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# Early stopping of Stochastic Gradient Descent

Stochastic Gradient Descent is an optimization technique which minimizes a loss function in a stochastic fashion, performing a gradient descent step sample by sample. In particular, it is a very efficient method to fit linear models.

As a stochastic method, the loss function is not necessarily decreasing at each iteration, and convergence is only guaranteed in expectation. For this reason, monitoring the convergence on the loss function can be difficult.

Another approach is to monitor convergence on a validation score. In this case, the input data is split into a training set and a validation set. The model is then fitted on the training set and the stopping criterion is based on the prediction score computed on the validation set. This enables us to find the least number of iterations which is sufficient to build a model that generalizes well to unseen data and reduces the chance of over-fitting the training data.

This early stopping strategy is activated if `early_stopping=True`; otherwise the stopping criterion only uses the training loss on the entire input data. To better control the early stopping strategy, we can specify a parameter `validation_fraction` which set the fraction of the input dataset that we keep aside to compute the validation score. The optimization will continue until the validation score did not improve by at least `tol` during the last `n_iter_no_change` iterations. The actual number of iterations is available at the attribute `n_iter_`.

This example illustrates how the early stopping can used in the `SGDClassifier` model to achieve almost the same accuracy as compared to a model built without early stopping. This can significantly reduce training time. Note that scores differ between the stopping criteria even from early iterations because some of the training data is held out with the validation stopping criterion.

    No stopping criterion: .................................................
    Training loss: .................................................
    Validation score: .................................................

    # Authors: Tom Dupre la Tour
    #
    # License: BSD 3 clause

    import sys
    import time

    import matplotlib.pyplot as plt
    import numpy as np
    import pandas as pd

    from sklearn import linear_model
    from sklearn.datasets import fetch_openml
    from sklearn.exceptions import ConvergenceWarning
    from sklearn.model_selection import train_test_split
    from sklearn.utils import shuffle
    from sklearn.utils._testing import ignore_warnings

    def load_mnist(n_samples=None, class_0="0", class_1="8"):
        """Load MNIST, select two classes, shuffle and return only n_samples."""
        # Load data from http://openml.org/d/554
        mnist = fetch_openml("mnist_784", version=1, as_frame=False, parser="pandas")

        # take only two classes for binary classification
        mask = np.logical_or(mnist.target == class_0, mnist.target == class_1)

        X, y = shuffle(mnist.data[mask], mnist.target[mask], random_state=42)
        if n_samples is not None:
            X, y = X[:n_samples], y[:n_samples]
        return X, y

    @ignore_warnings(category=ConvergenceWarning)
    def fit_and_score(estimator, max_iter, X_train, X_test, y_train, y_test):
        """Fit the estimator on the train set and score it on both sets"""
        estimator.set_params(max_iter=max_iter)
        estimator.set_params(random_state=0)

        start = time.time()
        estimator.fit(X_train, y_train)

        fit_time = time.time() - start
        n_iter = estimator.n_iter_
        train_score = estimator.score(X_train, y_train)
        test_score = estimator.score(X_test, y_test)

        return fit_time, n_iter, train_score, test_score

    # Define the estimators to compare
    estimator_dict = {
        "No stopping criterion": linear_model.SGDClassifier(n_iter_no_change=3),
        "Training loss": linear_model.SGDClassifier(
            early_stopping=False, n_iter_no_change=3, tol=0.1
        ),
        "Validation score": linear_model.SGDClassifier(
            early_stopping=True, n_iter_no_change=3, tol=0.0001, validation_fraction=0.2
        ),
    }

    # Load the dataset
    X, y = load_mnist(n_samples=10000)
    X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.5, random_state=0)

    results = []
    for estimator_name, estimator in estimator_dict.items():
        print(estimator_name + ": ", end="")
        for max_iter in range(1, 50):
            print(".", end="")
            sys.stdout.flush()

            fit_time, n_iter, train_score, test_score = fit_and_score(
                estimator, max_iter, X_train, X_test, y_train, y_test
            )

            results.append(
                (estimator_name, max_iter, fit_time, n_iter, train_score, test_score)
            )
        print("")

    # Transform the results in a pandas dataframe for easy plotting
    columns = [
        "Stopping criterion",
        "max_iter",
        "Fit time (sec)",
        "n_iter_",
        "Train score",
        "Test score",
    ]
    results_df = pd.DataFrame(results, columns=columns)

    # Define what to plot
    lines = "Stopping criterion"
    x_axis = "max_iter"
    styles = ["-.", "--", "-"]

    # First plot: train and test scores
    fig, axes = plt.subplots(nrows=1, ncols=2, sharey=True, figsize=(12, 4))
    for ax, y_axis in zip(axes, ["Train score", "Test score"]):
        for style, (criterion, group_df) in zip(styles, results_df.groupby(lines)):
            group_df.plot(x=x_axis, y=y_axis, label=criterion, ax=ax, style=style)
        ax.set_title(y_axis)
        ax.legend(title=lines)
    fig.tight_layout()

    # Second plot: n_iter and fit time
    fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(12, 4))
    for ax, y_axis in zip(axes, ["n_iter_", "Fit time (sec)"]):
        for style, (criterion, group_df) in zip(styles, results_df.groupby(lines)):
            group_df.plot(x=x_axis, y=y_axis, label=criterion, ax=ax, style=style)
        ax.set_title(y_axis)
        ax.legend(title=lines)
    fig.tight_layout()

    plt.show()

Total running time of the script: (1 minutes 12.653 seconds)

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