loss_explain.ipynb

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     "text": [
      "直接使用tensorflow.keras.loss.categorical_crossentropy:\n",
      "\t6筆資料中真實值對應的機率: 0.09150913, 0.05102272, 0.11195231, 0.18319334, 0.25989717, 0.4133129\n",
      "\t6筆資料中真實值對應的loss:  [2.39131654 2.97548426 2.18968228 1.69721319 1.34746916 0.88355029]\n",
      "\t這個epoch最終的loss:  1.9141192850880648\n",
      "\n",
      "根據公式計算:\n",
      "\t6筆資料中真實值對應的機率: 0.09150913, 0.05102272, 0.11195231, 0.18319334, 0.25989717, 0.4133129\n",
      "\t6筆資料中真實值對應的loss:  [2.3913165302577313, 2.975484255263029, 2.189682301938554, 1.6972131811147098, 1.3474692261973618, 0.883550345757009]\n",
      "\t這個epoch最終的loss:  1.9141193067547324\n",
      "\n",
      "最完美的情況:\n",
      "\t6筆資料中真實值對應的機率: [1, 1, 1, 1, 1, 1]\n",
      "\t6筆資料中真實值對應的loss:  [-0.0, -0.0, -0.0, -0.0, -0.0, -0.0]\n",
      "\t這個epoch最終的loss:  0.0\n",
      "\n",
      "模型認為每一個都有可能的情況:\n",
      "\t6筆資料中真實值對應的機率: [0.2, 0.2, 0.2, 0.2, 0.2, 0.2]\n",
      "\t6筆資料中真實值對應的loss:  [1.6094379124341003, 1.6094379124341003, 1.6094379124341003, 1.6094379124341003, 1.6094379124341003, 1.6094379124341003]\n",
      "\t這個epoch最終的loss:  1.6094379124341003\n",
      "\n",
      "每筆資料中真實值對應的機率剛好為歐拉常數的-1次方(約為0.36)的情況:\n",
      "\t6筆資料中真實值對應的機率:  [0.36787944117144233, 0.36787944117144233, 0.36787944117144233, 0.36787944117144233, 0.36787944117144233, 0.36787944117144233]\n",
      "\t6筆資料中真實值對應的loss:  [1.0, 1.0, 1.0, 1.0, 1.0, 1.0]\n",
      "\t這個epoch最終的loss:  1.0\n",
      "\n",
      "每筆資料的真實值對映的機率都不高於0.36時, 只要有一筆資料的機率 < 0.36, 這個epoch的loss就會 > 1:\n",
      "\t6筆資料中真實值對應的機率:  [0.36777944117144235, 0.36787944117144233, 0.36787944117144233, 0.36787944117144233, 0.36787944117144233, 0.36787944117144233]\n",
      "\t6筆資料中真實值對應的loss:  [1.0002718651348228, 1.0, 1.0, 1.0, 1.0, 1.0]\n",
      "\t這個epoch最終的loss:  1.0000453108558038\n",
      "\n",
      "loss非常大的範例情況:\n",
      "\t6筆資料中真實值對應的機率: [0.00000001, 0.00000001, 0.0000000001, 0.0000000001, 0.000000001, 0.000000001]\n",
      "\t6筆資料中真實值對應的loss:  [18.420680743952367, 18.420680743952367, 23.025850929940457, 23.025850929940457, 20.72326583694641, 20.72326583694641]\n",
      "\t這個epoch最終的loss:  20.723265836946414\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "from tensorflow.keras.losses import categorical_crossentropy\n",
    "\n",
    "# 假設這是某個epoch的真實值與預測結果, batch_size = 6, features = 5\n",
    "y_pred = np.array([[0.6268896, 0.25881532, 0.09150913, 0.02037693, 0.00240903],\n",
    "                   [0.09165806, 0.3015705, 0.2891071, 0.26664162, 0.05102272],\n",
    "                   [0.08631954, 0.11195231, 0.28462902, 0.30316573, 0.21393338],\n",
    "                   [0.11869003, 0.18319334, 0.18643036, 0.3534843, 0.15820198],\n",
    "                   [0.10678157, 0.18504941, 0.25989717, 0.26833403, 0.17993775],\n",
    "                   [0.11031441, 0.15937562, 0.18078934, 0.4133129, 0.13620767]])\n",
    "\n",
    "y_true = np.array([[0., 0., 1., 0., 0.],\n",
    "                   [0., 0., 0., 0., 1.],\n",
    "                   [0., 1., 0., 0., 0.],\n",
    "                   [0., 1., 0., 0., 0.],\n",
    "                   [0., 0., 1., 0., 0.],\n",
    "                   [0., 0., 0., 1., 0.]])\n",
    "\n",
    "\"\"\"\n",
    "則以第一筆資料為例, categorical_crossentropy的計算方式為:\n",
    "[ 0 * ln(0.6268896)\n",
    "+ 0 * ln(0.25881532)\n",
    "+ 1 * ln(0.09150913)\n",
    "+ 0 * ln(0.02037693)\n",
    "+ 0 * ln(0.00240903) ]\n",
    "* -1\n",
    "= 2.3913165302577313\n",
    "所以只要取真實值對應的那個數就好, 其他都是0\n",
    "對每一筆資料都會有對應的loss, 總共有batch_size筆資料\n",
    "# 對於這個epoch, loss會是batch_size筆資料的loss值取平均, 也就是模型在每個epoch中最終得出的loss值\n",
    "\"\"\"\n",
    "\n",
    "# 呼叫keras的api, 它使用的是自然對數\n",
    "print('直接使用tensorflow.keras.loss.categorical_crossentropy:')\n",
    "print('\\t6筆資料中真實值對應的機率: 0.09150913, 0.05102272, 0.11195231, 0.18319334, 0.25989717, 0.4133129')\n",
    "print('\\t6筆資料中真實值對應的loss: ', categorical_crossentropy(y_true, y_pred).numpy())\n",
    "print('\\t這個epoch最終的loss: ', categorical_crossentropy(y_true, y_pred).numpy().mean())\n",
    "print()\n",
    "\n",
    "# 手刻, math.log()就是自然對數; math.log(x, base)\n",
    "print('根據公式計算:')\n",
    "print('\\t6筆資料中真實值對應的機率: 0.09150913, 0.05102272, 0.11195231, 0.18319334, 0.25989717, 0.4133129')\n",
    "arr = [-math.log(0.09150913), -math.log(0.05102272), -math.log(0.11195231), -math.log(0.18319334), -math.log(0.25989717), -math.log(0.4133129)]\n",
    "print('\\t6筆資料中真實值對應的loss: ', arr)\n",
    "print('\\t這個epoch最終的loss: ', sum(arr) / 6)\n",
    "print()\n",
    "\n",
    "# 最完美情況: 每一筆都預測正確, 機率都是1, 其它都是0\n",
    "print('最完美的情況:')\n",
    "print('\\t6筆資料中真實值對應的機率: [1, 1, 1, 1, 1, 1]')\n",
    "arr = [-math.log(1), -math.log(1), -math.log(1), -math.log(1), -math.log(1), -math.log(1)]\n",
    "print('\\t6筆資料中真實值對應的loss: ', arr)\n",
    "print('\\t這個epoch最終的loss: ', sum(arr) / 6)\n",
    "print()\n",
    "\n",
    "# 最差情況: 每一個類別都有可能, 機率都是0.2\n",
    "print('模型認為每一個都有可能的情況:')\n",
    "print('\\t6筆資料中真實值對應的機率: [0.2, 0.2, 0.2, 0.2, 0.2, 0.2]')\n",
    "arr = [-math.log(0.2), -math.log(0.2), -math.log(0.2), -math.log(0.2), -math.log(0.2), -math.log(0.2)]\n",
    "print('\\t6筆資料中真實值對應的loss: ', arr)\n",
    "print('\\t這個epoch最終的loss: ', sum(arr) / 6)\n",
    "print()\n",
    "\n",
    "# 平均loss==1的情況, 機率為pow(math.e, -1)約為0.36\n",
    "print('每筆資料中真實值對應的機率剛好為歐拉常數的-1次方(約為0.36)的情況:')\n",
    "print('\\t6筆資料中真實值對應的機率: ', [pow(math.e, -1), pow(math.e, -1), pow(math.e, -1), pow(math.e, -1), pow(math.e, -1), pow(math.e, -1)])\n",
    "arr = [-math.log(pow(math.e, -1)), -math.log(pow(math.e, -1)), -math.log(pow(math.e, -1)), -math.log(pow(math.e, -1)), -math.log(pow(math.e, -1)), -math.log(pow(math.e, -1))]\n",
    "print('\\t6筆資料中真實值對應的loss: ', arr)\n",
    "print('\\t這個epoch最終的loss: ', sum(arr) / 6)\n",
    "print()\n",
    "\n",
    "# 也就是說, 只要每一筆資料中真實值對應的機率都 <= pow(math.e, -1), 且其中一筆資料 < pow(math.e, -1), 則平均loss就會 > 1, 此epoch的loss就會 > 1\n",
    "print('每筆資料的真實值對映的機率都不高於0.36時, 只要有一筆資料的機率 < 0.36, 這個epoch的loss就會 > 1:')\n",
    "print('\\t6筆資料中真實值對應的機率: ', [pow(math.e, -1)-0.0001, pow(math.e, -1), pow(math.e, -1), pow(math.e, -1), pow(math.e, -1), pow(math.e, -1)])\n",
    "arr = [-math.log(pow(math.e, -1)-0.0001), -math.log(pow(math.e, -1)), -math.log(pow(math.e, -1)), -math.log(pow(math.e, -1)), -math.log(pow(math.e, -1)), -math.log(pow(math.e, -1))]\n",
    "print('\\t6筆資料中真實值對應的loss: ', arr)\n",
    "print('\\t這個epoch最終的loss: ', sum(arr) / 6)\n",
    "print()\n",
    "\n",
    "# loss最差的情況: 當機率為0時, -ln(0)為無窮大(∞), loss也會是∞; 所以對於categorical crossentropy而言, 值域為[0, ∞)\n",
    "print('loss非常大的範例情況:')\n",
    "print('\\t6筆資料中真實值對應的機率: [0.00000001, 0.00000001, 0.0000000001, 0.0000000001, 0.000000001, 0.000000001]')\n",
    "arr = [-math.log(0.00000001), -math.log(0.00000001), -math.log(0.0000000001), -math.log(0.0000000001), -math.log(0.000000001), -math.log(0.000000001)] # 機率不能 <= 0, 不然會ValueError\n",
    "print('\\t6筆資料中真實值對應的loss: ', arr)\n",
    "print('\\t這個epoch最終的loss: ', sum(arr) / 6)"
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