Cross-validation¶

Optunity offers a simple interface to k-fold cross-validation. This is a statistical approach to measure a model’s generalization performance. In the context of hyperparameter search, cross-validation is used to estimate the performance of a hyperparameter tuple. The cross-validation routines we offer are optional and can be replaced by comparable routines from other packages or some other method to estimate generalization performance.

The fold generation procedure in Optunity allows for iterated cross-validation and is aware of both strata (data instances that must be spread across folds) and clusters (sets of instances that must assigned to a single fold). Please refer to optunity.cross_validated() for implementation and API details.

We will build examples step by step. The basic setup is a train and predict function along with some data to construct folds over:

from __future__ import print_function
import optunity as opt

def train(x, y, filler=''):
print(filler + 'Training data:')
for instance, label in zip(x, y):
print(filler + str(instance) + ' ' + str(label))

def predict(x, filler=''):
print(filler + 'Testing data:')
for instance in x:
print(filler + str(instance))

data = list(range(9))
labels = [0] * 9


The recommended way to perform cross-validation is using the optunity.cross_validation.cross_validated() function decorator. To use it, you must specify an objective function. This function should contain the logic that is placed in the inner loop in cross-validation (e.g. train a model, predict test set, compute score), with the following signature: f(x_train, y_train, x_test, y_test, hyperpar_1, hyperpar_2, ...) (argument names are important):

• x_train: training data
• y_train: training labels (optional)
• x_test: test data
• y_test: test labels (optional)
• hyperparameter names: the hyperparameters that must be optimized

The cross_validated decorator takes care of generating folds, partitioning the data, iterating over folds and aggregating partial results. After decoration, the arguments x_train, y_train, x_test and y_test will be bound (e.g. the decorated function does not take these as arguments). The decorated function will have hyperparameters as (keyword) arguments and returns a cross-validation result.

A simple code example:

@opt.cross_validated(x=data, y=labels, num_folds=3)
def cved(x_train, y_train, x_test, y_test):
train(x_train, y_train)
predict(x_test)
return 0.0

cved()


Reusing folds¶

If you want to compare different aspects of the learning approach (learning algorithms, score function, ...), it is a good idea to use the same cross-validation folds. This is very easy by using the cross_validated decorator without syntactic sugar. Lets say we want to compare an SVM with RBF kernel and polynomial kernel with the same cross-validation configuration:

import sklearn.svm as svm

def svm_rbf(x_train, y_train, x_test, y_test, C, gamma):
model = svm.SVC(kernel='rbf', C=C, gamma=gamma).fit(x_train, y_train)
y_pred = model.predict(x_test)
return opt.score_functions.accuracy(y_test, y_pred)

def svm_poly(x_train, y_train, x_test, y_test, C, d):
model = svm.SVC(kernel=’poly’, C=C, degree=d).fit(x_train, y_train)
y_pred = model.predict(x_test)
return opt.score_functions.accuracy(y_test, y_pred)

cv_decorator = opt.cross_validated(x=data, y=labels, num_folds=3)

svm_rbf_cv = cv_decorator(svm_rbf)
svm_poly_cv = cv_decorator(svm_poly)


In this example, the function svm_rbf_cv takes keyword arguments C and gamma while svm_poly_cv takes C and d. Both perform cross-validation on the same data, using the same folds.

Nested cross-validation¶

Nested cross-validation is a commonly used approach to estimate the generalization performance of a modeling process which includes model selection internally. A good summary is provided here.

Nested cv consists of two cross-validation procedures wrapped around eachother. The inner cv is used for model selection, the outer cv estimates generalization performance.

This can be done in a straightforward manner using Optunity:

@opt.cross_validated(x=data, y=labels, num_folds=3)
def nested_cv(x_train, y_train, x_test, y_test):

@opt.cross_validated(x=x_train, y=y_train, num_folds=3)
def inner_cv(x_train, y_train, x_test, y_test):
train(x_train, y_train, '...')
predict(x_test, '...')
return 0.0

inner_cv()
predict(x_test)
return 0.0

nested_cv()


The inner optunity.cross_validated() decorator has access to the train and test folds generated by the outer procedure (x_train and x_test). For notational simplicity we assume a problem without labels here.

Note

The inner folds are regenerated in every iteration (since we are redefining inner_cv each time). The inner folds will therefore be different each time. The outer folds remain static, unless regenerate_folds=True is passed.

A complete example of nested cross-validation is available in Basic: nested cross-validation.

Aggregators¶

Optunity’s cross-validation implementation allows you to specify an aggregator. This is the function that will be used to compute the cross-validation result based on the results of individual folds. The default function is mean. You can specify any function to compute another measure if desired (for instance min, max, ...).

Computing multiple performance measures during cross-validation¶

Sometimes it is desired to compute multiple performance measures using cross-validation. This is particularly useful for nested cross-validation. This is possible in Optunity by letting the wrapped function return multiple scores and using the optunity.cross_validation.list_mean() aggregator:

@optunity.cross_validated(x=data, y=labels, num_folds=3,
aggregator=optunity.cross_validation.list_mean)
def f(x_train, y_train, x_test, y_test):
model = train(x_train, y_train)
predictions = model.predict(x_test)
score_1 = score_1(y_test, predictions)
score_2 = score_2(y_test, predictions)
return score_1, score_2


For even more flexibility, you can use optunity.cross_validation.identity() as aggregator, which will return a list of return values for every cross-validation fold.