\n",
"Master in Computer Science
\n", "Master in Data Science and Economics\n", "
\n",
"\n",
"\n", "Master in Data Science and Economics\n", "
Algorithms for massive datasets
\n",
"\n",
"Gradient Boosting
\n",
"\n",
"aka Gradient Boosting Machines, GBM, ...
\n",
"\n",
"\n",
"Prof. Dario Malchiodi\n",
"\n",
"\n",
"![](\"img/logo-di.jpg\")
\n",
"![](\"img/logo-demm.jpg\")
\n",
"![](\"img/logo-unimi.jpg\")
\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"Background
\n",
"\n",
"- \n",
"
- Boosting refers in general to combination of weak learners\n", " into a more powerful composite model \n", "
- This combination is often done via building an ensemble of\n", " learners, each trained separately \n", "
- The ensemble is then queried aggregating the results of the single\n", " components (majority vote, averaging, ...) \n", "
Additive modeling
\n",
"\n",
"Decompose a function additively
\n", "\n", "$$F_M(x) = \\sum_{i=0}^M f_i(x)$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For instance...
"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"ExecuteTime": {
"end_time": "2022-05-25T11:54:43.118973Z",
"start_time": "2022-05-25T11:54:38.411607Z"
}
},
"outputs": [],
"source": [
"import io\n",
"import pandas as pd\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import math\n",
"\n",
"import warnings\n",
"warnings.filterwarnings(\"ignore\", category=UserWarning)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- \n",
"
- We will focus on regression (although the overall methodology applies\n", " also to classification) \n", "
- Here is the graph of a target function $F$ (the actual code producing the\n", " graph is hidden) \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"f = lambda x: 30 + x + np.sin(x)\n",
"x = np.linspace(-30, 30, 100)\n",
"plt.plot(x, f(x))\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
- \n",
"
- The simplest function we can use as an approximation of $F$ is a constant\n", " function $f_0$ \n", "
- We can fix the constant value according to the average value perceived\n", " in the graph, which is 30 \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"x = np.linspace(-30, 30, 100)\n",
"f_0 = np.vectorize(lambda x: 30)\n",
"plt.plot(x, f(x))\n",
"plt.plot(x, f_0(x))\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
- \n",
"
- $F$ approximately increases with $x$ \n", "
- so we can add to our approximation the term $f_1(x) = x$ \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"f_1 = lambda x: x\n",
"plt.plot(x, f(x))\n",
"plt.plot(x, f_0(x) + f_1(x))\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
- \n",
"
- Finally, we notice an oscillating behaviour \n", "
- We model it through a third term $f_2(x) = \\sin x$ \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"f_2 = lambda x: np.sin(x)\n",
"plt.plot(x, f(x))\n",
"plt.plot(x, f_0(x) + f_1(x) + f_2(x))\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We see how the composite result $F_2(x) = f_0(x) + f_1(x) + f_2(x)$ perfectly\n",
"superposes with $F$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"![](\"https://explained.ai/gradient-boosting/images/golf-dir-vector.png\"\n",)
\n",
"
Boosting
\n",
"\n",
"- \n",
"
- Given a set of examples $\\{ (x_1, y_1), \\dots, (x_n, y_n) \\}$ \n", "
- where for each $i$, $y_i = F(x_i)$ for some unknown, yet fixed $F$ \n", "
- Suppose that $F = F_M$ as above, for suitable $M$ (or that $F_M$ is\n", " a good approximation for $F$) \n", "
- That is, $y = F(x)$ can be predicted using $\\hat y = F_m(x)$ in an\n", " incremental tweaking process: $F_m(x) = F_{m-1}(x) + f_m(x)$ (having\n", " fixed an initial $f_0$) \n", "
- This is an example of boosting where $f_i$s are the weak learners\n", " and aggregation is done through summing. \n", "
Gradient boosting
\n",
"\n",
"- \n",
"
- Incrementally learn $f_i$s \n", "
- That is, let $$F_M(x) = f_0(x) + \\sum_{i=1}^M \\Delta_i(x)$$ \n", "
- \n",
"
- $f_0$ is an initial baseline learnt from data \n", "
- $\\Delta_1$ represents an attempt to improve $F_0(x) = f_0(x)$,\n", " obtained through learning the residue $F(x) - F_0(x)$ \n", "
- $\\Delta_2$ represents a further attempt to improve\n", " $F_1(x) = f_0(x) + \\Delta_1(x)$, learning the residue $F(x) - F_1(x)$ \n", "
- and so forth \n", "
An analogy
\n",
"\n",
"\n",
"[Source: explained.ai]
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"![explained.ai](\"https://explained.ai/gradient-boosting/images/golf-dir-vector.png\"){kind=link}
A working example
\n",
"\n",
"Let us consider a toy dataset associating the surface of a flat with its\n", " monthly rent
" ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "ExecuteTime": { "end_time": "2022-05-25T11:55:02.699545Z", "start_time": "2022-05-25T11:55:02.675467Z" } }, "outputs": [ { "data": { "text/html": [ "\n",
"\n",
"\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" sqfeet rent\n",
"0 750 1160\n",
"1 800 1200\n",
"2 850 1280\n",
"3 900 1450\n",
"4 950 2000"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data = pd.read_csv(io.StringIO('''sqfeet,rent\\n\n",
"750,1160\\n\n",
"800,1200\\n\n",
"850,1280\\n\n",
"900,1450\\n\n",
"950,2000'''))\n",
"\n",
"data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"| \n", " | sqfeet | \n", "rent | \n", "
|---|---|---|
| 0 | \n", "750 | \n", "1160 | \n", "
| 1 | \n", "800 | \n", "1200 | \n", "
| 2 | \n", "850 | \n", "1280 | \n", "
| 3 | \n", "900 | \n", "1450 | \n", "
| 4 | \n", "950 | \n", "2000 | \n", "
Each row contains
\n", "- \n",
"
- the observation of a feature $x_i$, in this case the surface \n", "
- the value for a label $y_i$, that is the rent \n", "
- in more general settings, $x_i$s will be vectors \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.plot(data['sqfeet'], data['rent'], 'o', c='orange')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
- \n",
"
- Let us choose as baseline $f_0$ the constant function whose value\n", " equals the average of labels \n", "
- We put $f_0$ inside a list
fs, which will also contain the subsequently\n", " generated $f_i$s \n",
"
- \n",
"
- The use of
fsallows us to define once and for all\n", " a functionFwhich is the equivalent of $F_M$\n", " regardless of the number of improvements \n",
" - For now, invoking
Fis equivalent to evaluating $F_0$ \n",
"
\n",
"\n",
"\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" sqfeet rent F_0\n",
"0 750 1160 1418.0\n",
"1 800 1200 1418.0\n",
"2 850 1280 1418.0\n",
"3 900 1450 1418.0\n",
"4 950 2000 1418.0"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def F(x):\n",
" return sum([f(x) for f in fs])\n",
"\n",
"data['F_0'] = data['sqfeet'].map(F)\n",
"\n",
"data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"| \n", " | sqfeet | \n", "rent | \n", "F_0 | \n", "
|---|---|---|---|
| 0 | \n", "750 | \n", "1160 | \n", "1418.0 | \n", "
| 1 | \n", "800 | \n", "1200 | \n", "1418.0 | \n", "
| 2 | \n", "850 | \n", "1280 | \n", "1418.0 | \n", "
| 3 | \n", "900 | \n", "1450 | \n", "1418.0 | \n", "
| 4 | \n", "950 | \n", "2000 | \n", "1418.0 | \n", "
- \n",
"
- Let us plot $f_0$ (or, analogously, $F_0$) against the dataset \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"x = np.linspace(data['sqfeet'].min(), data['sqfeet'].max(), 100)\n",
"y = list(map(F, x))\n",
"plt.plot(x, y)\n",
"plt.plot(data['sqfeet'], data['rent'], 'o', c='orange')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
""
]
},
"execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from sklearn import tree\n",
"import graphviz\n",
"\n",
"r = tree.DecisionTreeRegressor(max_depth=1)\n",
"r.fit(data[['sqfeet']], data['R_0'])\n",
"\n",
"graphviz.Source(tree.export_graphviz(r,\n",
" out_file=None,\n",
" feature_names=data.columns[0:1],\n",
" rounded=True,\n",
" impurity=False))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"(x)>, (x)>]"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fs"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/html": [
"
- \n",
"
- Let us compute the residuals, that is the difference between\n", " each label and the predicted value \n", "
\n",
"\n",
"\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" sqfeet rent F_0 R_0\n",
"0 750 1160 1418.0 -258.0\n",
"1 800 1200 1418.0 -218.0\n",
"2 850 1280 1418.0 -138.0\n",
"3 900 1450 1418.0 32.0\n",
"4 950 2000 1418.0 582.0"
]
},
"execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data['R_0'] = round(data['rent'] - data['F_0'], 1)\n",
"data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"| \n", " | sqfeet | \n", "rent | \n", "F_0 | \n", "R_0 | \n", "
|---|---|---|---|---|
| 0 | \n", "750 | \n", "1160 | \n", "1418.0 | \n", "-258.0 | \n", "
| 1 | \n", "800 | \n", "1200 | \n", "1418.0 | \n", "-218.0 | \n", "
| 2 | \n", "850 | \n", "1280 | \n", "1418.0 | \n", "-138.0 | \n", "
| 3 | \n", "900 | \n", "1450 | \n", "1418.0 | \n", "32.0 | \n", "
| 4 | \n", "950 | \n", "2000 | \n", "1418.0 | \n", "582.0 | \n", "
- \n",
"
- We want thus to learn to predict the
R_0column values\n", " starting from the corresponding observations \n",
" - We do so using regression tree stumps, that is regression\n", " trees having depth equal to 1 and splitting on a condition involving\n", " only one feature \n", "
How regression trees are built?
\n",
"\n",
"- \n",
"
- Split is chosen so that an impurity measure\n", " is optimized on the two subgroups of observations identifying\n", " the two leaves \n", "
In our case...
"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [],
"source": [
"R_0_left = data[data['sqfeet'] <= 925]['R_0']\n",
"R_0_right = data[data['sqfeet'] > 925]['R_0']"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0 -258.0\n",
"1 -218.0\n",
"2 -138.0\n",
"3 32.0\n",
"Name: R_0, dtype: float64"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"R_0_left"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4 582.0\n",
"Name: R_0, dtype: float64"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"R_0_right"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-145.5, 16491.666666666668)"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(R_0_left.mean(), R_0_left.var())"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(582.0, 0.0)"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(R_0_right.mean(), R_0_right.var(ddof=0))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Why regression tree stumps?
\n",
"\n",
"- \n",
"
- Because we want to use weak learners \n", "
- There are, of course, other possibilities \n", "
Let us continue
\n",
"\n",
"- \n",
"
- Let's build $f_1$ as the prediction done by the regression tree \n", "
- If we add $f_1$ as a second member of the list
fs... \n",
" - ...invoking $F(x)$ will be the equivalent of evaluating $F_1(x)$ \n", "
\n",
"\n",
"\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" sqfeet rent F_0 R_0 F_1\n",
"0 750 1160 1418.0 -258.0 1272.5\n",
"1 800 1200 1418.0 -218.0 1272.5\n",
"2 850 1280 1418.0 -138.0 1272.5\n",
"3 900 1450 1418.0 32.0 1272.5\n",
"4 950 2000 1418.0 582.0 2000.0"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data['F_1'] = round(data['sqfeet'].map(F), 1)\n",
"data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"| \n", " | sqfeet | \n", "rent | \n", "F_0 | \n", "R_0 | \n", "F_1 | \n", "
|---|---|---|---|---|---|
| 0 | \n", "750 | \n", "1160 | \n", "1418.0 | \n", "-258.0 | \n", "1272.5 | \n", "
| 1 | \n", "800 | \n", "1200 | \n", "1418.0 | \n", "-218.0 | \n", "1272.5 | \n", "
| 2 | \n", "850 | \n", "1280 | \n", "1418.0 | \n", "-138.0 | \n", "1272.5 | \n", "
| 3 | \n", "900 | \n", "1450 | \n", "1418.0 | \n", "32.0 | \n", "1272.5 | \n", "
| 4 | \n", "950 | \n", "2000 | \n", "1418.0 | \n", "582.0 | \n", "2000.0 | \n", "
- \n",
"
- Let us plot the prediction (blue stepwise curve) against the dataset... \n", "
- ...as well as the residuals \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.subplot(121)\n",
"y = list(map(F, x))\n",
"plt.plot(x, y)\n",
"plt.plot(data['sqfeet'], data['rent'], 'o')\n",
"\n",
"plt.subplot(122)\n",
"y = list(map(fs[-1], x))\n",
"plt.plot(x, y)\n",
"plt.plot(data['sqfeet'], data['R_0'], 'o')\n",
"\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
Adding improvements
\n",
"\n",
"- \n",
"
- Let us write a function that abstracts out the operations done so far \n", "
- \n",
"
- The function accepts as argument the index $i$ of $\\Delta_i$ \n", "
- computes $\\Delta_i$ \n", "
- draws the prediction and residual for the improved function \n", "
- (globally) updates the DataFrame adding columns for the residual and\n", " the improved function \n", "
- and returns the learnt regression tree stump \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"t = add_improvement(2)"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"execution_count": 97,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"
\n",
"\n",
"\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" sqfeet rent F_0 R_0 F_1 R_1 F_2\n",
"0 750 1160 1418.0 -258.0 1272.5 -112.5 1180.0\n",
"1 800 1200 1418.0 -218.0 1272.5 -72.5 1180.0\n",
"2 850 1280 1418.0 -138.0 1272.5 7.5 1334.2\n",
"3 900 1450 1418.0 32.0 1272.5 177.5 1334.2\n",
"4 950 2000 1418.0 582.0 2000.0 0.0 2061.7"
]
},
"execution_count": 98,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"| \n", " | sqfeet | \n", "rent | \n", "F_0 | \n", "R_0 | \n", "F_1 | \n", "R_1 | \n", "F_2 | \n", "
|---|---|---|---|---|---|---|---|
| 0 | \n", "750 | \n", "1160 | \n", "1418.0 | \n", "-258.0 | \n", "1272.5 | \n", "-112.5 | \n", "1180.0 | \n", "
| 1 | \n", "800 | \n", "1200 | \n", "1418.0 | \n", "-218.0 | \n", "1272.5 | \n", "-72.5 | \n", "1180.0 | \n", "
| 2 | \n", "850 | \n", "1280 | \n", "1418.0 | \n", "-138.0 | \n", "1272.5 | \n", "7.5 | \n", "1334.2 | \n", "
| 3 | \n", "900 | \n", "1450 | \n", "1418.0 | \n", "32.0 | \n", "1272.5 | \n", "177.5 | \n", "1334.2 | \n", "
| 4 | \n", "950 | \n", "2000 | \n", "1418.0 | \n", "582.0 | \n", "2000.0 | \n", "0.0 | \n", "2061.7 | \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"t = add_improvement(3)"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"
\n",
"\n",
"\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" sqfeet rent F_0 R_0 F_1 R_1 F_2 R_2 F_3\n",
"0 750 1160 1418.0 -258.0 1272.5 -112.5 1180.0 -20.0 1195.4\n",
"1 800 1200 1418.0 -218.0 1272.5 -72.5 1180.0 20.0 1195.4\n",
"2 850 1280 1418.0 -138.0 1272.5 7.5 1334.2 -54.2 1349.6\n",
"3 900 1450 1418.0 32.0 1272.5 177.5 1334.2 115.8 1349.6\n",
"4 950 2000 1418.0 582.0 2000.0 0.0 2061.7 -61.7 2000.0"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
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\n",
"text/plain": [
"| \n", " | sqfeet | \n", "rent | \n", "F_0 | \n", "R_0 | \n", "F_1 | \n", "R_1 | \n", "F_2 | \n", "R_2 | \n", "F_3 | \n", "
|---|---|---|---|---|---|---|---|---|---|
| 0 | \n", "750 | \n", "1160 | \n", "1418.0 | \n", "-258.0 | \n", "1272.5 | \n", "-112.5 | \n", "1180.0 | \n", "-20.0 | \n", "1195.4 | \n", "
| 1 | \n", "800 | \n", "1200 | \n", "1418.0 | \n", "-218.0 | \n", "1272.5 | \n", "-72.5 | \n", "1180.0 | \n", "20.0 | \n", "1195.4 | \n", "
| 2 | \n", "850 | \n", "1280 | \n", "1418.0 | \n", "-138.0 | \n", "1272.5 | \n", "7.5 | \n", "1334.2 | \n", "-54.2 | \n", "1349.6 | \n", "
| 3 | \n", "900 | \n", "1450 | \n", "1418.0 | \n", "32.0 | \n", "1272.5 | \n", "177.5 | \n", "1334.2 | \n", "115.8 | \n", "1349.6 | \n", "
| 4 | \n", "950 | \n", "2000 | \n", "1418.0 | \n", "582.0 | \n", "2000.0 | \n", "0.0 | \n", "2061.7 | \n", "-61.7 | \n", "2000.0 | \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"t = add_improvement(4)"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"
\n",
"\n",
"\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" sqfeet rent F_0 R_0 F_1 R_1 F_2 R_2 F_3 R_3 \\\n",
"0 750 1160 1418.0 -258.0 1272.5 -112.5 1180.0 -20.0 1195.4 -35.4 \n",
"1 800 1200 1418.0 -218.0 1272.5 -72.5 1180.0 20.0 1195.4 4.6 \n",
"2 850 1280 1418.0 -138.0 1272.5 7.5 1334.2 -54.2 1349.6 -69.6 \n",
"3 900 1450 1418.0 32.0 1272.5 177.5 1334.2 115.8 1349.6 100.4 \n",
"4 950 2000 1418.0 582.0 2000.0 0.0 2061.7 -61.7 2000.0 0.0 \n",
"\n",
" F_4 \n",
"0 1161.9 \n",
"1 1161.9 \n",
"2 1316.1 \n",
"3 1399.8 \n",
"4 2050.2 "
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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KpxPYNQBLUi3KtxaQ2UR0Sx4GajaWevnmAZhNSFdXOk1ATgCWpI5sArKO0CV3ApuNqYzDQM0mots1ALOx1d0HYBXVJZVnT2CzIpRtPwCzier2KCCzsdXKtx+A2YR0S6XfE9isrbIagBOAVY9HAZmNo2yLwZlNlJeCMBtHvY6bgKySsqUgio4i4wRgSapFh60FZB2jyzUAs7GFJ4JZRXXJO4KZjalWdwKwaupKaDVQJwBLUi1AbgOyCnInsNk4IoJuX/+tgrwUhNk4vBaQVVWX5C0hzcZSDzwRzCrJTUBm4/B+AFZV3hDGbBw1DwO1iurqwn0AVlGHdsI3V8JXurLbQzun9DZeCsKqKqUmIG8KbzPn0E7YvRVqx7PHx5/KHgOs2jKpt6q7E9gqyvMArJoe2zZ88W+oHc/KJ6keuAnIKslbQlo1HX96cuVjqNW9FpBVk+cBWDX1rphc+Rjq4Q1hrJq8JaRV01tuge7e08u6e7PySap7FJBVlLeEtGpatQXW7YDe8wBlt+t2TLoDGKBW91pAVk3dXelsCelRQDazVm2Z0gV/pGw56BmIxywxHgVkNo5aYvMAJH1O0k8k7ZX0DUkLm47dKOmgpCckXVpgmFYCKc0DcAKwJCU4D+BBYHVErAH+CbgRQNKFwGbgImAD8EVJ3YVFacnrTqgG4CYgO81grc6Bwy8WvlbJYC2tTuCI+E7Tw0eAP8jvbwTuiYgTwCFJB4F1wMOzHKKVhCTqiYwCGjcBSLodeD9wJCJWjzj2MeBzwOKIeD4vuxH4IFADrouIb+flFwN3AD3At4DrIxJJgzbkzv/zM/7r/zpQdBgAvGJ+sn+f/DHwP/L7S8kSQkN/XnYGSVuBrQArVkx+aKxVQ3dXOltCTuQbdgfwBeCu5kJJy4H3AE83lTVXh18PPCTpgoioAbeRnfyPkCWADcAD0/8n2Ex69oUBeud185db3l5oHF0Sa887Z1Z/p6SHgNe1OLQtIu7Nn7MNOAU0FjlqVU1p+e2OiB3ADoC1a9emcQWwWZfSRLBxE0BEfF/SyhaHbgU+DtzbVNayOizpZ8DZEfEwgKS7gCtwAkjOseODvPoV83jXm5YUHcqsi4h3j3Vc0tVkteH1TbXXfmB509OWAc+1J0KrgtIvBSHpcuDZiHhsxKGlwDNNjxvV4aX5/ZHlo73/Vkl7JO05evToVEK0KXphYJBzeucVHUZyJG0APgFcHhHNCx7dB2yWNF/SKuB8YHcRMVo5lKoGMJKkXmAb8HutDrcoizHKW3JVuTi/Pn6Shb1ziw4jRV8A5gMP5hPUHomID0XEfkl9wONkTUPX5k2eZi11JbQhzFR62d4ArAIey78Iy4AfSlrH6NXh/vz+yHJLzLHjgyxd2FN0GMmJiDeOcewWYPLrXVhH6u4SiVQAJt8EFBH7ImJJRKyMiJVkF/e3R8TPGaU6HBGHgZckXaIsa1zF6X0HlogXBgZdAzBro+6udGoA4yYASXeTjWl+k6R+SR8c7bkRsR9oVIf/ltOrw9cAfw0cBH6KO4CTU68HLxw/ycIe9wGYtUtKS0FMZBTQleMcXzniccvqcETsAVaPLLd0/L+Tp6gHrgGYtVGXvBqoJejY8UEAFnoUkFnbpDQKyAnAhvz6+EkAFva4BmDWLl3KOoFTWAjBCcCGvDBUA3ACMGuXxhpXKXQEOwHYkBcGnADM2m0oAbgGYCk5ljcBLfAoILO2aSxznsKKoE4ANqTRBLTAfQBmbdPY6c41AEvKCwODvHL+HObN8Wlh1i5DNQAnAEvJr4+f9F//Zm3W6ANIYS6AE4ANOXbcy0CYtVujBuBRQJYUrwNk1n5dCY0CSnbPvU50z+6neepXx8d/Ypscev5f+M1/9ZrCfr9ZJ+hOaBSQE0AiBk7WuOHr++gSzOkqrmL2thULC/vdZp0gpVFATgCJGBjMFk29+fKLuOo3VxYbjJm1zfA8gOITgPsAEtFIAGfN6S44EjNrJy8FYWcYOJkngHlOAGZV5qUg7Awv5zWAnrlOAGZV1mgC8mqgNmTACcCsIww3ARUcCE4AyWg0AfXM83+JWZXl13/3AdiwoU5g1wDMKs1rAdkZ3Adg1hk8CsjO8PJ0awCHdsI3V8JXurLbQztnLDazwuztg1tXw80Ls9u9fUVHNG1eCsLOMNQHMJUEcGgn7N4KtXwZieNPZY8BVm2ZoQjNZtnePrj/OhgcyB4feyZ7DLBmU3FxTVO3J4LZSAOD2ZCAnqnMA3hs2/DFv6F2PCs3K6tdnx6++DcMDmTlJeYmIDtDoxN4/lQ2Yzn+9OTKzcrgWP/kyktiaDnoBJqAnAAS8fJgjZ653Sg/OSald8Xkys3KYMGyyZWXxPCGMAUHghNAMgZO1qbW/APwllugu/f0su7erNysrNbfBHN7Ti+b25OVl1hjNVAPA7UhA4M1zprqXryrtsC6HdB7HqDsdt0OdwBbua3ZBJdthwXLAWW3l20vdQcwMFTLT6EJyKOAEjEwWJveQnCrtviCb9WzZlPpL/gjeRSQneFE3gdgaZP0MUkhaVFT2Y2SDkp6QtKlRcZn6UtpFJBrAIkYcAJInqTlwHuAp5vKLgQ2AxcBrwceknRBRNSKidJS56Ug7AzT6gS22XIr8HGg+Zu7EbgnIk5ExCHgILCuiOCsHLwaqJ1hYLDuheASJuly4NmIeGzEoaXAM02P+/OyVu+xVdIeSXuOHj3apkgtdd4T2M7w8mDNCaBgkh4CXtfi0Dbgk8DvtXpZi7KW3+yI2AHsAFi7dm3x334rREp7AjsBJGLgZI2eua6QFSki3t2qXNJvAKuAx/IhfMuAH0paR/YX//Kmpy8DnmtzqFZiKXUC+4qTCHcCpysi9kXEkohYGREryS76b4+InwP3AZslzZe0Cjgf2F1guJa4lJaCcA0gES9Pdx6AFSIi9kvqAx4HTgHXegSQjaWxHHQKewI7ASSgXg9OnKq7BlASeS2g+fEtgNfdsAlpTATzKCAD4OVT3g3MrFN0JTQKaNwEIOl2SUck/bip7HOSfiJpr6RvSFrYdKzlrEhJF0valx/brikte1lNwxvCOwGYVV3ZloK4A9gwouxBYHVErAH+CbgRzpgVuQH4oqTGVe02YCtZJ9n5Ld6zfGZoG8ahDeHnOAGYVV2pRgFFxPeBX40o+05EnMofPkI29A1GmRUp6Vzg7Ih4OLKej7uAK2bo31CMxjaMx58CYngbxikkgaH9gF0DMKu8RidwVZaC+GPggfz+aLMil+b3R5aX1wxuwzhwMt8O0n0AZpU33Alc8gQgaRvZ0LfGn72jzYqc8GzJ/H3TnzI/g9swNpqAnADMqm+oCajMNQBJVwPvB7bE8IDW0WZF9jPcTNRc3lJE7IiItRGxdvHixVMNsb1mcBvGRhNQzzwPyjKrupSWgpjSFUfSBuATwOUR0dwO0nJWZEQcBl6SdEk++ucq4N5pxl6sGdyGcagT2DUAs8rLKwBJzAMYdyKYpLuBdwKLJPUDnyIb9TMfeDAfzflIRHxonFmR15CNKOoh6zN4gDJr7L712Las2ad3RXbxn8KuXC87AZh1jO6EOoHHTQARcWWL4i+N8fyWsyIjYg+welLRjeLlwRo/euaFmXiradoAa5pGswbwz7+c9Ls8/tyLgPsAzDqBJKSSJIAUHXnxBJt3PFJ0GDNqTpc4u2du0WGY2SzolpIYBVTKBPC6X3+Nn6z9JPNPPsuJeUt5evk2nl/0B0WHNS1LXjWfV84v5X+HmU1SV5emNwpobx/s+jQc64cFy2D9TbBm06TfpnxXnEM7mffoh4bG4J91sp8LfvZRLljyqim1v5uZzbZuaeqjgPb2wf3XweBA9vjYM9ljmHQSKF8CGGsClhOAmZVAd5f46qP9fO+Jyc9z+vJLn+S1MXB64eBAViOofAKYwQlYZmZFuPZdb2Tfsy9M6bVLXny+9YFj/a3Lx1C+BNC7Il9/p0W5mVkJXPPON0z9xbcuy5p9Rlqw7MyycZRv6ukMTsAyMyud9TfB3J7Ty+b2ZOWTVL4EsGoLrNsBvecBym7X7XD7v5l1hjWb4LLtsGA5oOz2su0dMgoIsou9L/hm1qnWbJrSBX+k8tUAzMxsRjgBmJl1KCcAM7MO5QRgZtahnADMzDqUIoElScci6SjQYuYXAIuAUabFzapU4oB0YkklDhg7lvMiYta3nSvJeQ3pxJJKHFCOWM4HHo6IDS2ODUk+AYxF0p6IWOs4hqUSSypxQFqxTERK8aYSSypxQLVicROQmVmHcgIwM+tQZU8AO4oOIJdKHJBOLKnEAWnFMhEpxZtKLKnEARWKpdR9AGZmNnVlrwGYmdkUOQGYmXWoZBOApDdJ+lHTz4uSPizpZknPNpW/r+k1N0o6KOkJSZfOYCwfkbRf0o8l3S3pLEmvlvSgpCfz23PaHccYsRTxmVyfx7Bf0ofzsqI+k1axzPpnMsFYkzmv8/dO4txO5bzO37tzzu2ISP4H6AZ+DpwH3Ax8rMVzLgQeA+YDq4CfAt0z8LuXAoeAnvxxH/AfgM8CN+RlNwB/1s44xolltj+T1cCPgV6yJcUfIpt4UsRnMloss/qZlO28TuncTuW87sRzO9kawAjrgZ9GxGgzJwE2AvdExImIOAQcBNbN0O+fA/RImkP2n/Fc/vvuzI/fCVwxC3GMFsto2hXLm4FHIuJ4RJwC/g74fYr5TEaLZTTt/v+ZjKLPa0jn3E7hvIYOO7fLkgA2A3c3Pf4TSXsl3d5UFVsKNG+U2Z+XTUtEPAt8HngaOAwci4jvAK+NiMP5cw4DS9oZxzixwCx+JmR/lfyOpNdI6gXeByyngM9kjFhgdj+TqSjsvIZ0zu2EzmvosHM7+QQgaR5wOfDVvOg24A3AW8lOlj9vPLXFy6c9xjX/cDeSValeD7xC0gfGekk74hgnlln9TCLiAPBnwIPA35JVO0+NFXo74hgnlln9TCar6PM6jyGJczuV8xo679xOPgEA7wV+GBG/AIiIX0RELSLqwF8xXMXpZzg7Aixj7GrkRL0bOBQRRyNiEPg68FvALySdC5DfHmlzHKPGUsBnQkR8KSLeHhG/A/wKeJJiPpOWsRTxmUxS0ec1pHNuJ3NeQ2ed22VIAFfSVE1u/Cfkfp+smgRwH7BZ0nxJq8g6S3bPwO9/GrhEUq8kkbXbHsh/39X5c64G7m1zHKPGUsBngqQl+e0K4N+T/R8V8Zm0jKWIz2SSij6vIZ1zO5nzGjrs3J6J3up2/ZB1Bv0SWNBU9jfAPmBv/g8+t+nYNrKe7yeA985gHH8K/CT/oP+GrJf9NcAusr8OdgGvbnccY8RSxGfy98DjZNXS9XlZUZ9Jq1hm/TMp23md0rmdynndaee2l4IwM+tQZWgCMjOzNnACMDPrUE4AZmYdygnAzKxDOQGYmXUoJwAzsw7lBGBm1qH+Pyv3grLyAhnnAAAAAElFTkSuQmCC\n",
"text/plain": [
"| \n", " | sqfeet | \n", "rent | \n", "F_0 | \n", "R_0 | \n", "F_1 | \n", "R_1 | \n", "F_2 | \n", "R_2 | \n", "F_3 | \n", "R_3 | \n", "F_4 | \n", "
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | \n", "750 | \n", "1160 | \n", "1418.0 | \n", "-258.0 | \n", "1272.5 | \n", "-112.5 | \n", "1180.0 | \n", "-20.0 | \n", "1195.4 | \n", "-35.4 | \n", "1161.9 | \n", "
| 1 | \n", "800 | \n", "1200 | \n", "1418.0 | \n", "-218.0 | \n", "1272.5 | \n", "-72.5 | \n", "1180.0 | \n", "20.0 | \n", "1195.4 | \n", "4.6 | \n", "1161.9 | \n", "
| 2 | \n", "850 | \n", "1280 | \n", "1418.0 | \n", "-138.0 | \n", "1272.5 | \n", "7.5 | \n", "1334.2 | \n", "-54.2 | \n", "1349.6 | \n", "-69.6 | \n", "1316.1 | \n", "
| 3 | \n", "900 | \n", "1450 | \n", "1418.0 | \n", "32.0 | \n", "1272.5 | \n", "177.5 | \n", "1334.2 | \n", "115.8 | \n", "1349.6 | \n", "100.4 | \n", "1399.8 | \n", "
| 4 | \n", "950 | \n", "2000 | \n", "1418.0 | \n", "582.0 | \n", "2000.0 | \n", "0.0 | \n", "2061.7 | \n", "-61.7 | \n", "2000.0 | \n", "0.0 | \n", "2050.2 | \n", "
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"t = add_improvement(5)"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"t = add_improvement(6)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"F = gradient_boosting(data[['sqfeet']], data['rent'], 5, 1)\n",
"\n",
"plt.plot(x, list(map(F, x)))\n",
"plt.plot(data['sqfeet'], data['rent'], 'o', color='orange')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"result = [[get_mse(data[['sqfeet']], data['rent'], M, eta)\n",
" for M in Ms] for eta in etas]\n",
"\n",
"for r, eta in zip(result, etas):\n",
" plt.plot(Ms, r, label=eta, color='blue' if eta==0.8 else 'gray')\n",
"plt.ylim(0, 20000)\n",
"plt.xlim(1.5, 4)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" sqfeet rent F_0 R_0 F_1 R_1 F_2 R_2 F_3 R_3 \\\n",
"0 750 1160 1418.0 -258.0 1272.5 -112.5 1180.0 -20.0 1195.4 -35.4 \n",
"1 800 1200 1418.0 -218.0 1272.5 -72.5 1180.0 20.0 1195.4 4.6 \n",
"2 850 1280 1418.0 -138.0 1272.5 7.5 1334.2 -54.2 1349.6 -69.6 \n",
"3 900 1450 1418.0 32.0 1272.5 177.5 1334.2 115.8 1349.6 100.4 \n",
"4 950 2000 1418.0 582.0 2000.0 0.0 2061.7 -61.7 2000.0 0.0 \n",
"\n",
" F_4 R_4 F_5 \n",
"0 1161.9 -1.9 1174.5 \n",
"1 1161.9 38.1 1174.5 \n",
"2 1316.1 -36.1 1328.7 \n",
"3 1399.8 50.2 1412.3 \n",
"4 2050.2 -50.2 2000.0 "
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"| \n", " | sqfeet | \n", "rent | \n", "F_0 | \n", "R_0 | \n", "F_1 | \n", "R_1 | \n", "F_2 | \n", "R_2 | \n", "F_3 | \n", "R_3 | \n", "F_4 | \n", "R_4 | \n", "F_5 | \n", "
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | \n", "750 | \n", "1160 | \n", "1418.0 | \n", "-258.0 | \n", "1272.5 | \n", "-112.5 | \n", "1180.0 | \n", "-20.0 | \n", "1195.4 | \n", "-35.4 | \n", "1161.9 | \n", "-1.9 | \n", "1174.5 | \n", "
| 1 | \n", "800 | \n", "1200 | \n", "1418.0 | \n", "-218.0 | \n", "1272.5 | \n", "-72.5 | \n", "1180.0 | \n", "20.0 | \n", "1195.4 | \n", "4.6 | \n", "1161.9 | \n", "38.1 | \n", "1174.5 | \n", "
| 2 | \n", "850 | \n", "1280 | \n", "1418.0 | \n", "-138.0 | \n", "1272.5 | \n", "7.5 | \n", "1334.2 | \n", "-54.2 | \n", "1349.6 | \n", "-69.6 | \n", "1316.1 | \n", "-36.1 | \n", "1328.7 | \n", "
| 3 | \n", "900 | \n", "1450 | \n", "1418.0 | \n", "32.0 | \n", "1272.5 | \n", "177.5 | \n", "1334.2 | \n", "115.8 | \n", "1349.6 | \n", "100.4 | \n", "1399.8 | \n", "50.2 | \n", "1412.3 | \n", "
| 4 | \n", "950 | \n", "2000 | \n", "1418.0 | \n", "582.0 | \n", "2000.0 | \n", "0.0 | \n", "2061.7 | \n", "-61.7 | \n", "2000.0 | \n", "0.0 | \n", "2050.2 | \n", "-50.2 | \n", "2000.0 | \n", "
Incremental shrinkage
\n",
"\n",
"i.e., adding a learning rate $\\eta > 0$:
\n", "\n", "$$F_M(x) = f_0(x) + \\eta \\sum_{i=1}^M \\Delta_i(x)$$" ] }, { "cell_type": "code", "execution_count": 99, "metadata": {}, "outputs": [], "source": [ "def gradient_boosting(X, y, M, eta=1):\n", " base = y.mean()\n", " f0 = lambda x: base\n", " rs = [None] * M\n", " F = lambda x: f0(x)\n", "\n", " for i in range(1, M+1):\n", " fval = [np.round(F(x), 1) for x in X['sqfeet']]\n", " \n", " residual = round(y - fval, 1)\n", " rs[i-1] = tree.DecisionTreeRegressor(max_depth=1)\n", " rs[i-1].fit(X, residual)\n", " \n", " \n", " #deltafunc = lambda x: rs[i-1].predict([[x]])[0]\n", " F = lambda x: f0(x) + eta * sum([rs[j-1].predict([[x]])[0]\n", " for j in range(1, i)])\n", " \n", "\n", " return np.vectorize(lambda x: F(x))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A sanity check
\n",
"Let's use $\\eta = 1$
" ] }, { "cell_type": "code", "execution_count": 100, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "How do we evaluate our results
\n",
"\n",
"- \n",
"
- Using a loss function \n", "
- In this case, $\\mathrm{MSE}$:\n", " $$ L(\\mathbf x, \\mathbf y, F_m) = \\mathrm{MSE}(\\mathbf x, \\mathbf y, F_m)\n", " = \\frac{1}{n} \\sum_{i=1}^n \\left(y_i - F_m(x_i) \\right)^2 $$\n", " where $\\mathbf x$ and $\\mathbf y$ gather all observations and labels \n", "
- (we'll see that this is a suitable choice among various possibilities) \n", "
Hyperparameters
\n",
"\n",
"- \n",
"
- How do we choose $M$ and $\\eta$? \n", "
- Evluating our loss function over a grid \n", "
Stochastic gradient boosting
\n",
"\n",
"Don't use all data at each iteration:
\n", "- \n",
"
- subsample rows before creating each tree \n", "
- subsample columns before creating each tree \n", "
- subsample columns before considering each split \n", "
Aggressive subsampling (e.g., 50% of data) often proves\n", " to be effective
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Summing up
\n",
"\n",
"$$ F_M(x) = f_0(x) + \\eta \\sum_{i=1}^M \\Delta_i(x) $$\n",
"\n",
"- \n",
"
- at each iteration $m$ we learn residuals $y_i - F_{m-1}(x_i)$ \n", "
- thus we use both sign and magnitude \n", "
- risk: chasing outliers! \n", "
- possible solution: just learn $\\mathrm{sign} \\left( y_i - F_{m-1}(x_i) \\right)$ \n", "
- hint: don't evaluate using MSE, use MAE instead:\n", " $$ \\mathrm{MAE}(\\mathbf x, \\mathbf y, F_M) =\n", " \\frac{1}{n} \\sum_{i=1}^n \\left|y_i - F_{m-1}(x_i) \\right|$$ \n", "
- note: when learning regression tree stumps, choose splits according\n", " to squared error minimization (for sake of efficiency) \n", "
A note on MSE
\n",
"- \n",
"
- consider $L(\\mathbf x, \\mathbf y, F_M) = \\frac{1}{n}\\sum_i \\left( y_i - F_M(x_i) \\right)^2$ \n", "
- let $\\hat y_i := F_M(x_i)$, and note that $n$ is fixed \n", "
- $L(\\mathbf x, \\mathbf y, F_M) \\propto \\sum_i \\left( y_i - \\hat y_i \\right)^2$ \n", "
- for each $k$ we have $\\frac{\\partial}{\\partial \\hat y_k}L(\\mathbf y, \\hat {\\mathbf y}) \\propto -2(y_k - \\hat y_k)$ \n", "
- thus $\\nabla L(\\mathbf y, \\hat {\\mathbf y}) \\propto -2(\\mathbf y - \\hat {\\mathbf y})$ \n", "
A note on MAE
\n",
"- \n",
"
- consider $L(\\mathbf x, \\mathbf y, F_M) = \\frac{1}{n}\\sum_i \\left| y_i - F_M(x_i) \\right|$ \n", "
- $L(\\mathbf x, \\mathbf y, F_M) \\propto \\sum_i \\left| y_i - \\hat y_i \\right|$ \n", "
- for each $k$ we have $\\frac{\\partial}{\\partial \\hat y_k}L(\\mathbf y, \\hat {\\mathbf y}) \\propto -\\mathrm{sign}(y_k - \\hat y_k)$ \n", "
- thus $\\nabla L(\\mathbf y, \\hat {\\mathbf y}) \\propto - \\mathrm{sign}(\\mathbf y - \\hat {\\mathbf y})$ \n", "
Why «gradient» boosting?
\n",
"- \n",
"
- At each iteration of gradient boosting $F_m(\\mathbf x) = F_{m-1}(\\mathbf x) + \\eta \\Delta_m(\\mathbf x)$ \n", "
- Let $\\hat{\\mathbf y}_m := F_m(\\mathbf x)$, so that $\\hat{\\mathbf y}_m = \\hat{\\mathbf y}_{m-1} + \\eta \\Delta_m(\\mathbf x)$ \n", "
- Let $\\mathbf r_{m-1}$ denote the target residuals that we approximate via $\\Delta_m(\\mathbf x)$ \n", "
- Using this approximation we have $\\hat{\\mathbf y}_m = \\hat{\\mathbf y}_{m-1} + \\eta \\mathbf r_{m-1}$... \n", "
- ...that is, $\\hat{\\mathbf y}_t = \\hat{\\mathbf y}_{t-1} - \\eta (-\\mathbf r_{t-1})$ \n", "
- Compare this with the gradient descent update $\\mathbf x_t = \\mathbf x_{t-1} - \\eta \\nabla L(\\mathbf x_{t-1})$ \n", "
Thus at each iteration the next improvement is chosen implicitly optimizing\n", " via gradient descent either MSE or MAE.
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Gradient boosting in practice
\n",
"\n",
"- \n",
"
- XGBoost\n", " is a popular implementation of gradient boosting \n", "
- it provides portability and scalability (e.g., it\n", " runs on Hadoop and Spark) \n", "
- it proved successful in several Kaggle competitions \n", "
Sources
\n",
"\n",
"The analysed experiments are from T. Parr and J. Howard, How\n", " to explain gradient boosting.
" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.11" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": false, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": false, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }