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<li><a class="reference internal" href="#">Gradient Boosting Out-of-Bag estimates</a></li>
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<p class="first admonition-title">Note</p>
<p class="last">Click <a class="reference internal" href="#sphx-glr-download-auto-examples-ensemble-plot-gradient-boosting-oob-py"><span class="std std-ref">here</span></a> to download the full example code</p>
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<div class="sphx-glr-example-title section" id="gradient-boosting-out-of-bag-estimates">
<span id="sphx-glr-auto-examples-ensemble-plot-gradient-boosting-oob-py"></span><h1>Gradient Boosting Out-of-Bag estimates<a class="headerlink" href="#gradient-boosting-out-of-bag-estimates" title="Permalink to this headline">¶</a></h1>
<p>Out-of-bag (OOB) estimates can be a useful heuristic to estimate
the “optimal” number of boosting iterations.
OOB estimates are almost identical to cross-validation estimates but
they can be computed on-the-fly without the need for repeated model
fitting.
OOB estimates are only available for Stochastic Gradient Boosting
(i.e. <code class="docutils literal"><span class="pre">subsample</span> <span class="pre"><</span> <span class="pre">1.0</span></code>), the estimates are derived from the improvement
in loss based on the examples not included in the bootstrap sample
(the so-called out-of-bag examples).
The OOB estimator is a pessimistic estimator of the true
test loss, but remains a fairly good approximation for a small number of trees.</p>
<p>The figure shows the cumulative sum of the negative OOB improvements
as a function of the boosting iteration. As you can see, it tracks the test
loss for the first hundred iterations but then diverges in a
pessimistic way.
The figure also shows the performance of 3-fold cross validation which
usually gives a better estimate of the test loss
but is computationally more demanding.</p>
<img alt="../../_images/sphx_glr_plot_gradient_boosting_oob_001.png" class="sphx-glr-single-img" src="../../_images/sphx_glr_plot_gradient_boosting_oob_001.png" />
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none"><div class="highlight"><pre><span></span>Accuracy: 0.6840
</pre></div>
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<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="c1"># Author: Peter Prettenhofer <[email protected]></span>
<span class="c1">#</span>
<span class="c1"># License: BSD 3 clause</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">ensemble</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.model_selection.KFold.html#sklearn.model_selection.KFold" title="View documentation for sklearn.model_selection.KFold"><span class="n">KFold</span></a>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.model_selection.train_test_split.html#sklearn.model_selection.train_test_split" title="View documentation for sklearn.model_selection.train_test_split"><span class="n">train_test_split</span></a>
<span class="kn">from</span> <span class="nn">scipy.special</span> <span class="kn">import</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.expit.html#scipy.special.expit" title="View documentation for scipy.special.expit"><span class="n">expit</span></a>
<span class="c1"># Generate data (adapted from G. Ridgeway's gbm example)</span>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">random_state</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">13</span><span class="p">)</span>
<span class="n">x1</span> <span class="o">=</span> <span class="n">random_state</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">x2</span> <span class="o">=</span> <span class="n">random_state</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">x3</span> <span class="o">=</span> <span class="n">random_state</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.expit.html#scipy.special.expit" title="View documentation for scipy.special.expit"><span class="n">expit</span></a><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.sin.html#numpy.sin" title="View documentation for numpy.sin"><span class="n">np</span><span class="o">.</span><span class="n">sin</span></a><span class="p">(</span><span class="mi">3</span> <span class="o">*</span> <span class="n">x1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">x2</span> <span class="o">+</span> <span class="n">x3</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">random_state</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.c_.html#numpy.c_" title="View documentation for numpy.c_"><span class="n">np</span><span class="o">.</span><span class="n">c_</span></a><span class="p">[</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">,</span> <span class="n">x3</span><span class="p">]</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.train_test_split.html#sklearn.model_selection.train_test_split" title="View documentation for sklearn.model_selection.train_test_split"><span class="n">train_test_split</span></a><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">test_size</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
<span class="n">random_state</span><span class="o">=</span><span class="mi">9</span><span class="p">)</span>
<span class="c1"># Fit classifier with out-of-bag estimates</span>
<span class="n">params</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'n_estimators'</span><span class="p">:</span> <span class="mi">1200</span><span class="p">,</span> <span class="s1">'max_depth'</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="s1">'subsample'</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">,</span>
<span class="s1">'learning_rate'</span><span class="p">:</span> <span class="mf">0.01</span><span class="p">,</span> <span class="s1">'min_samples_leaf'</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="s1">'random_state'</span><span class="p">:</span> <span class="mi">3</span><span class="p">}</span>
<span class="n">clf</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="View documentation for sklearn.ensemble.GradientBoostingClassifier"><span class="n">ensemble</span><span class="o">.</span><span class="n">GradientBoostingClassifier</span></a><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">)</span>
<span class="n">clf</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="n">acc</span> <span class="o">=</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="k">print</span><span class="p">(</span><span class="s2">"Accuracy: {:.4f}"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">acc</span><span class="p">))</span>
<span class="n">n_estimators</span> <span class="o">=</span> <span class="n">params</span><span class="p">[</span><span class="s1">'n_estimators'</span><span class="p">]</span>
<span class="n">x</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.arange.html#numpy.arange" title="View documentation for numpy.arange"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="n">n_estimators</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">heldout_score</span><span class="p">(</span><span class="n">clf</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="sd">"""compute deviance scores on ``X_test`` and ``y_test``. """</span>
<span class="n">score</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html#numpy.zeros" title="View documentation for numpy.zeros"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_estimators</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">y_pred</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">staged_decision_function</span><span class="p">(</span><span class="n">X_test</span><span class="p">)):</span>
<span class="n">score</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">loss_</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
<span class="k">return</span> <span class="n">score</span>
<span class="k">def</span> <span class="nf">cv_estimate</span><span class="p">(</span><span class="n">n_splits</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="n">cv</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.KFold.html#sklearn.model_selection.KFold" title="View documentation for sklearn.model_selection.KFold"><span class="n">KFold</span></a><span class="p">(</span><span class="n">n_splits</span><span class="o">=</span><span class="n">n_splits</span><span class="p">)</span>
<span class="n">cv_clf</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="View documentation for sklearn.ensemble.GradientBoostingClassifier"><span class="n">ensemble</span><span class="o">.</span><span class="n">GradientBoostingClassifier</span></a><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">)</span>
<span class="n">val_scores</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html#numpy.zeros" title="View documentation for numpy.zeros"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_estimators</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="k">for</span> <span class="n">train</span><span class="p">,</span> <span class="n">test</span> <span class="ow">in</span> <span class="n">cv</span><span class="o">.</span><span class="n">split</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="n">cv_clf</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">train</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[</span><span class="n">train</span><span class="p">])</span>
<span class="n">val_scores</span> <span class="o">+=</span> <span class="n">heldout_score</span><span class="p">(</span><span class="n">cv_clf</span><span class="p">,</span> <span class="n">X_train</span><span class="p">[</span><span class="n">test</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[</span><span class="n">test</span><span class="p">])</span>
<span class="n">val_scores</span> <span class="o">/=</span> <span class="n">n_splits</span>
<span class="k">return</span> <span class="n">val_scores</span>
<span class="c1"># Estimate best n_estimator using cross-validation</span>
<span class="n">cv_score</span> <span class="o">=</span> <span class="n">cv_estimate</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="c1"># Compute best n_estimator for test data</span>
<span class="n">test_score</span> <span class="o">=</span> <span class="n">heldout_score</span><span class="p">(</span><span class="n">clf</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="c1"># negative cumulative sum of oob improvements</span>
<span class="n">cumsum</span> <span class="o">=</span> <span class="o">-</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.cumsum.html#numpy.cumsum" title="View documentation for numpy.cumsum"><span class="n">np</span><span class="o">.</span><span class="n">cumsum</span></a><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">oob_improvement_</span><span class="p">)</span>
<span class="c1"># min loss according to OOB</span>
<span class="n">oob_best_iter</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmin.html#numpy.argmin" title="View documentation for numpy.argmin"><span class="n">np</span><span class="o">.</span><span class="n">argmin</span></a><span class="p">(</span><span class="n">cumsum</span><span class="p">)]</span>
<span class="c1"># min loss according to test (normalize such that first loss is 0)</span>
<span class="n">test_score</span> <span class="o">-=</span> <span class="n">test_score</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">test_best_iter</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmin.html#numpy.argmin" title="View documentation for numpy.argmin"><span class="n">np</span><span class="o">.</span><span class="n">argmin</span></a><span class="p">(</span><span class="n">test_score</span><span class="p">)]</span>
<span class="c1"># min loss according to cv (normalize such that first loss is 0)</span>
<span class="n">cv_score</span> <span class="o">-=</span> <span class="n">cv_score</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">cv_best_iter</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmin.html#numpy.argmin" title="View documentation for numpy.argmin"><span class="n">np</span><span class="o">.</span><span class="n">argmin</span></a><span class="p">(</span><span class="n">cv_score</span><span class="p">)]</span>
<span class="c1"># color brew for the three curves</span>
<span class="n">oob_color</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span> <span class="o">/</span> <span class="mf">256.0</span><span class="p">,</span> <span class="p">(</span><span class="mi">190</span><span class="p">,</span> <span class="mi">174</span><span class="p">,</span> <span class="mi">212</span><span class="p">)))</span>
<span class="n">test_color</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span> <span class="o">/</span> <span class="mf">256.0</span><span class="p">,</span> <span class="p">(</span><span class="mi">127</span><span class="p">,</span> <span class="mi">201</span><span class="p">,</span> <span class="mi">127</span><span class="p">)))</span>
<span class="n">cv_color</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span> <span class="o">/</span> <span class="mf">256.0</span><span class="p">,</span> <span class="p">(</span><span class="mi">253</span><span class="p">,</span> <span class="mi">192</span><span class="p">,</span> <span class="mi">134</span><span class="p">)))</span>
<span class="c1"># plot curves and vertical lines for best iterations</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="View documentation for matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">cumsum</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'OOB loss'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">oob_color</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="View documentation for matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">test_score</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'Test loss'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">test_color</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="View documentation for matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">cv_score</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'CV loss'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">cv_color</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.axvline.html#matplotlib.pyplot.axvline" title="View documentation for matplotlib.pyplot.axvline"><span class="n">plt</span><span class="o">.</span><span class="n">axvline</span></a><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">oob_best_iter</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">oob_color</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.axvline.html#matplotlib.pyplot.axvline" title="View documentation for matplotlib.pyplot.axvline"><span class="n">plt</span><span class="o">.</span><span class="n">axvline</span></a><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">test_best_iter</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">test_color</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.axvline.html#matplotlib.pyplot.axvline" title="View documentation for matplotlib.pyplot.axvline"><span class="n">plt</span><span class="o">.</span><span class="n">axvline</span></a><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">cv_best_iter</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">cv_color</span><span class="p">)</span>
<span class="c1"># add three vertical lines to xticks</span>
<span class="n">xticks</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xticks.html#matplotlib.pyplot.xticks" title="View documentation for matplotlib.pyplot.xticks"><span class="n">plt</span><span class="o">.</span><span class="n">xticks</span></a><span class="p">()</span>
<span class="n">xticks_pos</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html#numpy.array" title="View documentation for numpy.array"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">(</span><span class="n">xticks</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span> <span class="o">+</span>
<span class="p">[</span><span class="n">oob_best_iter</span><span class="p">,</span> <span class="n">cv_best_iter</span><span class="p">,</span> <span class="n">test_best_iter</span><span class="p">])</span>
<span class="n">xticks_label</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html#numpy.array" title="View documentation for numpy.array"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="nb">int</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">xticks</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span> <span class="o">+</span>
<span class="p">[</span><span class="s1">'OOB'</span><span class="p">,</span> <span class="s1">'CV'</span><span class="p">,</span> <span class="s1">'Test'</span><span class="p">])</span>
<span class="n">ind</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.argsort.html#numpy.argsort" title="View documentation for numpy.argsort"><span class="n">np</span><span class="o">.</span><span class="n">argsort</span></a><span class="p">(</span><span class="n">xticks_pos</span><span class="p">)</span>
<span class="n">xticks_pos</span> <span class="o">=</span> <span class="n">xticks_pos</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span>
<span class="n">xticks_label</span> <span class="o">=</span> <span class="n">xticks_label</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xticks.html#matplotlib.pyplot.xticks" title="View documentation for matplotlib.pyplot.xticks"><span class="n">plt</span><span class="o">.</span><span class="n">xticks</span></a><span class="p">(</span><span class="n">xticks_pos</span><span class="p">,</span> <span class="n">xticks_label</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="View documentation for matplotlib.pyplot.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'upper right'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s1">'normalized loss'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="View documentation for matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s1">'number of iterations'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="View documentation for matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
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