{
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"cell_type": "markdown",
"metadata": {},
"source": [
"![](../docs/banner.png)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Appendix B: Logistic Loss"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Imports\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"from sklearn.preprocessing import StandardScaler\n",
"import plotly.express as px"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1. Logistic Regression Refresher\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Logistic Regression is a classification model where we calculate the probability of an observation belonging to a class as:\n",
"\n",
"$$z=w^Tx$$\n",
"\n",
"$$\\hat{y} = \\frac{1}{(1+\\exp(-z))}$$\n",
"\n",
"And then assign that observation to a class based on some threshold (usually 0.5):\n",
"\n",
"$$\\text{Class }\\hat{y}=\\left\\{\n",
"\\begin{array}{ll}\n",
" 0, & \\hat{y}\\le0.5 \\\\\n",
" 1, & \\hat{y}>0.5 \\\\\n",
"\\end{array} \n",
"\\right.$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2. Motivating the Loss Function\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- Below is the mean squared error as a loss function for optimizing linear regression:\n",
"\n",
"$$f(w)=\\frac{1}{n}\\sum^{n}_{i=1}(\\hat{y}-y_i))^2$$\n",
"\n",
"- That won't work for logistic regression classification problems because it ends up being \"non-convex\" (which basically means there are multiple minima)\n",
"- Instead we use the following loss function:\n",
"\n",
"$$f(w)=-\\frac{1}{n}\\sum_{i=1}^ny_i\\log\\left(\\frac{1}{1 + \\exp(-w^Tx_i)}\\right) + (1 - y_i)\\log\\left(1 - \\frac{1}{1 + \\exp(-w^Tx_i)}\\right)$$\n",
"\n",
"- This function is called the \"log loss\" or \"binary cross entropy\"\n",
"- I want to visually show you the differences in these two functions, and then we'll discuss why that loss functions works\n",
"- Recall the Pokemon dataset from [Chapter 1](chapter1_gradient-descent.ipynb), I'm going to load that in again (and standardize the data while I'm at it):"
]
},
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"cell_type": "code",
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"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" name | \n",
" attack | \n",
" defense | \n",
" speed | \n",
" capture_rt | \n",
" legendary | \n",
"
\n",
" \n",
" \n",
" \n",
" 0 | \n",
" Bulbasaur | \n",
" 49 | \n",
" 49 | \n",
" 45 | \n",
" 45 | \n",
" 0 | \n",
"
\n",
" \n",
" 1 | \n",
" Ivysaur | \n",
" 62 | \n",
" 63 | \n",
" 60 | \n",
" 45 | \n",
" 0 | \n",
"
\n",
" \n",
" 2 | \n",
" Venusaur | \n",
" 100 | \n",
" 123 | \n",
" 80 | \n",
" 45 | \n",
" 0 | \n",
"
\n",
" \n",
" 3 | \n",
" Charmander | \n",
" 52 | \n",
" 43 | \n",
" 65 | \n",
" 45 | \n",
" 0 | \n",
"
\n",
" \n",
" 4 | \n",
" Charmeleon | \n",
" 64 | \n",
" 58 | \n",
" 80 | \n",
" 45 | \n",
" 0 | \n",
"
\n",
" \n",
"
\n",
"
"
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"text/plain": [
" name attack defense speed capture_rt legendary\n",
"0 Bulbasaur 49 49 45 45 0\n",
"1 Ivysaur 62 63 60 45 0\n",
"2 Venusaur 100 123 80 45 0\n",
"3 Charmander 52 43 65 45 0\n",
"4 Charmeleon 64 58 80 45 0"
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},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
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"source": [
"df = pd.read_csv(\"data/pokemon.csv\", usecols=['name', 'defense', 'attack', 'speed', 'capture_rt', 'legendary'])\n",
"x = StandardScaler().fit_transform(df.drop(columns=[\"name\", \"legendary\"]))\n",
"X = np.hstack((np.ones((len(x), 1)), x))\n",
"y = df['legendary'].to_numpy()\n",
"df.head()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- The goal here is to use the features (but not \"name\", that's just there for illustration purposes) to predict the target \"legendary\" (which takes values of 0/No and 1/Yes).\n",
"- So we have 4 features meaning that our logistic regression model will have 5 parameters that need to be estimated (4 feature coefficients and 1 intercept)\n",
"- At this point let's define our loss functions:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def sigmoid(w, x):\n",
" \"\"\"Sigmoid function (i.e., logistic regression predictions).\"\"\"\n",
" return 1 / (1 + np.exp(-x @ w))\n",
"\n",
"\n",
"def mse(w, x, y):\n",
" \"\"\"Mean squared error.\"\"\"\n",
" return np.mean((sigmoid(w, x) - y) ** 2)\n",
"\n",
"\n",
"def logistic_loss(w, x, y):\n",
" \"\"\"Logistic loss.\"\"\"\n",
" return -np.mean(y * np.log(sigmoid(w, x)) + (1 - y) * np.log(1 - sigmoid(w, x)))"
]
},
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"cell_type": "markdown",
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"source": [
"- For a moment, let's assume a value for all the parameters execpt for $w_1$\n",
"- We will then calculate the mean squared error for different values of $w_1$ as in the code below"
]
},
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"cell_type": "code",
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" w1 mse log\n",
"0 -3.0 0.451184 1.604272\n",
"1 -2.9 0.446996 1.571701\n",
"2 -2.8 0.442773 1.539928\n",
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"w1_arr = np.arange(-3, 6.1, 0.1)\n",
"losses = pd.DataFrame({\"w1\": w1_arr,\n",
" \"mse\": [mse([0.5, w1, -0.5, 0.5, -2], X, y) for w1 in w1_arr],\n",
" \"log\": [logistic_loss([0.5, w1, -0.5, 0.5, -2], X, y) for w1 in w1_arr]})\n",
"losses.head()"
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"\n",
" \n",
" \n",
"
\n",
" \n",
"
"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = px.line(losses.melt(id_vars=\"w1\", var_name=\"loss\"), x=\"w1\", y=\"value\", color=\"loss\", facet_col=\"loss\", facet_col_spacing=0.1)\n",
"fig.update_yaxes(matches=None, showticklabels=True, col=2)\n",
"fig.update_xaxes(matches=None, showticklabels=True, col=2)\n",
"fig.update_layout(width=800, height=400)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- This is a pretty simple dataset but you can already see the \"non-convexity\" of the MSE loss function.\n",
"- If you want a more mathematical description of the logistic loss function, check out [Chapter 3 of Neural Networks and Deep Learning by Michael Nielsen](http://neuralnetworksanddeeplearning.com/chap3.html) or [this Youtube video by Andrew Ng](https://www.youtube.com/watch?v=HIQlmHxI6-0)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 3. Breaking Down the Log Loss Function\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- So we saw the log loss before:\n",
"\n",
"$$f(w)=-\\frac{1}{n}\\sum_{i=1}^ny_i\\log\\left(\\frac{1}{1 + \\exp(-w^Tx_i)}\\right) + (1 - y_i)\\log\\left(1 - \\frac{1}{1 + \\exp(-w^Tx_i)}\\right)$$\n",
"\n",
"- It looks complicated but it's actually quite simple. Let's break it down.\n",
"- Recall that we have a binary classification task here so $y_i$ can only be 0 or 1."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### When `y = 1`"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- When $y_i = 1$ we are left with:\n",
"\n",
"$$f(w)=-\\frac{1}{n}\\sum_{i=1}^n\\log\\left(\\frac{1}{1 + \\exp(-w^Tx_i)}\\right)$$\n",
"\n",
"- That looks fine!\n",
"- With $y_i = 1$, if $\\hat{y_i} = \\frac{1}{1 + \\exp(-w^Tx_i)}$ is also close to 1 we want the loss to be small, if it is close to 0 we want the loss to be large, that's where the `log()` comes in:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"y = 1\n",
"y_hat_small = 0.05\n",
"y_hat_large = 0.95"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.995732273553991"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"-np.log(y_hat_small)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.05129329438755058"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"-np.log(y_hat_large)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### When `y = 0`"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- When $y_i = 1$ we are left with:\n",
"\n",
"$$f(w)=-\\frac{1}{n}\\sum_{i=1}^n\\log\\left(1 - \\frac{1}{1 + \\exp(-w^Tx_i)}\\right)$$\n",
"\n",
"- With $y_i = 0$, if $\\hat{y_i} = \\frac{1}{1 + \\exp(-w^Tx_i)}$ is also close to 0 we want the loss to be small, if it is close to 1 we want the loss to be large, that's where the `log()` comes in:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"y = 0\n",
"y_hat_small = 0.05\n",
"y_hat_large = 0.95"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.05129329438755058"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"-np.log(1 - y_hat_small)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.99573227355399"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"-np.log(1 - y_hat_large)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Plot Log Loss"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- We know that our predictions from logistic regression $\\hat{y}$ are limited between 0 and 1 thanks to the sigmoid function\n",
"- So let's plot the losses because it's interesting to see how the worse our predictions are, the worse the loss is (i.e., if $y=1$ and our model predicts $\\hat{y}=0.05$, the penalty is exponentially bigger than if the prediction was $\\hat{y}=0.90$)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
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],
"title": {
"text": "loss"
},
"type": "linear"
}
}
},
"image/png": 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",
"text/html": [
"\n",
" \n",
" \n",
"
\n",
" \n",
"
"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"y_hat = np.arange(0.01, 1.00, 0.01)\n",
"log_loss = pd.DataFrame({\"y_hat\": y_hat,\n",
" \"y=0\": -np.log(1 - y_hat),\n",
" \"y=1\": -np.log(y_hat)}).melt(id_vars=\"y_hat\", var_name=\"y\", value_name=\"loss\")\n",
"fig = px.line(log_loss, x=\"y_hat\", y=\"loss\", color=\"y\")\n",
"fig.update_layout(width=500, height=400)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4. Log Loss Gradient\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- In [Chapter 1](chapter1_gradient-descent.ipynb) we used the gradient of the log loss to implement gradient descent\n",
"- Here's the log loss and it's gradient:\n",
"\n",
"$$f(w)=-\\frac{1}{n}\\sum_{i=1}^ny_i\\log\\left(\\frac{1}{1 + \\exp(-w^Tx_i)}\\right) + (1 - y_i)\\log\\left(1 - \\frac{1}{1 + \\exp(-w^Tx_i)}\\right)$$\n",
"\n",
"$$\\frac{\\partial f(w)}{\\partial w}=\\frac{1}{n}\\sum_{i=1}^nx_i\\left(\\frac{1}{1 + \\exp(-w^Tx_i)} - y_i)\\right)$$\n",
"\n",
"- Let's derive that now.\n",
"- We'll denote:\n",
"\n",
"$$z = -w^Tx_i$$\n",
"\n",
"$$\\sigma(z) = \\frac{1}{1 + \\exp(z)}$$\n",
"\n",
"- Such that:\n",
"\n",
"$$f(w)=-\\frac{1}{n}\\sum_{i=1}^ny_i\\log\\sigma(z) + (1 - y_i)\\log(1 - \\sigma(z))$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- Okay let's do it:\n",
"\n",
"$$\n",
"\\begin{equation}\n",
"\\begin{split}\n",
"\\frac{\\partial f(w)}{\\partial w} & =-\\frac{1}{n}\\sum_{i=1}^ny_i \\times \\frac{1}{\\sigma(z)} \\times \\frac{\\partial \\sigma(z)}{\\partial w} + (1 - y_i) \\times \\frac{1}{1 - \\sigma(z)} \\times -\\frac{\\partial \\sigma(z)}{\\partial w} \\\\\n",
"& =-\\frac{1}{n}\\sum_{i=1}^n\\left(\\frac{y_i}{\\sigma(z)} - \\frac{1 - y_i}{1 - \\sigma(z)}\\right)\\frac{\\partial \\sigma(z)}{\\partial w} \\\\\n",
"& =\\frac{1}{n}\\sum_{i=1}^n \\frac{\\sigma(z)-y_i}{\\sigma(z)(1 - \\sigma(z))}\\frac{\\partial \\sigma(z)}{\\partial w}\n",
"\\end{split}\n",
"\\end{equation}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- Now we just need to work out $\\frac{\\partial \\sigma(z)}{\\partial w}$, I'll mostly skip this part but there's an intuitive derivation [here](https://medium.com/analytics-vidhya/derivative-of-log-loss-function-for-logistic-regression-9b832f025c2d), it's just about using the chain rule:\n",
"\n",
"$$\n",
"\\begin{equation}\n",
"\\begin{split}\n",
"\\frac{\\partial \\sigma(z)}{\\partial w} & = \\frac{\\partial \\sigma(z)}{\\partial z} \\times \\frac{\\partial z}{\\partial w}\\\\\n",
"& = \\sigma(z)(1-\\sigma(z))x_i \\\\\n",
"\\end{split}\n",
"\\end{equation}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- So finally:\n",
"\n",
"$$\n",
"\\begin{equation}\n",
"\\begin{split}\n",
"\\frac{\\partial f(w)}{\\partial w} & =\\frac{1}{n}\\sum_{i=1}^n \\frac{\\sigma(z)-y_i}{\\sigma(z)(1 - \\sigma(z))} \\times \\sigma(z)(1-\\sigma(z))x_i \\\\\n",
"& = \\frac{1}{n}\\sum_{i=1}^nx_i(\\sigma(z)-y_i) \\\\\n",
"& = \\frac{1}{n}\\sum_{i=1}^nx_i\\left(\\frac{1}{1 + \\exp(-w^Tx_i)} - y_i)\\right)\n",
"\\end{split}\n",
"\\end{equation}\n",
"$$"
]
}
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