neural_network/numpy_nn/cross_entropy_5.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "9aa453ff",
   "metadata": {},
   "source": [
    "e ** x = b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ee6590f9",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "a0481e20",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.6486586255873816\n",
      "5.199999999999999\n"
     ]
    }
   ],
   "source": [
    "b = 5.2\n",
    "print(np.log(b))\n",
    "print(math.e ** 1.6486586255873816)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64590073",
   "metadata": {},
   "source": [
    "交叉熵损失函数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a45646b",
   "metadata": {},
   "source": [
    "class:  3  \n",
    "label:  0  \n",
    "one-hot:  [1, 0, 0]  \n",
    "prediction: [0.7, 0.1, 0.2]  \n",
    "$$L = -\\sum_jy_ilog(\\hat{y}) = -(1\\cdot\\log(0.7) + 0\\cdot\\log(0.1) + 0\\cdot\\log(0.2))$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "bbf3bd0e",
   "metadata": {},
   "outputs": [],
   "source": [
    "softmax_output = [0.7, 0.1, 0.2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "44c25853",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.35667494393873245\n"
     ]
    }
   ],
   "source": [
    "target_output = [1, 0, 0]#独热编码\n",
    "loss = -(math.log(softmax_output[0]) * target_output[0] +\n",
    "         math.log(softmax_output[1]) * target_output[1] +\n",
    "         math.log(softmax_output[2]) * target_output[2])\n",
    "print(loss)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a9cebb9",
   "metadata": {},
   "source": [
    "简化运算"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "d07da7f4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.35667494393873245"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loss = -(math.log(softmax_output[0]) * target_output[0])\n",
    "loss"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00ca4e6a",
   "metadata": {},
   "source": [
    "多个数据"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "3b3b7290",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.7 0.5 0.9]\n",
      "[0.7 0.5 0.9]\n"
     ]
    }
   ],
   "source": [
    "softmax_outputs = np.array([[0.7, 0.1, 0.2],\n",
    "                            [0.1, 0.5, 0.4],\n",
    "                            [0.02, 0.9, 0.08]])\n",
    "class_targets = [0, 1, 1]\n",
    "print(softmax_outputs[[0, 1, 2],class_targets])\n",
    "print(softmax_outputs[range(len(softmax_outputs)),class_targets])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "e639666d",
   "metadata": {},
   "outputs": [],
   "source": [
    "neg_log = -np.log(softmax_outputs[range(len(softmax_outputs)),class_targets])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "636b4d4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.35667494, 0.69314718, 0.10536052])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "neg_log"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "ff5eee5d",
   "metadata": {},
   "outputs": [],
   "source": [
    "avg_loss = np.mean(neg_log)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "ab993d9f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.38506088005216804"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "avg_loss"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d026b79",
   "metadata": {},
   "source": [
    "考虑np.log(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "841620c0",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "D:\\envs\\stark-lin\\lib\\site-packages\\ipykernel_launcher.py:1: RuntimeWarning: divide by zero encountered in log\n",
      "  \"\"\"Entry point for launching an IPython kernel.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-inf"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.log(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45c2d27e",
   "metadata": {},
   "source": [
    "对输出进行裁剪"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "e7642073",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(16.11809565095832, 1.0000000494736474e-07)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-np.log(1e-7), -np.log(1-1e-7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "7ace302a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16.11809565095832"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_pred = 0\n",
    "y_pred_clip = np.clip(y_pred, 1e-7, 1-1e-7)\n",
    "-np.log(y_pred_clip)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a3083d8f",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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