Model Complexity Influence

Demonstrate how model complexity influences both prediction accuracy and computational performance.

The dataset is the Boston Housing dataset (resp. 20 Newsgroups) for regression (resp. classification).

For each class of models we make the model complexity vary through the choice of relevant model parameters and measure the influence on both computational performance (latency) and predictive power (MSE or Hamming Loss).

  • ../../_images/plot_model_complexity_influence_001.png
  • ../../_images/plot_model_complexity_influence_002.png
  • ../../_images/plot_model_complexity_influence_003.png

Script output:

Benchmarking SGDClassifier(alpha=0.001, average=False, class_weight=None, epsilon=0.1,
       eta0=0.0, fit_intercept=True, l1_ratio=0.25,
       learning_rate='optimal', loss='modified_huber', n_iter=5, n_jobs=1,
       penalty='elasticnet', power_t=0.5, random_state=None, shuffle=True,
       verbose=0, warm_start=False)
Complexity: 4454 | Hamming Loss (Misclassification Ratio): 0.2501 | Pred. Time: 0.030249s

Benchmarking SGDClassifier(alpha=0.001, average=False, class_weight=None, epsilon=0.1,
       eta0=0.0, fit_intercept=True, l1_ratio=0.5, learning_rate='optimal',
       loss='modified_huber', n_iter=5, n_jobs=1, penalty='elasticnet',
       power_t=0.5, random_state=None, shuffle=True, verbose=0,
       warm_start=False)
Complexity: 1624 | Hamming Loss (Misclassification Ratio): 0.2923 | Pred. Time: 0.022773s

Benchmarking SGDClassifier(alpha=0.001, average=False, class_weight=None, epsilon=0.1,
       eta0=0.0, fit_intercept=True, l1_ratio=0.75,
       learning_rate='optimal', loss='modified_huber', n_iter=5, n_jobs=1,
       penalty='elasticnet', power_t=0.5, random_state=None, shuffle=True,
       verbose=0, warm_start=False)
Complexity: 873 | Hamming Loss (Misclassification Ratio): 0.3191 | Pred. Time: 0.018224s

Benchmarking SGDClassifier(alpha=0.001, average=False, class_weight=None, epsilon=0.1,
       eta0=0.0, fit_intercept=True, l1_ratio=0.9, learning_rate='optimal',
       loss='modified_huber', n_iter=5, n_jobs=1, penalty='elasticnet',
       power_t=0.5, random_state=None, shuffle=True, verbose=0,
       warm_start=False)
Complexity: 655 | Hamming Loss (Misclassification Ratio): 0.3252 | Pred. Time: 0.016595s

Benchmarking NuSVR(C=1000.0, cache_size=200, coef0=0.0, degree=3, gamma=3.0517578125e-05,
   kernel='rbf', max_iter=-1, nu=0.1, shrinking=True, tol=0.001,
   verbose=False)
Complexity: 69 | MSE: 31.8133 | Pred. Time: 0.000387s

Benchmarking NuSVR(C=1000.0, cache_size=200, coef0=0.0, degree=3, gamma=3.0517578125e-05,
   kernel='rbf', max_iter=-1, nu=0.25, shrinking=True, tol=0.001,
   verbose=False)
Complexity: 136 | MSE: 25.6140 | Pred. Time: 0.000747s

Benchmarking NuSVR(C=1000.0, cache_size=200, coef0=0.0, degree=3, gamma=3.0517578125e-05,
   kernel='rbf', max_iter=-1, nu=0.5, shrinking=True, tol=0.001,
   verbose=False)
Complexity: 243 | MSE: 22.3315 | Pred. Time: 0.001296s

Benchmarking NuSVR(C=1000.0, cache_size=200, coef0=0.0, degree=3, gamma=3.0517578125e-05,
   kernel='rbf', max_iter=-1, nu=0.75, shrinking=True, tol=0.001,
   verbose=False)
Complexity: 350 | MSE: 21.3679 | Pred. Time: 0.001860s

Benchmarking NuSVR(C=1000.0, cache_size=200, coef0=0.0, degree=3, gamma=3.0517578125e-05,
   kernel='rbf', max_iter=-1, nu=0.9, shrinking=True, tol=0.001,
   verbose=False)
Complexity: 404 | MSE: 21.0915 | Pred. Time: 0.002109s

Benchmarking GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=3, max_features=None,
             max_leaf_nodes=None, min_impurity_split=1e-07,
             min_samples_leaf=1, min_samples_split=2,
             min_weight_fraction_leaf=0.0, n_estimators=10, presort='auto',
             random_state=None, subsample=1.0, verbose=0, warm_start=False)
Complexity: 10 | MSE: 28.9793 | Pred. Time: 0.000105s

Benchmarking GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=3, max_features=None,
             max_leaf_nodes=None, min_impurity_split=1e-07,
             min_samples_leaf=1, min_samples_split=2,
             min_weight_fraction_leaf=0.0, n_estimators=50, presort='auto',
             random_state=None, subsample=1.0, verbose=0, warm_start=False)
Complexity: 50 | MSE: 8.3398 | Pred. Time: 0.000199s

Benchmarking GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=3, max_features=None,
             max_leaf_nodes=None, min_impurity_split=1e-07,
             min_samples_leaf=1, min_samples_split=2,
             min_weight_fraction_leaf=0.0, n_estimators=100,
             presort='auto', random_state=None, subsample=1.0, verbose=0,
             warm_start=False)
Complexity: 100 | MSE: 7.0096 | Pred. Time: 0.000304s

Benchmarking GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=3, max_features=None,
             max_leaf_nodes=None, min_impurity_split=1e-07,
             min_samples_leaf=1, min_samples_split=2,
             min_weight_fraction_leaf=0.0, n_estimators=200,
             presort='auto', random_state=None, subsample=1.0, verbose=0,
             warm_start=False)
Complexity: 200 | MSE: 6.1836 | Pred. Time: 0.000426s

Benchmarking GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=3, max_features=None,
             max_leaf_nodes=None, min_impurity_split=1e-07,
             min_samples_leaf=1, min_samples_split=2,
             min_weight_fraction_leaf=0.0, n_estimators=500,
             presort='auto', random_state=None, subsample=1.0, verbose=0,
             warm_start=False)
Complexity: 500 | MSE: 6.3426 | Pred. Time: 0.001167s

Python source code: plot_model_complexity_influence.py


print(__doc__)

# Author: Eustache Diemert <eustache@diemert.fr>
# License: BSD 3 clause

import time
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1.parasite_axes import host_subplot
from mpl_toolkits.axisartist.axislines import Axes
from scipy.sparse.csr import csr_matrix

from sklearn import datasets
from sklearn.utils import shuffle
from sklearn.metrics import mean_squared_error
from sklearn.svm.classes import NuSVR
from sklearn.ensemble.gradient_boosting import GradientBoostingRegressor
from sklearn.linear_model.stochastic_gradient import SGDClassifier
from sklearn.metrics import hamming_loss

###############################################################################
# Routines


# initialize random generator
np.random.seed(0)


def generate_data(case, sparse=False):
    """Generate regression/classification data."""
    bunch = None
    if case == 'regression':
        bunch = datasets.load_boston()
    elif case == 'classification':
        bunch = datasets.fetch_20newsgroups_vectorized(subset='all')
    X, y = shuffle(bunch.data, bunch.target)
    offset = int(X.shape[0] * 0.8)
    X_train, y_train = X[:offset], y[:offset]
    X_test, y_test = X[offset:], y[offset:]
    if sparse:
        X_train = csr_matrix(X_train)
        X_test = csr_matrix(X_test)
    else:
        X_train = np.array(X_train)
        X_test = np.array(X_test)
    y_test = np.array(y_test)
    y_train = np.array(y_train)
    data = {'X_train': X_train, 'X_test': X_test, 'y_train': y_train,
            'y_test': y_test}
    return data


def benchmark_influence(conf):
    """
    Benchmark influence of :changing_param: on both MSE and latency.
    """
    prediction_times = []
    prediction_powers = []
    complexities = []
    for param_value in conf['changing_param_values']:
        conf['tuned_params'][conf['changing_param']] = param_value
        estimator = conf['estimator'](**conf['tuned_params'])
        print("Benchmarking %s" % estimator)
        estimator.fit(conf['data']['X_train'], conf['data']['y_train'])
        conf['postfit_hook'](estimator)
        complexity = conf['complexity_computer'](estimator)
        complexities.append(complexity)
        start_time = time.time()
        for _ in range(conf['n_samples']):
            y_pred = estimator.predict(conf['data']['X_test'])
        elapsed_time = (time.time() - start_time) / float(conf['n_samples'])
        prediction_times.append(elapsed_time)
        pred_score = conf['prediction_performance_computer'](
            conf['data']['y_test'], y_pred)
        prediction_powers.append(pred_score)
        print("Complexity: %d | %s: %.4f | Pred. Time: %fs\n" % (
            complexity, conf['prediction_performance_label'], pred_score,
            elapsed_time))
    return prediction_powers, prediction_times, complexities


def plot_influence(conf, mse_values, prediction_times, complexities):
    """
    Plot influence of model complexity on both accuracy and latency.
    """
    plt.figure(figsize=(12, 6))
    host = host_subplot(111, axes_class=Axes)
    plt.subplots_adjust(right=0.75)
    par1 = host.twinx()
    host.set_xlabel('Model Complexity (%s)' % conf['complexity_label'])
    y1_label = conf['prediction_performance_label']
    y2_label = "Time (s)"
    host.set_ylabel(y1_label)
    par1.set_ylabel(y2_label)
    p1, = host.plot(complexities, mse_values, 'b-', label="prediction error")
    p2, = par1.plot(complexities, prediction_times, 'r-',
                    label="latency")
    host.legend(loc='upper right')
    host.axis["left"].label.set_color(p1.get_color())
    par1.axis["right"].label.set_color(p2.get_color())
    plt.title('Influence of Model Complexity - %s' % conf['estimator'].__name__)
    plt.show()


def _count_nonzero_coefficients(estimator):
    a = estimator.coef_.toarray()
    return np.count_nonzero(a)

###############################################################################
# main code
regression_data = generate_data('regression')
classification_data = generate_data('classification', sparse=True)
configurations = [
    {'estimator': SGDClassifier,
     'tuned_params': {'penalty': 'elasticnet', 'alpha': 0.001, 'loss':
                      'modified_huber', 'fit_intercept': True},
     'changing_param': 'l1_ratio',
     'changing_param_values': [0.25, 0.5, 0.75, 0.9],
     'complexity_label': 'non_zero coefficients',
     'complexity_computer': _count_nonzero_coefficients,
     'prediction_performance_computer': hamming_loss,
     'prediction_performance_label': 'Hamming Loss (Misclassification Ratio)',
     'postfit_hook': lambda x: x.sparsify(),
     'data': classification_data,
     'n_samples': 30},
    {'estimator': NuSVR,
     'tuned_params': {'C': 1e3, 'gamma': 2 ** -15},
     'changing_param': 'nu',
     'changing_param_values': [0.1, 0.25, 0.5, 0.75, 0.9],
     'complexity_label': 'n_support_vectors',
     'complexity_computer': lambda x: len(x.support_vectors_),
     'data': regression_data,
     'postfit_hook': lambda x: x,
     'prediction_performance_computer': mean_squared_error,
     'prediction_performance_label': 'MSE',
     'n_samples': 30},
    {'estimator': GradientBoostingRegressor,
     'tuned_params': {'loss': 'ls'},
     'changing_param': 'n_estimators',
     'changing_param_values': [10, 50, 100, 200, 500],
     'complexity_label': 'n_trees',
     'complexity_computer': lambda x: x.n_estimators,
     'data': regression_data,
     'postfit_hook': lambda x: x,
     'prediction_performance_computer': mean_squared_error,
     'prediction_performance_label': 'MSE',
     'n_samples': 30},
]
for conf in configurations:
    prediction_performances, prediction_times, complexities = \
        benchmark_influence(conf)
    plot_influence(conf, prediction_performances, prediction_times,
                   complexities)

Total running time of the example: 54.53 seconds ( 0 minutes 54.53 seconds)