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%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

The issues associated with validation and cross-validation are some of the most important aspects of the practice of machine learning. Selecting the optimal model for your data is vital, and is a piece of the problem that is not often appreciated by machine learning practitioners.

Of core importance is the following question:

If our estimator is underperforming, how should we move forward?

Use simpler or more complicated model?
Add more features to each observed data point?
Add more training samples?

The answer is often counter-intuitive. In particular, sometimes using a more complicated model will give worse results. Also, sometimes adding training data will not improve your results. The ability to determine what steps will improve your model is what separates the successful machine learning practitioners from the unsuccessful.

Learning Curves and Validation Curves

One way to address this issue is to use what are often called Learning Curves. Given a particular dataset and a model we'd like to fit (e.g. using feature creation and linear regression), we'd like to tune our value of the hyperparameter kernel to give us the best fit. We can visualize the different regimes with the following plot, modified from the sklearn examples here



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import numpy as np
import matplotlib.pyplot as plt
from sklearn.pipeline import Pipeline
from sklearn.svm import SVR
from sklearn import cross_validation

np.random.seed(0)

n_samples = 200
kernels = ['linear', 'poly', 'rbf']

true_fun = lambda X: X ** 3
X = np.sort(5 * (np.random.rand(n_samples) - .5))
y = true_fun(X) + .01 * np.random.randn(n_samples)

plt.figure(figsize=(14, 5))
for i in range(len(kernels)):
    ax = plt.subplot(1, len(kernels), i + 1)
    plt.setp(ax, xticks=(), yticks=())
    model = SVR(kernel=kernels[i], C=5)
    model.fit(X[:, np.newaxis], y)

    # Evaluate the models using crossvalidation
    scores = cross_validation.cross_val_score(model,
        X[:, np.newaxis], y, scoring="mean_squared_error", cv=10)

    X_test = np.linspace(3 * -.5, 3 * .5, 100)
    plt.plot(X_test, model.predict(X_test[:, np.newaxis]), label="Model")
    plt.plot(X_test, true_fun(X_test), label="True function")
    plt.scatter(X, y, label="Samples")
    plt.xlabel("x")
    plt.ylabel("y")
    plt.xlim((-3 * .5, 3 * .5))
    plt.ylim((-1, 1))
    plt.legend(loc="best")
    plt.title("Kernel {}\nMSE = {:.2e}(+/- {:.2e})".format(
        kernels[i], -scores.mean(), scores.std()))
plt.show()

Learning Curves

What the right model for a dataset is depends critically on how much data we have. More data allows us to be more confident about building a complex model. Lets built some intuition on why that is. Look at the following datasets:



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import numpy as np
import matplotlib.pyplot as plt
from sklearn import cross_validation

np.random.seed(0)
n_samples = 200

true_fun = lambda X: X ** 3
X = np.sort(5 * (np.random.rand(n_samples) - .5))
y = true_fun(X) + .02 * np.random.randn(n_samples)
X = X[:, None]
y = y
f, axarr = plt.subplots(1, 3)
axarr[0].scatter(X[::20], y[::20])
axarr[0].set_xlim((-3 * .5, 3 * .5))
axarr[0].set_ylim((-1, 1))
axarr[1].scatter(X[::10], y[::10])
axarr[1].set_xlim((-3 * .5, 3 * .5))
axarr[1].set_ylim((-1, 1))
axarr[2].scatter(X, y)
axarr[2].set_xlim((-3 * .5, 3 * .5))
axarr[2].set_ylim((-1, 1))
plt.show()

They all come from the same underlying process. But if you were asked to make a prediction, you would be more likely to draw a straight line for the left-most one, as there are only very few datapoints, and no real rule is apparent. For the dataset in the middle, some structure is recognizable, though the exact shape of the true function is maybe not obvious. With even more data on the right hand side, you would probably be very comfortable with drawing a curved line with a lot of certainty.

A great way to explore how a model fit evolves with different dataset sizes are learning curves. A learning curve plots the validation error for a given model against different training set sizes.

But first, take a moment to think about what we're going to see:

Questions:

As the number of training samples are increased, what do you expect to see for the training error? For the validation error?
Would you expect the training error to be higher or lower than the validation error? Would you ever expect this to change?

We can run the following code to plot the learning curve for a kernel = linear model:



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from sklearn.learning_curve import learning_curve
from sklearn.svm import SVR
training_sizes, train_scores, test_scores = learning_curve(SVR(kernel='linear'), X, y, cv=10, scoring="mean_squared_error",
                                                           train_sizes=[.6, .7, .8, .9, 1.])
# Use the negative because we want to minimize squared error
plt.plot(training_sizes, -train_scores.mean(axis=1), label="training scores")
plt.plot(training_sizes, -test_scores.mean(axis=1), label="test scores")
plt.ylim((0, 50))
plt.legend(loc='best')

You can see that for the model with kernel = linear, the validation score doesn't really decrease as more data is given.

Notice that the validation error generally decreases with a growing training set, while the training error generally increases with a growing training set. From this we can infer that as the training size increases, they will converge to a single value.

From the above discussion, we know that kernel = linear underfits the data. This is indicated by the fact that both the training and validation errors are very high. When confronted with this type of learning curve, we can expect that adding more training data will not help matters: both lines will converge to a relatively high error.

When the learning curves have converged to a high error, we have an underfitting model.

An underfitting model can be improved by:

Using a more sophisticated model (i.e. in this case, increase complexity of the kernel parameter)
Gather more features for each sample.
Decrease regularization in a regularized model.

A underfitting model cannot be improved, however, by increasing the number of training samples (do you see why?)

Now let's look at an overfit model:



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from sklearn.learning_curve import learning_curve
from sklearn.svm import SVR
training_sizes, train_scores, test_scores = learning_curve(SVR(kernel='rbf'), X, y, cv=10, scoring="mean_squared_error",
                                                           train_sizes=[.6, .7, .8, .9, 1.])
# Use the negative because we want to minimize squared error
plt.plot(training_sizes, -train_scores.mean(axis=1), label="training scores")
plt.plot(training_sizes, -test_scores.mean(axis=1), label="test scores")
plt.ylim((0, 50))
plt.legend(loc='best')

Here we show the learning curve for kernel = rbf. From the above discussion, we know that kernel = rbf is an estimator which mildly overfits the data. This is indicated by the fact that the training error is much less than the validation error. As we add more samples to this training set, the training error will continue to climb, while the cross-validation error will continue to decrease, until they meet in the middle. We can infer that adding more data will allow the estimator to very closely match the best possible cross-validation error.

When the learning curves have not yet converged with our full training set, it indicates an overfit model.

An overfitting model can be improved by:

Gathering more training samples.
Using a less-sophisticated model (i.e. in this case, make kernel less complex with kernel = poly)
Increasing regularization (parameter C for SVM/SVR).

In particular, gathering more features for each sample will not help the results.

Summary

We’ve seen above that an under-performing algorithm can be due to two possible situations: underfitting and overfitting. Using the technique of learning curves, we can train on progressively larger subsets of the data, evaluating the training error and cross-validation error to determine whether our algorithm is overfitting or underfitting. But what do we do with this information?

Underfitting

If our algorithm is underfitting, the following actions might help:

Add more features. It may be helpful to make use of information as additional features. For example, to predict housing prices features such as the neighborhood the house is in, the year the house was built, the size of the lot, etc. can help the model by giving new dimensions to help differentiate houses. Adding these features to the training and test sets can improve the fit.
Use a more sophisticated model. Adding complexity to the model can help improve the fit. For a SVR fit, this can be accomplished by increasing the kernel complexity (generally linear << poly << rbf). Each learning technique has its own methods of adding complexity.
Use fewer samples. Though this will not improve the classification, an underfitting algorithm can attain nearly the same error with a smaller training sample. For algorithms which are computationally expensive, reducing the training sample size can lead to very large improvements in speed.
Decrease regularization. Regularization is a technique used to impose simplicity in some machine learning models, by adding a penalty term that depends on the characteristics of the parameters. If a model is underfitting, decreasing the regularization can lead to better results.

Overfitting

If our algorithm shows signs of overfitting, the following actions might help:

Use fewer features. Using a feature selection technique may be useful, and decrease the overfitting of the estimator.
Use a simpler model. Model complexity and overfitting go hand-in-hand. For example, models like random forests tend to overfit much more than linear models and SVMs.
Use more training samples. Adding training samples can reduce the effect of overfitting.
Increase Regularization. Regularization is designed to prevent overfitting. So increasing regularization can lead to better results for overfitting models.

These choices become very important in real-world situations, as data collection usually costs time and energy. If the model is underfitting, then spending weeks or months collecting more data could be a colossal waste of time! However, more data (usually) gives us a better view of the true nature of the problem, so these issues should always be carefully considered before going on a "data foraging expedition".



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