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<ul>
<li><a class="reference internal" href="#">1.11. Ensembles: Gradient boosting, random forests, bagging, voting, stacking</a><ul>
<li><a class="reference internal" href="#gradient-boosted-trees">1.11.1. Gradient-boosted trees</a><ul>
<li><a class="reference internal" href="#histogram-based-gradient-boosting">1.11.1.1. Histogram-Based Gradient Boosting</a><ul>
<li><a class="reference internal" href="#usage">1.11.1.1.1. Usage</a></li>
<li><a class="reference internal" href="#missing-values-support">1.11.1.1.2. Missing values support</a></li>
<li><a class="reference internal" href="#sample-weight-support">1.11.1.1.3. Sample weight support</a></li>
<li><a class="reference internal" href="#categorical-features-support">1.11.1.1.4. Categorical Features Support</a></li>
<li><a class="reference internal" href="#monotonic-constraints">1.11.1.1.5. Monotonic Constraints</a></li>
<li><a class="reference internal" href="#interaction-constraints">1.11.1.1.6. Interaction constraints</a></li>
<li><a class="reference internal" href="#low-level-parallelism">1.11.1.1.7. Low-level parallelism</a></li>
<li><a class="reference internal" href="#why-it-s-faster">1.11.1.1.8. Why it’s faster</a></li>
</ul>
</li>
<li><a class="reference internal" href="#gradientboostingclassifier-and-gradientboostingregressor">1.11.1.2. <code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code> and <code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a><ul>
<li><a class="reference internal" href="#fitting-additional-weak-learners">1.11.1.2.1. Fitting additional weak-learners</a></li>
<li><a class="reference internal" href="#controlling-the-tree-size">1.11.1.2.2. Controlling the tree size</a></li>
<li><a class="reference internal" href="#mathematical-formulation">1.11.1.2.3. Mathematical formulation</a></li>
<li><a class="reference internal" href="#loss-functions">1.11.1.2.4. Loss Functions</a></li>
<li><a class="reference internal" href="#shrinkage-via-learning-rate">1.11.1.2.5. Shrinkage via learning rate</a></li>
<li><a class="reference internal" href="#subsampling">1.11.1.2.6. Subsampling</a></li>
<li><a class="reference internal" href="#interpretation-with-feature-importance">1.11.1.2.7. Interpretation with feature importance</a></li>
</ul>
</li>
</ul>
</li>
<li><a class="reference internal" href="#random-forests-and-other-randomized-tree-ensembles">1.11.2. Random forests and other randomized tree ensembles</a><ul>
<li><a class="reference internal" href="#random-forests">1.11.2.1. Random Forests</a></li>
<li><a class="reference internal" href="#extremely-randomized-trees">1.11.2.2. Extremely Randomized Trees</a></li>
<li><a class="reference internal" href="#parameters">1.11.2.3. Parameters</a></li>
<li><a class="reference internal" href="#parallelization">1.11.2.4. Parallelization</a></li>
<li><a class="reference internal" href="#feature-importance-evaluation">1.11.2.5. Feature importance evaluation</a></li>
<li><a class="reference internal" href="#totally-random-trees-embedding">1.11.2.6. Totally Random Trees Embedding</a></li>
</ul>
</li>
<li><a class="reference internal" href="#bagging-meta-estimator">1.11.3. Bagging meta-estimator</a></li>
<li><a class="reference internal" href="#voting-classifier">1.11.4. Voting Classifier</a><ul>
<li><a class="reference internal" href="#majority-class-labels-majority-hard-voting">1.11.4.1. Majority Class Labels (Majority/Hard Voting)</a></li>
<li><a class="reference internal" href="#id24">1.11.4.2. Usage</a></li>
<li><a class="reference internal" href="#weighted-average-probabilities-soft-voting">1.11.4.3. Weighted Average Probabilities (Soft Voting)</a></li>
<li><a class="reference internal" href="#id25">1.11.4.4. Usage</a></li>
</ul>
</li>
<li><a class="reference internal" href="#voting-regressor">1.11.5. Voting Regressor</a><ul>
<li><a class="reference internal" href="#id27">1.11.5.1. Usage</a></li>
</ul>
</li>
<li><a class="reference internal" href="#stacked-generalization">1.11.6. Stacked generalization</a></li>
<li><a class="reference internal" href="#adaboost">1.11.7. AdaBoost</a><ul>
<li><a class="reference internal" href="#id35">1.11.7.1. Usage</a></li>
</ul>
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<div class="sk-page-content container-fluid body px-md-3" role="main">
<section id="ensembles-gradient-boosting-random-forests-bagging-voting-stacking">
<span id="ensemble"></span><h1><span class="section-number">1.11. </span>Ensembles: Gradient boosting, random forests, bagging, voting, stacking<a class="headerlink" href="#ensembles-gradient-boosting-random-forests-bagging-voting-stacking" title="Link to this heading">¶</a></h1>
<p><strong>Ensemble methods</strong> combine the predictions of several
base estimators built with a given learning algorithm in order to improve
generalizability / robustness over a single estimator.</p>
<p>Two very famous examples of ensemble methods are <a class="reference internal" href="#gradient-boosting"><span class="std std-ref">gradient-boosted trees</span></a> and <a class="reference internal" href="#forest"><span class="std std-ref">random forests</span></a>.</p>
<p>More generally, ensemble models can be applied to any base learner beyond
trees, in averaging methods such as <a class="reference internal" href="#bagging"><span class="std std-ref">Bagging methods</span></a>,
<a class="reference internal" href="#stacking"><span class="std std-ref">model stacking</span></a>, or <a class="reference internal" href="#voting-classifier"><span class="std std-ref">Voting</span></a>, or in
boosting, as <a class="reference internal" href="#adaboost"><span class="std std-ref">AdaBoost</span></a>.</p>
<nav class="contents local" id="contents">
<ul class="simple">
<li><p><a class="reference internal" href="#gradient-boosted-trees" id="id36">Gradient-boosted trees</a></p></li>
<li><p><a class="reference internal" href="#random-forests-and-other-randomized-tree-ensembles" id="id37">Random forests and other randomized tree ensembles</a></p></li>
<li><p><a class="reference internal" href="#bagging-meta-estimator" id="id38">Bagging meta-estimator</a></p></li>
<li><p><a class="reference internal" href="#voting-classifier" id="id39">Voting Classifier</a></p></li>
<li><p><a class="reference internal" href="#voting-regressor" id="id40">Voting Regressor</a></p></li>
<li><p><a class="reference internal" href="#stacked-generalization" id="id41">Stacked generalization</a></p></li>
<li><p><a class="reference internal" href="#adaboost" id="id42">AdaBoost</a></p></li>
</ul>
</nav>
<section id="gradient-boosted-trees">
<span id="gradient-boosting"></span><h2><a class="toc-backref" href="#id36" role="doc-backlink"><span class="section-number">1.11.1. </span>Gradient-boosted trees</a><a class="headerlink" href="#gradient-boosted-trees" title="Link to this heading">¶</a></h2>
<p><a class="reference external" href="https://en.wikipedia.org/wiki/Gradient_boosting">Gradient Tree Boosting</a>
or Gradient Boosted Decision Trees (GBDT) is a generalization
of boosting to arbitrary differentiable loss functions, see the seminal work of
<a class="reference internal" href="#friedman2001" id="id1"><span>[Friedman2001]</span></a>. GBDT is an excellent model for both regression and
classification, in particular for tabular data.</p>
<aside class="topic">
<p class="topic-title"><a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> vs <a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a></p>
<p>Scikit-learn provides two implementations of gradient-boosted trees:
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a> vs
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> for classification, and the
corresponding classes for regression. The former can be <strong>orders of
magnitude faster</strong> than the latter when the number of samples is
larger than tens of thousands of samples.</p>
<p>Missing values and categorical data are natively supported by the
Hist… version, removing the need for additional preprocessing such as
imputation.</p>
<p><a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a>, might be preferred for small sample
sizes since binning may lead to split points that are too approximate
in this setting.</p>
</aside>
<section id="histogram-based-gradient-boosting">
<span id="id2"></span><h3><span class="section-number">1.11.1.1. </span>Histogram-Based Gradient Boosting<a class="headerlink" href="#histogram-based-gradient-boosting" title="Link to this heading">¶</a></h3>
<p>Scikit-learn 0.21 introduced two new implementations of
gradient boosted trees, namely <a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a>
and <a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a>, inspired by
<a class="reference external" href="https://github.com/Microsoft/LightGBM">LightGBM</a> (See <a class="reference internal" href="#lightgbm" id="id3"><span>[LightGBM]</span></a>).</p>
<p>These histogram-based estimators can be <strong>orders of magnitude faster</strong>
than <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a> when the number of samples is larger
than tens of thousands of samples.</p>
<p>They also have built-in support for missing values, which avoids the need
for an imputer.</p>
<p>These fast estimators first bin the input samples <code class="docutils literal notranslate"><span class="pre">X</span></code> into
integer-valued bins (typically 256 bins) which tremendously reduces the
number of splitting points to consider, and allows the algorithm to
leverage integer-based data structures (histograms) instead of relying on
sorted continuous values when building the trees. The API of these
estimators is slightly different, and some of the features from
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> and <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a>
are not yet supported, for instance some loss functions.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/inspection/plot_partial_dependence.html#sphx-glr-auto-examples-inspection-plot-partial-dependence-py"><span class="std std-ref">Partial Dependence and Individual Conditional Expectation Plots</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/ensemble/plot_forest_hist_grad_boosting_comparison.html#sphx-glr-auto-examples-ensemble-plot-forest-hist-grad-boosting-comparison-py"><span class="std std-ref">Comparing Random Forests and Histogram Gradient Boosting models</span></a></p></li>
</ul>
</aside>
<section id="usage">
<h4><span class="section-number">1.11.1.1.1. </span>Usage<a class="headerlink" href="#usage" title="Link to this heading">¶</a></h4>
<p>Most of the parameters are unchanged from
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> and <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a>.
One exception is the <code class="docutils literal notranslate"><span class="pre">max_iter</span></code> parameter that replaces <code class="docutils literal notranslate"><span class="pre">n_estimators</span></code>, and
controls the number of iterations of the boosting process:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">HistGradientBoostingClassifier</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_hastie_10_2</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_hastie_10_2</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:</span><span class="mi">2000</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="mi">2000</span><span class="p">:]</span>
<span class="gp">>>> </span><span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">y</span><span class="p">[:</span><span class="mi">2000</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="mi">2000</span><span class="p">:]</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">HistGradientBoostingClassifier</span><span class="p">(</span><span class="n">max_iter</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">)</span>
<span class="go">0.8965</span>
</pre></div>
</div>
<p>Available losses for regression are ‘squared_error’,
‘absolute_error’, which is less sensitive to outliers, and
‘poisson’, which is well suited to model counts and frequencies. For
classification, ‘log_loss’ is the only option. For binary classification it uses the
binary log loss, also known as binomial deviance or binary cross-entropy. For
<code class="docutils literal notranslate"><span class="pre">n_classes</span> <span class="pre">>=</span> <span class="pre">3</span></code>, it uses the multi-class log loss function, with multinomial deviance
and categorical cross-entropy as alternative names. The appropriate loss version is
selected based on <a class="reference internal" href="../glossary.html#term-y"><span class="xref std std-term">y</span></a> passed to <a class="reference internal" href="../glossary.html#term-fit"><span class="xref std std-term">fit</span></a>.</p>
<p>The size of the trees can be controlled through the <code class="docutils literal notranslate"><span class="pre">max_leaf_nodes</span></code>,
<code class="docutils literal notranslate"><span class="pre">max_depth</span></code>, and <code class="docutils literal notranslate"><span class="pre">min_samples_leaf</span></code> parameters.</p>
<p>The number of bins used to bin the data is controlled with the <code class="docutils literal notranslate"><span class="pre">max_bins</span></code>
parameter. Using less bins acts as a form of regularization. It is generally
recommended to use as many bins as possible (255), which is the default.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">l2_regularization</span></code> parameter acts as a regularizer for the loss function,
and corresponds to <span class="math notranslate nohighlight">\(\lambda\)</span> in the following expression (see equation (2)
in <a class="reference internal" href="#xgboost" id="id4"><span>[XGBoost]</span></a>):</p>
<div class="math notranslate nohighlight">
\[\mathcal{L}(\phi) = \sum_i l(\hat{y}_i, y_i) + \frac12 \sum_k \lambda ||w_k||^2\]</div>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Details on l2 regularization</strong>:
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p>It is important to notice that the loss term <span class="math notranslate nohighlight">\(l(\hat{y}_i, y_i)\)</span> describes
only half of the actual loss function except for the pinball loss and absolute
error.</p>
<p>The index <span class="math notranslate nohighlight">\(k\)</span> refers to the k-th tree in the ensemble of trees. In the
case of regression and binary classification, gradient boosting models grow one
tree per iteration, then <span class="math notranslate nohighlight">\(k\)</span> runs up to <code class="docutils literal notranslate"><span class="pre">max_iter</span></code>. In the case of
multiclass classification problems, the maximal value of the index <span class="math notranslate nohighlight">\(k\)</span> is
<code class="docutils literal notranslate"><span class="pre">n_classes</span></code> <span class="math notranslate nohighlight">\(\times\)</span> <code class="docutils literal notranslate"><span class="pre">max_iter</span></code>.</p>
<p>If <span class="math notranslate nohighlight">\(T_k\)</span> denotes the number of leaves in the k-th tree, then <span class="math notranslate nohighlight">\(w_k\)</span>
is a vector of length <span class="math notranslate nohighlight">\(T_k\)</span>, which contains the leaf values of the form <code class="docutils literal notranslate"><span class="pre">w</span>
<span class="pre">=</span> <span class="pre">-sum_gradient</span> <span class="pre">/</span> <span class="pre">(sum_hessian</span> <span class="pre">+</span> <span class="pre">l2_regularization)</span></code> (see equation (5) in
<a class="reference internal" href="#xgboost" id="id5"><span>[XGBoost]</span></a>).</p>
<p>The leaf values <span class="math notranslate nohighlight">\(w_k\)</span> are derived by dividing the sum of the gradients of
the loss function by the combined sum of hessians. Adding the regularization to
the denominator penalizes the leaves with small hessians (flat regions),
resulting in smaller updates. Those <span class="math notranslate nohighlight">\(w_k\)</span> values contribute then to the
model’s prediction for a given input that ends up in the corresponding leaf. The
final prediction is the sum of the base prediction and the contributions from
each tree. The result of that sum is then transformed by the inverse link
function depending on the choice of the loss function (see
<a class="reference internal" href="#gradient-boosting-formulation"><span class="std std-ref">Mathematical formulation</span></a>).</p>
<p>Notice that the original paper <a class="reference internal" href="#xgboost" id="id6"><span>[XGBoost]</span></a> introduces a term <span class="math notranslate nohighlight">\(\gamma\sum_k
T_k\)</span> that penalizes the number of leaves (making it a smooth version of
<code class="docutils literal notranslate"><span class="pre">max_leaf_nodes</span></code>) not presented here as it is not implemented in scikit-learn;
whereas <span class="math notranslate nohighlight">\(\lambda\)</span> penalizes the magnitude of the individual tree
predictions before being rescaled by the learning rate, see
<a class="reference internal" href="#gradient-boosting-shrinkage"><span class="std std-ref">Shrinkage via learning rate</span></a>.</p>
<p></div>
</details></p>
<p>Note that <strong>early-stopping is enabled by default if the number of samples is
larger than 10,000</strong>. The early-stopping behaviour is controlled via the
<code class="docutils literal notranslate"><span class="pre">early_stopping</span></code>, <code class="docutils literal notranslate"><span class="pre">scoring</span></code>, <code class="docutils literal notranslate"><span class="pre">validation_fraction</span></code>,
<code class="docutils literal notranslate"><span class="pre">n_iter_no_change</span></code>, and <code class="docutils literal notranslate"><span class="pre">tol</span></code> parameters. It is possible to early-stop
using an arbitrary <a class="reference internal" href="../glossary.html#term-scorer"><span class="xref std std-term">scorer</span></a>, or just the training or validation loss.
Note that for technical reasons, using a callable as a scorer is significantly slower
than using the loss. By default, early-stopping is performed if there are at least
10,000 samples in the training set, using the validation loss.</p>
</section>
<section id="missing-values-support">
<span id="nan-support-hgbt"></span><h4><span class="section-number">1.11.1.1.2. </span>Missing values support<a class="headerlink" href="#missing-values-support" title="Link to this heading">¶</a></h4>
<p><a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a> have built-in support for missing
values (NaNs).</p>
<p>During training, the tree grower learns at each split point whether samples
with missing values should go to the left or right child, based on the
potential gain. When predicting, samples with missing values are assigned to
the left or right child consequently:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">HistGradientBoostingClassifier</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">])</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">gbdt</span> <span class="o">=</span> <span class="n">HistGradientBoostingClassifier</span><span class="p">(</span><span class="n">min_samples_leaf</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">gbdt</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="go">array([0, 0, 1, 1])</span>
</pre></div>
</div>
<p>When the missingness pattern is predictive, the splits can be performed on
whether the feature value is missing or not:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">])</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">gbdt</span> <span class="o">=</span> <span class="n">HistGradientBoostingClassifier</span><span class="p">(</span><span class="n">min_samples_leaf</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">learning_rate</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">gbdt</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="go">array([0, 1, 0, 0, 1])</span>
</pre></div>
</div>
<p>If no missing values were encountered for a given feature during training,
then samples with missing values are mapped to whichever child has the most
samples.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/ensemble/plot_hgbt_regression.html#sphx-glr-auto-examples-ensemble-plot-hgbt-regression-py"><span class="std std-ref">Features in Histogram Gradient Boosting Trees</span></a></p></li>
</ul>
</aside>
</section>
<section id="sample-weight-support">
<span id="sw-hgbdt"></span><h4><span class="section-number">1.11.1.1.3. </span>Sample weight support<a class="headerlink" href="#sample-weight-support" title="Link to this heading">¶</a></h4>
<p><a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a> support sample weights during
<a class="reference internal" href="../glossary.html#term-fit"><span class="xref std std-term">fit</span></a>.</p>
<p>The following toy example demonstrates that samples with a sample weight of zero are ignored:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="gp">>>> </span><span class="c1"># ignore the first 2 training samples by setting their weight to 0</span>
<span class="gp">>>> </span><span class="n">sample_weight</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">gb</span> <span class="o">=</span> <span class="n">HistGradientBoostingClassifier</span><span class="p">(</span><span class="n">min_samples_leaf</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">gb</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">sample_weight</span><span class="o">=</span><span class="n">sample_weight</span><span class="p">)</span>
<span class="go">HistGradientBoostingClassifier(...)</span>
<span class="gp">>>> </span><span class="n">gb</span><span class="o">.</span><span class="n">predict</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="go">array([1])</span>
<span class="gp">>>> </span><span class="n">gb</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="go">0.99...</span>
</pre></div>
</div>
<p>As you can see, the <code class="docutils literal notranslate"><span class="pre">[1,</span> <span class="pre">0]</span></code> is comfortably classified as <code class="docutils literal notranslate"><span class="pre">1</span></code> since the first
two samples are ignored due to their sample weights.</p>
<p>Implementation detail: taking sample weights into account amounts to
multiplying the gradients (and the hessians) by the sample weights. Note that
the binning stage (specifically the quantiles computation) does not take the
weights into account.</p>
</section>
<section id="categorical-features-support">
<span id="categorical-support-gbdt"></span><h4><span class="section-number">1.11.1.1.4. </span>Categorical Features Support<a class="headerlink" href="#categorical-features-support" title="Link to this heading">¶</a></h4>
<p><a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a> have native support for categorical
features: they can consider splits on non-ordered, categorical data.</p>
<p>For datasets with categorical features, using the native categorical support
is often better than relying on one-hot encoding
(<a class="reference internal" href="generated/sklearn.preprocessing.OneHotEncoder.html#sklearn.preprocessing.OneHotEncoder" title="sklearn.preprocessing.OneHotEncoder"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneHotEncoder</span></code></a>), because one-hot encoding
requires more tree depth to achieve equivalent splits. It is also usually
better to rely on the native categorical support rather than to treat
categorical features as continuous (ordinal), which happens for ordinal-encoded
categorical data, since categories are nominal quantities where order does not
matter.</p>
<p>To enable categorical support, a boolean mask can be passed to the
<code class="docutils literal notranslate"><span class="pre">categorical_features</span></code> parameter, indicating which feature is categorical. In
the following, the first feature will be treated as categorical and the
second feature as numerical:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">gbdt</span> <span class="o">=</span> <span class="n">HistGradientBoostingClassifier</span><span class="p">(</span><span class="n">categorical_features</span><span class="o">=</span><span class="p">[</span><span class="kc">True</span><span class="p">,</span> <span class="kc">False</span><span class="p">])</span>
</pre></div>
</div>
<p>Equivalently, one can pass a list of integers indicating the indices of the
categorical features:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">gbdt</span> <span class="o">=</span> <span class="n">HistGradientBoostingClassifier</span><span class="p">(</span><span class="n">categorical_features</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
</pre></div>
</div>
<p>When the input is a DataFrame, it is also possible to pass a list of column
names:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">gbdt</span> <span class="o">=</span> <span class="n">HistGradientBoostingClassifier</span><span class="p">(</span><span class="n">categorical_features</span><span class="o">=</span><span class="p">[</span><span class="s2">"site"</span><span class="p">,</span> <span class="s2">"manufacturer"</span><span class="p">])</span>
</pre></div>
</div>
<p>Finally, when the input is a DataFrame we can use
<code class="docutils literal notranslate"><span class="pre">categorical_features="from_dtype"</span></code> in which case all columns with a categorical
<code class="docutils literal notranslate"><span class="pre">dtype</span></code> will be treated as categorical features.</p>
<p>The cardinality of each categorical feature must be less than the <code class="docutils literal notranslate"><span class="pre">max_bins</span></code>
parameter. For an example using histogram-based gradient boosting on categorical
features, see
<a class="reference internal" href="../auto_examples/ensemble/plot_gradient_boosting_categorical.html#sphx-glr-auto-examples-ensemble-plot-gradient-boosting-categorical-py"><span class="std std-ref">Categorical Feature Support in Gradient Boosting</span></a>.</p>
<p>If there are missing values during training, the missing values will be
treated as a proper category. If there are no missing values during training,
then at prediction time, missing values are mapped to the child node that has
the most samples (just like for continuous features). When predicting,
categories that were not seen during fit time will be treated as missing
values.</p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Split finding with categorical features</strong>:
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p>The canonical way of considering
categorical splits in a tree is to consider
all of the <span class="math notranslate nohighlight">\(2^{K - 1} - 1\)</span> partitions, where <span class="math notranslate nohighlight">\(K\)</span> is the number of
categories. This can quickly become prohibitive when <span class="math notranslate nohighlight">\(K\)</span> is large.
Fortunately, since gradient boosting trees are always regression trees (even
for classification problems), there exist a faster strategy that can yield
equivalent splits. First, the categories of a feature are sorted according to
the variance of the target, for each category <code class="docutils literal notranslate"><span class="pre">k</span></code>. Once the categories are
sorted, one can consider <em>continuous partitions</em>, i.e. treat the categories
as if they were ordered continuous values (see Fisher <a class="reference internal" href="#fisher1958" id="id7"><span>[Fisher1958]</span></a> for a
formal proof). As a result, only <span class="math notranslate nohighlight">\(K - 1\)</span> splits need to be considered
instead of <span class="math notranslate nohighlight">\(2^{K - 1} - 1\)</span>. The initial sorting is a
<span class="math notranslate nohighlight">\(\mathcal{O}(K \log(K))\)</span> operation, leading to a total complexity of
<span class="math notranslate nohighlight">\(\mathcal{O}(K \log(K) + K)\)</span>, instead of <span class="math notranslate nohighlight">\(\mathcal{O}(2^K)\)</span>.</p>
<p></div>
</details></p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/ensemble/plot_gradient_boosting_categorical.html#sphx-glr-auto-examples-ensemble-plot-gradient-boosting-categorical-py"><span class="std std-ref">Categorical Feature Support in Gradient Boosting</span></a></p></li>
</ul>
</aside>
</section>
<section id="monotonic-constraints">
<span id="monotonic-cst-gbdt"></span><h4><span class="section-number">1.11.1.1.5. </span>Monotonic Constraints<a class="headerlink" href="#monotonic-constraints" title="Link to this heading">¶</a></h4>
<p>Depending on the problem at hand, you may have prior knowledge indicating
that a given feature should in general have a positive (or negative) effect
on the target value. For example, all else being equal, a higher credit
score should increase the probability of getting approved for a loan.
Monotonic constraints allow you to incorporate such prior knowledge into the
model.</p>
<p>For a predictor <span class="math notranslate nohighlight">\(F\)</span> with two features:</p>
<ul>
<li><p>a <strong>monotonic increase constraint</strong> is a constraint of the form:</p>
<div class="math notranslate nohighlight">
\[x_1 \leq x_1' \implies F(x_1, x_2) \leq F(x_1', x_2)\]</div>
</li>
<li><p>a <strong>monotonic decrease constraint</strong> is a constraint of the form:</p>
<div class="math notranslate nohighlight">
\[x_1 \leq x_1' \implies F(x_1, x_2) \geq F(x_1', x_2)\]</div>
</li>
</ul>
<p>You can specify a monotonic constraint on each feature using the
<code class="docutils literal notranslate"><span class="pre">monotonic_cst</span></code> parameter. For each feature, a value of 0 indicates no
constraint, while 1 and -1 indicate a monotonic increase and
monotonic decrease constraint, respectively:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">HistGradientBoostingRegressor</span>
<span class="go">... # monotonic increase, monotonic decrease, and no constraint on the 3 features</span>
<span class="gp">>>> </span><span class="n">gbdt</span> <span class="o">=</span> <span class="n">HistGradientBoostingRegressor</span><span class="p">(</span><span class="n">monotonic_cst</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
</pre></div>
</div>
<p>In a binary classification context, imposing a monotonic increase (decrease) constraint means that higher values of the feature are supposed
to have a positive (negative) effect on the probability of samples
to belong to the positive class.</p>
<p>Nevertheless, monotonic constraints only marginally constrain feature effects on the output.
For instance, monotonic increase and decrease constraints cannot be used to enforce the
following modelling constraint:</p>
<div class="math notranslate nohighlight">
\[x_1 \leq x_1' \implies F(x_1, x_2) \leq F(x_1', x_2')\]</div>
<p>Also, monotonic constraints are not supported for multiclass classification.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Since categories are unordered quantities, it is not possible to enforce
monotonic constraints on categorical features.</p>
</div>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/ensemble/plot_monotonic_constraints.html#sphx-glr-auto-examples-ensemble-plot-monotonic-constraints-py"><span class="std std-ref">Monotonic Constraints</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/ensemble/plot_hgbt_regression.html#sphx-glr-auto-examples-ensemble-plot-hgbt-regression-py"><span class="std std-ref">Features in Histogram Gradient Boosting Trees</span></a></p></li>
</ul>
</aside>
</section>
<section id="interaction-constraints">
<span id="interaction-cst-hgbt"></span><h4><span class="section-number">1.11.1.1.6. </span>Interaction constraints<a class="headerlink" href="#interaction-constraints" title="Link to this heading">¶</a></h4>
<p>A priori, the histogram gradient boosted trees are allowed to use any feature
to split a node into child nodes. This creates so called interactions between
features, i.e. usage of different features as split along a branch. Sometimes,
one wants to restrict the possible interactions, see <a class="reference internal" href="#mayer2022" id="id8"><span>[Mayer2022]</span></a>. This can be
done by the parameter <code class="docutils literal notranslate"><span class="pre">interaction_cst</span></code>, where one can specify the indices
of features that are allowed to interact.
For instance, with 3 features in total, <code class="docutils literal notranslate"><span class="pre">interaction_cst=[{0},</span> <span class="pre">{1},</span> <span class="pre">{2}]</span></code>
forbids all interactions.
The constraints <code class="docutils literal notranslate"><span class="pre">[{0,</span> <span class="pre">1},</span> <span class="pre">{1,</span> <span class="pre">2}]</span></code> specifies two groups of possibly
interacting features. Features 0 and 1 may interact with each other, as well
as features 1 and 2. But note that features 0 and 2 are forbidden to interact.
The following depicts a tree and the possible splits of the tree:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span> 1 <- Both constraint groups could be applied from now on
/ \
1 2 <- Left split still fulfills both constraint groups.
/ \ / \ Right split at feature 2 has only group {1, 2} from now on.
</pre></div>
</div>
<p>LightGBM uses the same logic for overlapping groups.</p>
<p>Note that features not listed in <code class="docutils literal notranslate"><span class="pre">interaction_cst</span></code> are automatically
assigned an interaction group for themselves. With again 3 features, this
means that <code class="docutils literal notranslate"><span class="pre">[{0}]</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">[{0},</span> <span class="pre">{1,</span> <span class="pre">2}]</span></code>.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/inspection/plot_partial_dependence.html#sphx-glr-auto-examples-inspection-plot-partial-dependence-py"><span class="std std-ref">Partial Dependence and Individual Conditional Expectation Plots</span></a></p></li>
</ul>
</aside>
<aside class="topic">
<p class="topic-title">References</p>
<div role="list" class="citation-list">
<div class="citation" id="mayer2022" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">Mayer2022</a><span class="fn-bracket">]</span></span>
<p>M. Mayer, S.C. Bourassa, M. Hoesli, and D.F. Scognamiglio.
2022. <a class="reference external" href="https://doi.org/10.3390/jrfm15050193">Machine Learning Applications to Land and Structure Valuation</a>.
Journal of Risk and Financial Management 15, no. 5: 193</p>
</div>
</div>
</aside>
</section>
<section id="low-level-parallelism">
<h4><span class="section-number">1.11.1.1.7. </span>Low-level parallelism<a class="headerlink" href="#low-level-parallelism" title="Link to this heading">¶</a></h4>
<p><a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a> use OpenMP
for parallelization through Cython. For more details on how to control the
number of threads, please refer to our <a class="reference internal" href="../computing/parallelism.html#parallelism"><span class="std std-ref">Parallelism</span></a> notes.</p>
<p>The following parts are parallelized:</p>
<ul class="simple">
<li><p>mapping samples from real values to integer-valued bins (finding the bin
thresholds is however sequential)</p></li>
<li><p>building histograms is parallelized over features</p></li>
<li><p>finding the best split point at a node is parallelized over features</p></li>
<li><p>during fit, mapping samples into the left and right children is
parallelized over samples</p></li>
<li><p>gradient and hessians computations are parallelized over samples</p></li>
<li><p>predicting is parallelized over samples</p></li>
</ul>
</section>
<section id="why-it-s-faster">
<span id="id9"></span><h4><span class="section-number">1.11.1.1.8. </span>Why it’s faster<a class="headerlink" href="#why-it-s-faster" title="Link to this heading">¶</a></h4>
<p>The bottleneck of a gradient boosting procedure is building the decision
trees. Building a traditional decision tree (as in the other GBDTs
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> and <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a>)
requires sorting the samples at each node (for
each feature). Sorting is needed so that the potential gain of a split point
can be computed efficiently. Splitting a single node has thus a complexity
of <span class="math notranslate nohighlight">\(\mathcal{O}(n_\text{features} \times n \log(n))\)</span> where <span class="math notranslate nohighlight">\(n\)</span>
is the number of samples at the node.</p>
<p><a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a>, in contrast, do not require sorting the
feature values and instead use a data-structure called a histogram, where the
samples are implicitly ordered. Building a histogram has a
<span class="math notranslate nohighlight">\(\mathcal{O}(n)\)</span> complexity, so the node splitting procedure has a
<span class="math notranslate nohighlight">\(\mathcal{O}(n_\text{features} \times n)\)</span> complexity, much smaller
than the previous one. In addition, instead of considering <span class="math notranslate nohighlight">\(n\)</span> split
points, we consider only <code class="docutils literal notranslate"><span class="pre">max_bins</span></code> split points, which might be much
smaller.</p>
<p>In order to build histograms, the input data <code class="docutils literal notranslate"><span class="pre">X</span></code> needs to be binned into
integer-valued bins. This binning procedure does require sorting the feature
values, but it only happens once at the very beginning of the boosting process
(not at each node, like in <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a>).</p>
<p>Finally, many parts of the implementation of
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a> are parallelized.</p>
<aside class="topic">
<p class="topic-title">References</p>
<div role="list" class="citation-list">
<div class="citation" id="xgboost" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>XGBoost<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id4">1</a>,<a role="doc-backlink" href="#id5">2</a>,<a role="doc-backlink" href="#id6">3</a>)</span>
<p>Tianqi Chen, Carlos Guestrin, <a class="reference external" href="https://arxiv.org/abs/1603.02754">“XGBoost: A Scalable Tree
Boosting System”</a></p>
</div>
<div class="citation" id="lightgbm" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">LightGBM</a><span class="fn-bracket">]</span></span>
<p>Ke et. al. <a class="reference external" href="https://papers.nips.cc/paper/6907-lightgbm-a-highly-efficient-gradient-boosting-decision-tree">“LightGBM: A Highly Efficient Gradient
BoostingDecision Tree”</a></p>
</div>
<div class="citation" id="fisher1958" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id7">Fisher1958</a><span class="fn-bracket">]</span></span>
<p>Fisher, W.D. (1958). <a class="reference external" href="http://csiss.ncgia.ucsb.edu/SPACE/workshops/2004/SAC/files/fisher.pdf">“On Grouping for Maximum Homogeneity”</a>
Journal of the American Statistical Association, 53, 789-798.</p>
</div>
</div>
</aside>
</section>
</section>
<section id="gradientboostingclassifier-and-gradientboostingregressor">
<h3><span class="section-number">1.11.1.2. </span><a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> and <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a><a class="headerlink" href="#gradientboostingclassifier-and-gradientboostingregressor" title="Link to this heading">¶</a></h3>
<p>The usage and the parameters of <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a> are described below. The 2 most important
parameters of these estimators are <code class="docutils literal notranslate"><span class="pre">n_estimators</span></code> and <code class="docutils literal notranslate"><span class="pre">learning_rate</span></code>.</p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Classification</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p><a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> supports both binary and multi-class
classification.
The following example shows how to fit a gradient boosting classifier
with 100 decision stumps as weak learners:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_hastie_10_2</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">GradientBoostingClassifier</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_hastie_10_2</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:</span><span class="mi">2000</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="mi">2000</span><span class="p">:]</span>
<span class="gp">>>> </span><span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">y</span><span class="p">[:</span><span class="mi">2000</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="mi">2000</span><span class="p">:]</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">GradientBoostingClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">)</span>
<span class="go">0.913...</span>
</pre></div>
</div>
<p>The number of weak learners (i.e. regression trees) is controlled by the
parameter <code class="docutils literal notranslate"><span class="pre">n_estimators</span></code>; <a class="reference internal" href="#gradient-boosting-tree-size"><span class="std std-ref">The size of each tree</span></a> can be controlled either by setting the tree
depth via <code class="docutils literal notranslate"><span class="pre">max_depth</span></code> or by setting the number of leaf nodes via
<code class="docutils literal notranslate"><span class="pre">max_leaf_nodes</span></code>. The <code class="docutils literal notranslate"><span class="pre">learning_rate</span></code> is a hyper-parameter in the range
(0.0, 1.0] that controls overfitting via <a class="reference internal" href="#gradient-boosting-shrinkage"><span class="std std-ref">shrinkage</span></a> .</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Classification with more than 2 classes requires the induction
of <code class="docutils literal notranslate"><span class="pre">n_classes</span></code> regression trees at each iteration,
thus, the total number of induced trees equals
<code class="docutils literal notranslate"><span class="pre">n_classes</span> <span class="pre">*</span> <span class="pre">n_estimators</span></code>. For datasets with a large number
of classes we strongly recommend to use
<a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingClassifier.html#sklearn.ensemble.HistGradientBoostingClassifier" title="sklearn.ensemble.HistGradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingClassifier</span></code></a> as an alternative to
<a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a> .</p>
</div>
<p></div>
</details></p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Regression</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p><a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a> supports a number of
<a class="reference internal" href="#gradient-boosting-loss"><span class="std std-ref">different loss functions</span></a>
for regression which can be specified via the argument
<code class="docutils literal notranslate"><span class="pre">loss</span></code>; the default loss function for regression is squared error
(<code class="docutils literal notranslate"><span class="pre">'squared_error'</span></code>).</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_friedman1</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">GradientBoostingRegressor</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_friedman1</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">1200</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:</span><span class="mi">200</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="mi">200</span><span class="p">:]</span>
<span class="gp">>>> </span><span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">y</span><span class="p">[:</span><span class="mi">200</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="mi">200</span><span class="p">:]</span>
<span class="gp">>>> </span><span class="n">est</span> <span class="o">=</span> <span class="n">GradientBoostingRegressor</span><span class="p">(</span>
<span class="gp">... </span> <span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">loss</span><span class="o">=</span><span class="s1">'squared_error'</span>
<span class="gp">... </span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">est</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">))</span>
<span class="go">5.00...</span>
</pre></div>
</div>
<p>The figure below shows the results of applying <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a>
with least squares loss and 500 base learners to the diabetes dataset
(<a class="reference internal" href="generated/sklearn.datasets.load_diabetes.html#sklearn.datasets.load_diabetes" title="sklearn.datasets.load_diabetes"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.datasets.load_diabetes</span></code></a>).
The plot shows the train and test error at each iteration.
The train error at each iteration is stored in the
<code class="docutils literal notranslate"><span class="pre">train_score_</span></code> attribute of the gradient boosting model.
The test error at each iterations can be obtained
via the <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor.staged_predict" title="sklearn.ensemble.GradientBoostingRegressor.staged_predict"><code class="xref py py-meth docutils literal notranslate"><span class="pre">staged_predict</span></code></a> method which returns a
generator that yields the predictions at each stage. Plots like these can be used
to determine the optimal number of trees (i.e. <code class="docutils literal notranslate"><span class="pre">n_estimators</span></code>) by early stopping.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/ensemble/plot_gradient_boosting_regression.html"><img alt="../_images/sphx_glr_plot_gradient_boosting_regression_001.png" src="../_images/sphx_glr_plot_gradient_boosting_regression_001.png" style="width: 450.0px; height: 450.0px;" />
</a>
</figure>
<p></div>
</details></p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/ensemble/plot_gradient_boosting_regression.html#sphx-glr-auto-examples-ensemble-plot-gradient-boosting-regression-py"><span class="std std-ref">Gradient Boosting regression</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/ensemble/plot_gradient_boosting_oob.html#sphx-glr-auto-examples-ensemble-plot-gradient-boosting-oob-py"><span class="std std-ref">Gradient Boosting Out-of-Bag estimates</span></a></p></li>
</ul>
</aside>
<section id="fitting-additional-weak-learners">
<span id="gradient-boosting-warm-start"></span><h4><span class="section-number">1.11.1.2.1. </span>Fitting additional weak-learners<a class="headerlink" href="#fitting-additional-weak-learners" title="Link to this heading">¶</a></h4>
<p>Both <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="sklearn.ensemble.GradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingRegressor</span></code></a> and <a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">GradientBoostingClassifier</span></code></a>
support <code class="docutils literal notranslate"><span class="pre">warm_start=True</span></code> which allows you to add more estimators to an already
fitted model.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">_</span> <span class="o">=</span> <span class="n">est</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">warm_start</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="c1"># set warm_start and new nr of trees</span>
<span class="gp">>>> </span><span class="n">_</span> <span class="o">=</span> <span class="n">est</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span> <span class="c1"># fit additional 100 trees to est</span>
<span class="gp">>>> </span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">est</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">))</span>
<span class="go">3.84...</span>
</pre></div>
</div>
</section>
<section id="controlling-the-tree-size">
<span id="gradient-boosting-tree-size"></span><h4><span class="section-number">1.11.1.2.2. </span>Controlling the tree size<a class="headerlink" href="#controlling-the-tree-size" title="Link to this heading">¶</a></h4>
<p>The size of the regression tree base learners defines the level of variable
interactions that can be captured by the gradient boosting model. In general,
a tree of depth <code class="docutils literal notranslate"><span class="pre">h</span></code> can capture interactions of order <code class="docutils literal notranslate"><span class="pre">h</span></code> .
There are two ways in which the size of the individual regression trees can
be controlled.</p>
<p>If you specify <code class="docutils literal notranslate"><span class="pre">max_depth=h</span></code> then complete binary trees
of depth <code class="docutils literal notranslate"><span class="pre">h</span></code> will be grown. Such trees will have (at most) <code class="docutils literal notranslate"><span class="pre">2**h</span></code> leaf nodes
and <code class="docutils literal notranslate"><span class="pre">2**h</span> <span class="pre">-</span> <span class="pre">1</span></code> split nodes.</p>
<p>Alternatively, you can control the tree size by specifying the number of
leaf nodes via the parameter <code class="docutils literal notranslate"><span class="pre">max_leaf_nodes</span></code>. In this case,
trees will be grown using best-first search where nodes with the highest improvement
in impurity will be expanded first.
A tree with <code class="docutils literal notranslate"><span class="pre">max_leaf_nodes=k</span></code> has <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">-</span> <span class="pre">1</span></code> split nodes and thus can
model interactions of up to order <code class="docutils literal notranslate"><span class="pre">max_leaf_nodes</span> <span class="pre">-</span> <span class="pre">1</span></code> .</p>
<p>We found that <code class="docutils literal notranslate"><span class="pre">max_leaf_nodes=k</span></code> gives comparable results to <code class="docutils literal notranslate"><span class="pre">max_depth=k-1</span></code>
but is significantly faster to train at the expense of a slightly higher
training error.
The parameter <code class="docutils literal notranslate"><span class="pre">max_leaf_nodes</span></code> corresponds to the variable <code class="docutils literal notranslate"><span class="pre">J</span></code> in the
chapter on gradient boosting in <a class="reference internal" href="#friedman2001" id="id10"><span>[Friedman2001]</span></a> and is related to the parameter
<code class="docutils literal notranslate"><span class="pre">interaction.depth</span></code> in R’s gbm package where <code class="docutils literal notranslate"><span class="pre">max_leaf_nodes</span> <span class="pre">==</span> <span class="pre">interaction.depth</span> <span class="pre">+</span> <span class="pre">1</span></code> .</p>
</section>
<section id="mathematical-formulation">
<span id="gradient-boosting-formulation"></span><h4><span class="section-number">1.11.1.2.3. </span>Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Link to this heading">¶</a></h4>
<p>We first present GBRT for regression, and then detail the classification
case.</p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Regression</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p>GBRT regressors are additive models whose prediction <span class="math notranslate nohighlight">\(\hat{y}_i\)</span> for a
given input <span class="math notranslate nohighlight">\(x_i\)</span> is of the following form:</p>
<div class="math notranslate nohighlight">
\[\hat{y}_i = F_M(x_i) = \sum_{m=1}^{M} h_m(x_i)\]</div>
<p>where the <span class="math notranslate nohighlight">\(h_m\)</span> are estimators called <em>weak learners</em> in the context
of boosting. Gradient Tree Boosting uses <a class="reference internal" href="tree.html#tree"><span class="std std-ref">decision tree regressors</span></a> of fixed size as weak learners. The constant M corresponds to the
<code class="docutils literal notranslate"><span class="pre">n_estimators</span></code> parameter.</p>
<p>Similar to other boosting algorithms, a GBRT is built in a greedy fashion:</p>
<div class="math notranslate nohighlight">
\[F_m(x) = F_{m-1}(x) + h_m(x),\]</div>
<p>where the newly added tree <span class="math notranslate nohighlight">\(h_m\)</span> is fitted in order to minimize a sum
of losses <span class="math notranslate nohighlight">\(L_m\)</span>, given the previous ensemble <span class="math notranslate nohighlight">\(F_{m-1}\)</span>:</p>
<div class="math notranslate nohighlight">
\[h_m = \arg\min_{h} L_m = \arg\min_{h} \sum_{i=1}^{n}
l(y_i, F_{m-1}(x_i) + h(x_i)),\]</div>
<p>where <span class="math notranslate nohighlight">\(l(y_i, F(x_i))\)</span> is defined by the <code class="docutils literal notranslate"><span class="pre">loss</span></code> parameter, detailed
in the next section.</p>
<p>By default, the initial model <span class="math notranslate nohighlight">\(F_{0}\)</span> is chosen as the constant that
minimizes the loss: for a least-squares loss, this is the empirical mean of
the target values. The initial model can also be specified via the <code class="docutils literal notranslate"><span class="pre">init</span></code>
argument.</p>
<p>Using a first-order Taylor approximation, the value of <span class="math notranslate nohighlight">\(l\)</span> can be
approximated as follows:</p>
<div class="math notranslate nohighlight">
\[l(y_i, F_{m-1}(x_i) + h_m(x_i)) \approx
l(y_i, F_{m-1}(x_i))
+ h_m(x_i)
\left[ \frac{\partial l(y_i, F(x_i))}{\partial F(x_i)} \right]_{F=F_{m - 1}}.\]</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Briefly, a first-order Taylor approximation says that
<span class="math notranslate nohighlight">\(l(z) \approx l(a) + (z - a) \frac{\partial l}{\partial z}(a)\)</span>.
Here, <span class="math notranslate nohighlight">\(z\)</span> corresponds to <span class="math notranslate nohighlight">\(F_{m - 1}(x_i) + h_m(x_i)\)</span>, and
<span class="math notranslate nohighlight">\(a\)</span> corresponds to <span class="math notranslate nohighlight">\(F_{m-1}(x_i)\)</span></p>
</div>
<p>The quantity <span class="math notranslate nohighlight">\(\left[ \frac{\partial l(y_i, F(x_i))}{\partial F(x_i)}
\right]_{F=F_{m - 1}}\)</span> is the derivative of the loss with respect to its
second parameter, evaluated at <span class="math notranslate nohighlight">\(F_{m-1}(x)\)</span>. It is easy to compute for
any given <span class="math notranslate nohighlight">\(F_{m - 1}(x_i)\)</span> in a closed form since the loss is
differentiable. We will denote it by <span class="math notranslate nohighlight">\(g_i\)</span>.</p>
<p>Removing the constant terms, we have:</p>
<div class="math notranslate nohighlight">
\[h_m \approx \arg\min_{h} \sum_{i=1}^{n} h(x_i) g_i\]</div>
<p>This is minimized if <span class="math notranslate nohighlight">\(h(x_i)\)</span> is fitted to predict a value that is
proportional to the negative gradient <span class="math notranslate nohighlight">\(-g_i\)</span>. Therefore, at each
iteration, <strong>the estimator</strong> <span class="math notranslate nohighlight">\(h_m\)</span> <strong>is fitted to predict the negative
gradients of the samples</strong>. The gradients are updated at each iteration.
This can be considered as some kind of gradient descent in a functional
space.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>For some losses, e.g. <code class="docutils literal notranslate"><span class="pre">'absolute_error'</span></code> where the gradients
are <span class="math notranslate nohighlight">\(\pm 1\)</span>, the values predicted by a fitted <span class="math notranslate nohighlight">\(h_m\)</span> are not
accurate enough: the tree can only output integer values. As a result, the
leaves values of the tree <span class="math notranslate nohighlight">\(h_m\)</span> are modified once the tree is
fitted, such that the leaves values minimize the loss <span class="math notranslate nohighlight">\(L_m\)</span>. The
update is loss-dependent: for the absolute error loss, the value of
a leaf is updated to the median of the samples in that leaf.</p>
</div>
<p></div>
</details></p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Classification</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p>Gradient boosting for classification is very similar to the regression case.
However, the sum of the trees <span class="math notranslate nohighlight">\(F_M(x_i) = \sum_m h_m(x_i)\)</span> is not
homogeneous to a prediction: it cannot be a class, since the trees predict
continuous values.</p>
<p>The mapping from the value <span class="math notranslate nohighlight">\(F_M(x_i)\)</span> to a class or a probability is
loss-dependent. For the log-loss, the probability that
<span class="math notranslate nohighlight">\(x_i\)</span> belongs to the positive class is modeled as <span class="math notranslate nohighlight">\(p(y_i = 1 |
x_i) = \sigma(F_M(x_i))\)</span> where <span class="math notranslate nohighlight">\(\sigma\)</span> is the sigmoid or expit function.</p>
<p>For multiclass classification, K trees (for K classes) are built at each of
the <span class="math notranslate nohighlight">\(M\)</span> iterations. The probability that <span class="math notranslate nohighlight">\(x_i\)</span> belongs to class
k is modeled as a softmax of the <span class="math notranslate nohighlight">\(F_{M,k}(x_i)\)</span> values.</p>
<p>Note that even for a classification task, the <span class="math notranslate nohighlight">\(h_m\)</span> sub-estimator is
still a regressor, not a classifier. This is because the sub-estimators are
trained to predict (negative) <em>gradients</em>, which are always continuous
quantities.</p>
<p></div>
</details></p>
</section>
<section id="loss-functions">
<span id="gradient-boosting-loss"></span><h4><span class="section-number">1.11.1.2.4. </span>Loss Functions<a class="headerlink" href="#loss-functions" title="Link to this heading">¶</a></h4>
<p>The following loss functions are supported and can be specified using
the parameter <code class="docutils literal notranslate"><span class="pre">loss</span></code>:</p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Regression</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<blockquote>
<div><ul class="simple">
<li><p>Squared error (<code class="docutils literal notranslate"><span class="pre">'squared_error'</span></code>): The natural choice for regression
due to its superior computational properties. The initial model is
given by the mean of the target values.</p></li>
<li><p>Absolute error (<code class="docutils literal notranslate"><span class="pre">'absolute_error'</span></code>): A robust loss function for
regression. The initial model is given by the median of the
target values.</p></li>
<li><p>Huber (<code class="docutils literal notranslate"><span class="pre">'huber'</span></code>): Another robust loss function that combines
least squares and least absolute deviation; use <code class="docutils literal notranslate"><span class="pre">alpha</span></code> to
control the sensitivity with regards to outliers (see <a class="reference internal" href="#friedman2001" id="id11"><span>[Friedman2001]</span></a> for
more details).</p></li>
<li><p>Quantile (<code class="docutils literal notranslate"><span class="pre">'quantile'</span></code>): A loss function for quantile regression.
Use <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">alpha</span> <span class="pre"><</span> <span class="pre">1</span></code> to specify the quantile. This loss function
can be used to create prediction intervals
(see <a class="reference internal" href="../auto_examples/ensemble/plot_gradient_boosting_quantile.html#sphx-glr-auto-examples-ensemble-plot-gradient-boosting-quantile-py"><span class="std std-ref">Prediction Intervals for Gradient Boosting Regression</span></a>).</p></li>
</ul>
</div></blockquote>
<p></div>
</details></p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Classification</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<blockquote>
<div><ul class="simple">
<li><p>Binary log-loss (<code class="docutils literal notranslate"><span class="pre">'log-loss'</span></code>): The binomial
negative log-likelihood loss function for binary classification. It provides
probability estimates. The initial model is given by the
log odds-ratio.</p></li>
<li><p>Multi-class log-loss (<code class="docutils literal notranslate"><span class="pre">'log-loss'</span></code>): The multinomial
negative log-likelihood loss function for multi-class classification with
<code class="docutils literal notranslate"><span class="pre">n_classes</span></code> mutually exclusive classes. It provides
probability estimates. The initial model is given by the
prior probability of each class. At each iteration <code class="docutils literal notranslate"><span class="pre">n_classes</span></code>
regression trees have to be constructed which makes GBRT rather
inefficient for data sets with a large number of classes.</p></li>
<li><p>Exponential loss (<code class="docutils literal notranslate"><span class="pre">'exponential'</span></code>): The same loss function
as <a class="reference internal" href="generated/sklearn.ensemble.AdaBoostClassifier.html#sklearn.ensemble.AdaBoostClassifier" title="sklearn.ensemble.AdaBoostClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">AdaBoostClassifier</span></code></a>. Less robust to mislabeled
examples than <code class="docutils literal notranslate"><span class="pre">'log-loss'</span></code>; can only be used for binary
classification.</p></li>
</ul>
</div></blockquote>
<p></div>
</details></p>
</section>
<section id="shrinkage-via-learning-rate">
<span id="gradient-boosting-shrinkage"></span><h4><span class="section-number">1.11.1.2.5. </span>Shrinkage via learning rate<a class="headerlink" href="#shrinkage-via-learning-rate" title="Link to this heading">¶</a></h4>
<p><a class="reference internal" href="#friedman2001" id="id12"><span>[Friedman2001]</span></a> proposed a simple regularization strategy that scales
the contribution of each weak learner by a constant factor <span class="math notranslate nohighlight">\(\nu\)</span>:</p>
<div class="math notranslate nohighlight">
\[F_m(x) = F_{m-1}(x) + \nu h_m(x)\]</div>
<p>The parameter <span class="math notranslate nohighlight">\(\nu\)</span> is also called the <strong>learning rate</strong> because
it scales the step length the gradient descent procedure; it can
be set via the <code class="docutils literal notranslate"><span class="pre">learning_rate</span></code> parameter.</p>
<p>The parameter <code class="docutils literal notranslate"><span class="pre">learning_rate</span></code> strongly interacts with the parameter
<code class="docutils literal notranslate"><span class="pre">n_estimators</span></code>, the number of weak learners to fit. Smaller values
of <code class="docutils literal notranslate"><span class="pre">learning_rate</span></code> require larger numbers of weak learners to maintain
a constant training error. Empirical evidence suggests that small
values of <code class="docutils literal notranslate"><span class="pre">learning_rate</span></code> favor better test error. <a class="reference internal" href="#htf" id="id13"><span>[HTF]</span></a>