partial C}{\\partial B}　\n= (B+1)'\n= 1\n$$\n\n連鎖律より，（出力側の微分値が入力されたと考えることもでき，）\n\n$$\n\\frac{\\partial D}{\\partial B}\n= \\frac{\\partial D}{\\partial C}\\frac{\\partial C}{\\partial B}\n= (-0.533) \\cdot 1\n= -0.533\n$$\n\n- A -> B:\n\n偏微分する．\n\n$$\n\\frac{\\partial B}{\\partial A}　\n= (\\exp{A})'\n= \\exp{A}\n= \\exp{-1}\n= 0.368\n$$\n\n同様に，連鎖律より，\n\n$$\n\\frac{\\partial D}{\\partial A}\n= \\frac{\\partial D}{\\partial B}\\frac{\\partial B}{\\partial A}\n= -0.533 \\cdot 0.368\n= -0.196\n$$\n\n- x -> A:\n\n偏微分する．\n\n$$\n\\frac{\\partial A}{\\partial x}　\n= (-x)'\n= -1\n$$\n\n同様に，連鎖律より，\n\n$$\n\\frac{\\partial D}{\\partial x}\n= \\frac{\\partial D}{\\partial A}\\frac{\\partial A}{\\partial x}\n= (-0.196) \\cdot (-1)\n= 0.196\n$$\n\n出力側の微分値と入力された値との偏微分の2つがわかれば，実際の出力との微分が求められる．\n"
},
{
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"cell_type": "code",
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d=\"M237.91,123.63h4.33a2.64,2.64,0,0,1,2.23,2.9,5.24,5.24,0,0,1-.74,2.7,5.52,5.52,0,0,1-1.81,1.93,3.9,3.9,0,0,1-1.56.63l-2.27,0H237c-1.17,0-1.06,0-1.09-.08a.07.07,0,0,1,0-.05.81.81,0,0,1,0-.17c0-.18.08-.21.12-.24h.36c.58,0,.69-.05.76-.19s1.69-6.71,1.69-6.77,0,0,0-.06a2.25,2.25,0,0,0-.73-.07h-.22c-.09-.07-.11-.08-.05-.32S237.84,123.7,237.91,123.63Zm1.06,7.64h.38A2.88,2.88,0,0,0,241,131a3.63,3.63,0,0,0,1.79-2.21,15.15,15.15,0,0,0,.5-2.07,4.4,4.4,0,0,0,0-.59,1.73,1.73,0,0,0-1.57-1.91h-1.62a.23.23,0,0,0-.18.11c0,.06-1.72,6.85-1.72,6.91A2,2,0,0,0,239,131.27Z\" style=\"fill:#ed1c24\"/><path d=\"M244.91,134.59c0-.18,4.19-11.61,4.21-11.64a.23.23,0,0,1,.44.12c0,.08-4.15,11.49-4.22,11.63a.25.25,0,0,1-.2.12A.24.24,0,0,1,244.91,134.59Z\" style=\"fill:#ed1c24\"/><path 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d=\"M319.91,123.63h4.33a2.64,2.64,0,0,1,2.23,2.9,5.24,5.24,0,0,1-.74,2.7,5.52,5.52,0,0,1-1.81,1.93,3.9,3.9,0,0,1-1.56.63l-2.27,0H319c-1.17,0-1.06,0-1.09-.08a.07.07,0,0,1,0-.05.81.81,0,0,1,0-.17c.05-.18.08-.21.12-.24h.36c.58,0,.69-.05.76-.19s1.69-6.71,1.69-6.77,0,0,0-.06a2.25,2.25,0,0,0-.73-.07h-.22c-.09-.07-.11-.08,0-.32S319.84,123.7,319.91,123.63Zm1.06,7.64h.38A2.88,2.88,0,0,0,323,131a3.63,3.63,0,0,0,1.79-2.21,15.15,15.15,0,0,0,.5-2.07,4.4,4.4,0,0,0,.05-.59,1.73,1.73,0,0,0-1.57-1.91h-1.62a.23.23,0,0,0-.18.11c0,.06-1.72,6.85-1.72,6.91A2,2,0,0,0,321,131.27Z\" style=\"fill:#ed1c24\"/><path d=\"M326.91,134.59c0-.18,4.19-11.61,4.21-11.64a.23.23,0,0,1,.44.12c0,.08-4.15,11.49-4.22,11.63a.25.25,0,0,1-.2.12A.24.24,0,0,1,326.91,134.59Z\" style=\"fill:#ed1c24\"/><path d=\"M334.13,125.26a1.21,1.21,0,0,1,.05-.27,2.28,2.28,0,0,1,1.67-1.68,1.67,1.67,0,0,1,.38-.06h.61a2.85,2.85,0,0,1,2.19,3.08,5.76,5.76,0,0,1-.27,1.69,6.49,6.49,0,0,1-1.7,3.16,3.15,3.15,0,0,1-2.15.9,1.93,1.93,0,0,1-1.8-.94,2.15,2.15,0,0,1-.34-.8,1.4,1.4,0,0,1,0-.43,2.15,2.15,0,0,1,.07-.68,3.71,3.71,0,0,1,1.18-2.05,3.34,3.34,0,0,1,1.61-.9,2.81,2.81,0,0,1,.5,0,1.86,1.86,0,0,1,.6.08,1.57,1.57,0,0,1,1,1.09l0-.13a6.68,6.68,0,0,0,.24-1.64,1.77,1.77,0,0,0-1-1.8,1.63,1.63,0,0,0-.66-.14,1.79,1.79,0,0,0-1.45.77c.3,0,.49.19.49.48a.76.76,0,0,1-.74.77A.47.47,0,0,1,334.13,125.26Zm2.29,1.44h-.19a1.74,1.74,0,0,0-1.33.55,2.15,2.15,0,0,0-.36.48,5.44,5.44,0,0,0-.5,1.45,5.35,5.35,0,0,0-.22,1.26c0,.76.42,1.14,1.2,1.14h.22a2.27,2.27,0,0,0,1.48-1.08,5.4,5.4,0,0,0,.46-1.13,3.45,3.45,0,0,0,.22-1.2C337.4,127.37,337.1,126.83,336.42,126.7Z\" style=\"fill:#ed1c24\"/><path d=\"M345.6,123.92a3.59,3.59,0,0,0-2.35,1.07,4.81,4.81,0,0,0-1,1.5,9.81,9.81,0,0,0-.63,2.42c0,.09,0,.23,0,.36a2.21,2.21,0,0,0,.54,1.66,2,2,0,0,0,1.49.6c1.38,0,2.43-1.41,2.78-2.59.05-.2.05-.2.29-.2s.25,0,.28.07,0,.11-.08.35a4.34,4.34,0,0,1-2.87,2.84,2.67,2.67,0,0,1-.71.07H343a2.85,2.85,0,0,1-2.43-2.23,4.31,4.31,0,0,1-.11-.93c0-.2,0-.4,0-.54a6.14,6.14,0,0,1,1.36-3.05c.14-.17.59-.62.75-.76a4.94,4.94,0,0,1,2.62-1.17h.69a2,2,0,0,1,1.29.84s.05.07.06.07l.44-.45a5.89,5.89,0,0,1,.47-.47h.07c.09,0,.17,0,.17.08s-.06.3-.38,1.64c-.23.89-.42,1.65-.44,1.68s0,.1-.27.1h-.19c-.05-.05-.08-.05-.08-.16a.55.55,0,0,0,0-.21,4.83,4.83,0,0,0,.05-.59C347.07,124.81,346.67,123.92,345.6,123.92Z\" style=\"fill:#ed1c24\"/></svg>\n"
},
"metadata": {}
}
]
},
{
"metadata": {
"code_folding": [
82
],
"scrolled": true,
"trusted": false
},
"cell_type": "code",
"source": "import scipy as sp\n\nclass FB:\n def __init__(self): \n pass\n \n class Sigmoid:\n def __init__(self):\n self._output = None\n \n def forward(self, input):\n self._output = 1 / (1 + sp.exp(- input))\n return self._output\n\n def backward(self, diff_input=1):\n sig = self._output \n return sig * (1 - sig) * diff_input\n \n class X_nthPower:\n def __init__(self, nthPower = 1):\n self._output = None\n self._nthPower = nthPower\n \n def forward(self, input):\n self._output = input**self._nthPower\n return self._output\n\n def backward(self, diff_input=1):\n '''\n x^n' = n x^{n-1} \n x^1 = (x^n)^(1/n)\n '''\n x = self._output ** (1/self._nthPower) \n return self._nthPower * self._output / x * diff_input\n \n class X_addCons:\n def __init__(self, constant = 0):\n self._output = None\n self._constant = constant\n\n def forward(self, input):\n '''\n X+c\n '''\n self._output = input + self._constant\n return self._output\n\n def backward(self, diff_input=1):\n return 1 * diff_input\n \n class X_scaleCons:\n def __init__(self, constant = 1):\n self._output = None\n self._constant = constant\n\n def forward(self, input):\n '''\n c x\n '''\n \n self._output = self._constant * input\n return self._output\n\n def backward(self, diff_input=1):\n return self._constant * diff_input\n \n \n class Exp:\n def __init__(self):\n self._output = None\n\n def forward(self, input):\n '''\n exp(x)\n '''\n self._output = sp.exp(input)\n return self._output\n\n def backward(self, diff_input=1):\n return self._output * diff_input \n \n \n class _Network:\n def __init__(self, units):\n '''\n units: list\n '''\n self._units = units\n \n def forward(self,input):\n x = input\n for u in self._units:\n fw = u.forward(x)\n x = fw\n \n return fw\n \n def backward(self):\n diff_x = 1\n for u in self._units[::-1]:\n bw = u.backward(diff_x)\n diff_x = bw\n return bw\n \n def createNetwork(self, units):\n return self._Network(units)\n \nfb = FB()\nsig = fb.Sigmoid()\nsig.forward(1)",
"execution_count": 194,
"outputs": [
{
"data": {
"text/plain": "0.7310585786300049"
},
"execution_count": 194,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "sig.backward()",
"execution_count": 195,
"outputs": [
{
"data": {
"text/plain": "0.19661193324148185"
},
"execution_count": 195,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "x = 1\nfb_xScaleMinus1 = fb.X_scaleCons(-1)\nfw1 = fb_xScaleMinus1.forward(1)\nfw1",
"execution_count": 196,
"outputs": [
{
"data": {
"text/plain": "-1"
},
"execution_count": 196,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "fb_exp = fb.Exp()\nfw2 = fb_exp.forward(fw1)\nfw2",
"execution_count": 197,
"outputs": [
{
"data": {
"text/plain": "0.36787944117144233"
},
"execution_count": 197,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "fb_xadd1 = fb.X_addCons(constant=1)\nfw3 = fb_xadd1.forward(fw2)\nfw3",
"execution_count": 198,
"outputs": [
{
"data": {
"text/plain": "1.3678794411714423"
},
"execution_count": 198,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "fb_xpowerMinus1 = fb.X_nthPower(nthPower=-1)\nfb_xpowerMinus1.forward(fw3)",
"execution_count": 199,
"outputs": [
{
"data": {
"text/plain": "0.7310585786300049"
},
"execution_count": 199,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# backward\nbw1 = fb_xpowerMinus1.backward(1)\nbw1",
"execution_count": 200,
"outputs": [
{
"data": {
"text/plain": "-0.534446645388523"
},
"execution_count": 200,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "bw2 = fb_xadd1.backward(bw1)\nbw2",
"execution_count": 201,
"outputs": [
{
"data": {
"text/plain": "-0.534446645388523"
},
"execution_count": 201,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "bw3 = fb_exp.backward(bw2)\nbw3",
"execution_count": 202,
"outputs": [
{
"data": {
"text/plain": "-0.19661193324148188"
},
"execution_count": 202,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "bw4 = fb_xScaleMinus1.backward(bw3)\nbw4",
"execution_count": 203,
"outputs": [
{
"data": {
"text/plain": "0.19661193324148188"
},
"execution_count": 203,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "n = fb.createNetwork([fb_xScaleMinus1, fb_exp, fb_xadd1, fb_xpowerMinus1])\nn",
"execution_count": 204,
"outputs": [
{
"data": {
"text/plain": "<__main__.FB._Network at 0x214a53a5cc0>"
},
"execution_count": 204,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "n.forward(1)",
"execution_count": 205,
"outputs": [
{
"data": {
"text/plain": "0.7310585786300049"
},
"execution_count": 205,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "n.backward()",
"execution_count": 206,
"outputs": [
{
"data": {
"text/plain": "0.19661193324148188"
},
"execution_count": 206,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "sig.backward()",
"execution_count": 207,
"outputs": [
{
"data": {
"text/plain": "0.19661193324148185"
},
"execution_count": 207,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# forward\n\nf_l = [lambda x: -x, lambda x: sp.exp(x), lambda x: 1+x, lambda x: 1/x]\nf_l",
"execution_count": 10,
"outputs": [
{
"data": {
"text/plain": "[<function __main__.<lambda>(x)>,\n <function __main__.<lambda>(x)>,\n <function __main__.<lambda>(x)>,\n <function __main__.<lambda>(x)>]"
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "def forward(f_l, x):\n o = []\n for f in f_l:\n if o:\n x = o[-1]\n o.append(f(x))\n return o\n\nforward(f_l, 1)",
"execution_count": 13,
"outputs": [
{
"data": {
"text/plain": "[-1, 0.36787944117144233, 1.3678794411714423, 0.7310585786300049]"
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# backward\ndiff_l = [lambda x: -1, lambda x: sp.exp(x), lambda x: 1, lambda x: -1 / x**2]\n\ndef backward(diff_l, f_l, x):\n forward_o = forward(f_l, x)\n \n grad = []\n for diff, inp in zip(diff_l[::-1], forward_o[-2::-1] + [x]):\n print(diff, inp)\n grad.append(diff(inp))\n return grad, sp.prod(grad)\n\nbackward(diff_l, f_l, 1)",
"execution_count": 30,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "<function <lambda> at 0x0000021C9D8B8488> 1.3678794411714423\n<function <lambda> at 0x0000021C9D8B86A8> 0.36787944117144233\n<function <lambda> at 0x0000021C9D8B8268> -1\n<function <lambda> at 0x0000021C9C32FA60> 1\n"
},
{
"data": {
"text/plain": "([-0.534446645388523, 1, 0.36787944117144233, -1], 0.19661193324148188)"
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "-1/(1.37)**2",
"execution_count": 27,
"outputs": [
{
"data": {
"text/plain": "-0.5327934359848686"
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "class Backprop:\n def __init__(self, f_list, diff_list, x=0):\n self._f_list = list(f_list)\n self._diff_list = list(diff_list)\n self._input = x\n self._forward_o = []\n self._grad_o = []\n \n def get_forward(self):\n return self._forward_o\n \n def forward(self, **kwargs):\n self._forward_o = []\n for key, value in kwargs.items():\n if(key == \"x\"):\n self._input = value\n \n x = self._input\n for f in self._f_list:\n if self._forward_o:\n x = self._forward_o[-1]\n self._forward_o.append(f(x))\n \n def backward(self):\n self._grad_o = []\n for diff, inp in zip(self._diff_list[::-1], self._forward_o[-2::-1] + [self._input]):\n self._grad_o.insert(0, diff(inp))\n \n return sp.prod(self._grad_o)\n \n def get_backward(self):\n return self._grad_o, sp.prod(self._grad_o)\n\n \n\nf_l = [lambda x: -x, lambda x: sp.exp(x), lambda x: 1+x, lambda x: 1/x]\n# backward\ndiff_l = [lambda x: -1, lambda x: sp.exp(x), lambda x: 1, lambda x: -1 / x**2]\nb = Backprop(f_l, diff_l, x=1)\nb.forward()\nb.backward()",
"execution_count": 38,
"outputs": [
{
"data": {
"text/plain": "0.19661193324148188"
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "b.get_backward()",
"execution_count": 39,
"outputs": [
{
"data": {
"text/plain": "([-1, 0.36787944117144233, 1, -0.534446645388523], 0.19661193324148188)"
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "b.forward(x = 2)\nb.get_forward()",
"execution_count": 41,
"outputs": [
{
"data": {
"text/plain": "[-2, 0.1353352832366127, 1.1353352832366128, 0.8807970779778823]"
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "b.backward()",
"execution_count": 42,
"outputs": [
{
"data": {
"text/plain": "0.1049935854035065"
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "diff_sig(2)",
"execution_count": 44,
"outputs": [
{
"data": {
"text/plain": "0.10499358540350662"
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## backprop"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$$\n\\vec{x} = \\begin{pmatrix}\nx_1 \\\\\nx_2\n\\end{pmatrix}\n$$\n\n重み行列\n\n$$\nW = \\begin{pmatrix}\nw_{11} & w_{12} \\\\\nw_{21} & w_{22}\n\\end{pmatrix}\n$$\n\n$$\n\\vec{y}\n= \n\\begin{pmatrix}\ny_1 \\\\\ny_2\n\\end{pmatrix}\n= W \\vec{x}\n= \\begin{pmatrix}\nw_{11} & w_{12} \\\\\nw_{21} & w_{22}\n\\end{pmatrix}\n\\begin{pmatrix}\nx_1 \\\\\nx_2\n\\end{pmatrix}\n$$\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "![](https://camo.qiitausercontent.com/4cf8ed949c07a9f217915bda597941ce1f46c654/68747470733a2f2f71696974612d696d6167652d73746f72652e73332e616d617a6f6e6177732e636f6d2f302f3132333935312f36613330313964662d316233362d633233632d633332612d6266393266313864313665332e706e67)"
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "sp.random.randn(2, 2)",
"execution_count": 208,
"outputs": [
{
"data": {
"text/plain": "array([[-0.8050283 , 0.6395666 ],\n [ 2.042945 , -1.04805116]])"
},
"execution_count": 208,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "v = sp.matrix([[1], [2]])\nv",
"execution_count": 213,
"outputs": [
{
"data": {
"text/plain": "matrix([[1],\n [2]])"
},
"execution_count": 213,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "m = sp.matrix([[1, 2], \n [3, 4]])\nm * v\nsp.dot(m, v)",
"execution_count": 216,
"outputs": [
{
"data": {
"text/plain": "matrix([[ 5],\n [11]])"
},
"execution_count": 216,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "import scipy as sp\n\nclass NN:\n def __init__(self, n_i, n_o, lr=0.01, init_weight=None): \n if init_weight is not None:\n self.weight = init_weight.reshape((n_o, n_i))\n else: \n self.weight = sp.random.randn(n_o, n_i)\n \n self.inputs = None\n self.grad = sp.zeros([n_i, n_o])\n self.lr = lr\n \n def forward(self, inputs):\n self.inputs = inputs.reshape(-1, 1) # 入力の値を保持\n return sp.dot(self.weight, self.inputs)\n \n def backward(self, diff_inputs):\n self.grad = sp.dot(diff_inputs.reshape((-1, 1)), self.inputs.reshape((1, -1)))\n return sp.dot(diff_inputs.reshape((1, -1)), self.weight)# dz/dx\n \n class Sigmoid:\n def __init__(self):\n self._output = None\n \n def forward(self, input):\n self._output = 1 / (1 + sp.exp(- input))\n return self._output\n\n def backward(self, diff_input=1):\n sig = self._output \n return sig * (1 - sig) * diff_input\n \n class X_nthPower:\n def __init__(self, nthPower = 1):\n self._output = None\n self._nthPower = nthPower\n \n def forward(self, input):\n self._output = input**self._nthPower\n return self._output\n\n def backward(self, diff_input=1):\n '''\n x^n' = n x^{n-1} \n x^1 = (x^n)^(1/n)\n '''\n x = self._output ** (1/self._nthPower) \n return self._nthPower * self._output / x * diff_input\n \n class X_addCons:\n def __init__(self, constant = 0):\n self._output = None\n self._constant = constant\n\n def forward(self, input):\n '''\n X+c\n '''\n self._output = input + self._constant\n return self._output\n\n def backward(self, diff_input=1):\n return 1 * diff_input\n \n class X_scaleCons:\n def __init__(self, constant = 1):\n self._output = None\n self._constant = constant\n\n def forward(self, input):\n '''\n c x\n '''\n \n self._output = self._constant * input\n return self._output\n\n def backward(self, diff_input=1):\n return self._constant * diff_input\n \n \n class Exp:\n def __init__(self):\n self._output = None\n\n def forward(self, input):\n '''\n exp(x)\n '''\n self._output = sp.exp(input)\n return self._output\n\n def backward(self, diff_input=1):\n return self._output * diff_input \n \n \n class _Network:\n def __init__(self, units):\n '''\n units: list\n '''\n self._units = units\n \n def forward(self,input):\n x = input\n for u in self._units:\n fw = u.forward(x)\n x = fw\n \n return fw\n \n def backward(self):\n diff_x = 1\n for u in self._units[::-1]:\n bw = u.backward(diff_x)\n diff_x = bw\n return bw\n \n def createNetwork(self, units):\n return self._Network(units) \n ",
"execution_count": 13,
"outputs": []
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "init_w = sp.matrix([[0.1, 0.1],\n [0.2, 0.2]])\nfc1 = NN(2, 2, 0.1, init_w)\nfc1",
"execution_count": 14,
"outputs": [
{
"data": {
"text/plain": "<__main__.NN at 0x234f3ffd9e8>"
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "x = sp.matrix([[1],\n [2]])\nfc1.forward(x)",
"execution_count": 15,
"outputs": [
{
"data": {
"text/plain": "matrix([[0.3],\n [0.6]])"
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "fc1.backward(sp.matrix([[0.3],\n [0.3]]))",
"execution_count": 16,
"outputs": [
{
"data": {
"text/plain": "matrix([[0.09, 0.09]])"
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "init_w2 = sp.matrix([[0.3, 0.3]])\nfc2 = NN(2, 1, 0.1, init_w2)\nfc2.forward(fc1.forward(x))",
"execution_count": 17,
"outputs": [
{
"data": {
"text/plain": "matrix([[0.27]])"
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "",
"execution_count": null,
"outputs": []
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "",
"execution_count": null,
"outputs": []
}
],
"metadata": {
"_draft": {
"nbviewer_url": "https://gist.github.com/1e9d338b8a724f7e99c5e445e0b6c421"
},
"gist": {
"id": "1e9d338b8a724f7e99c5e445e0b6c421",
"data": {
"description": "逆誤差伝播",
"public": true
}
},
"hide_input": false,
"kernelspec": {
"name": "python3",
"display_name": "Python 3",
"language": "python"
},
"language_info": {
"name": "python",
"version": "3.7.0",
"mimetype": "text/x-python",
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"pygments_lexer": "ipython3",
"nbconvert_exporter": "python",
"file_extension": ".py"
},
"toc": {
"nav_menu": {},
"number_sections": true,
"sideBar": true,
"skip_h1_title": false,
"base_numbering": 1,
"title_cell": "Table of Contents",
"title_sidebar": "Contents",
"toc_cell": false,
"toc_position": {
"height": "calc(100% - 180px)",
"left": "10px",
"top": "150px",
"width": "165px"
},
"toc_section_display": true,
"toc_window_display": true
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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