# Model Complexity Influence#

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

We will be using two datasets:
• Diabetes dataset for regression. This dataset consists of 10 measurements taken from diabetes patients. The task is to predict disease progression;

• The 20 newsgroups text dataset for classification. This dataset consists of newsgroup posts. The task is to predict on which topic (out of 20 topics) the post is written about.

We will model the complexity influence on three different estimators:

We make the model complexity vary through the choice of relevant model parameters in each of our selected models. Next, we will measure the influence on both computational performance (latency) and predictive power (MSE or Hamming Loss).

```# Authors: Eustache Diemert <eustache@diemert.fr>
#          Maria Telenczuk <https://github.com/maikia>
#          Guillaume Lemaitre <g.lemaitre58@gmail.com>
# License: BSD 3 clause

import time

import matplotlib.pyplot as plt
import numpy as np

from sklearn import datasets
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.linear_model import SGDClassifier
from sklearn.metrics import hamming_loss, mean_squared_error
from sklearn.model_selection import train_test_split
from sklearn.svm import NuSVR

# Initialize random generator
np.random.seed(0)
```

## Load the data#

First we load both datasets.

Note

We are using `fetch_20newsgroups_vectorized` to download 20 newsgroups dataset. It returns ready-to-use features.

Note

`X` of the 20 newsgroups dataset is a sparse matrix while `X` of diabetes dataset is a numpy array.

```def generate_data(case):
"""Generate regression/classification data."""
if case == "regression":
X, y = datasets.load_diabetes(return_X_y=True)
train_size = 0.8
elif case == "classification":
X, y = datasets.fetch_20newsgroups_vectorized(subset="all", return_X_y=True)
train_size = 0.4  # to make the example run faster

X_train, X_test, y_train, y_test = train_test_split(
X, y, train_size=train_size, random_state=0
)

data = {"X_train": X_train, "X_test": X_test, "y_train": y_train, "y_test": y_test}
return data

regression_data = generate_data("regression")
classification_data = generate_data("classification")
```

## Benchmark influence#

Next, we can calculate the influence of the parameters on the given estimator. In each round, we will set the estimator with the new value of `changing_param` and we will be collecting the prediction times, prediction performance and complexities to see how those changes affect the estimator. We will calculate the complexity using `complexity_computer` passed as a parameter.

```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
```

## Choose parameters#

We choose the parameters for each of our estimators by making a dictionary with all the necessary values. `changing_param` is the name of the parameter which will vary in each estimator. Complexity will be defined by the `complexity_label` and calculated using `complexity_computer`. Also note that depending on the estimator type we are passing different data.

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

configurations = [
{
"estimator": SGDClassifier,
"tuned_params": {
"penalty": "elasticnet",
"alpha": 0.001,
"loss": "modified_huber",
"fit_intercept": True,
"tol": 1e-1,
"n_iter_no_change": 2,
},
"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": 5,
},
{
"estimator": NuSVR,
"tuned_params": {"C": 1e3, "gamma": 2**-15},
"changing_param": "nu",
"changing_param_values": [0.05, 0.1, 0.2, 0.35, 0.5],
"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": 15,
},
{
"estimator": GradientBoostingRegressor,
"tuned_params": {
"loss": "squared_error",
"learning_rate": 0.05,
"max_depth": 2,
},
"changing_param": "n_estimators",
"changing_param_values": [10, 25, 50, 75, 100],
"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": 15,
},
]
```

## Run the code and plot the results#

We defined all the functions required to run our benchmark. Now, we will loop over the different configurations that we defined previously. Subsequently, we can analyze the plots obtained from the benchmark: Relaxing the `L1` penalty in the SGD classifier reduces the prediction error but leads to an increase in the training time. We can draw a similar analysis regarding the training time which increases with the number of support vectors with a Nu-SVR. However, we observed that there is an optimal number of support vectors which reduces the prediction error. Indeed, too few support vectors lead to an under-fitted model while too many support vectors lead to an over-fitted model. The exact same conclusion can be drawn for the gradient-boosting model. The only the difference with the Nu-SVR is that having too many trees in the ensemble is not as detrimental.

```def plot_influence(conf, mse_values, prediction_times, complexities):
"""
Plot influence of model complexity on both accuracy and latency.
"""

fig = plt.figure()
fig.subplots_adjust(right=0.75)

# first axes (prediction error)
ax1 = fig.add_subplot(111)
line1 = ax1.plot(complexities, mse_values, c="tab:blue", ls="-")[0]
ax1.set_xlabel("Model Complexity (%s)" % conf["complexity_label"])
y1_label = conf["prediction_performance_label"]
ax1.set_ylabel(y1_label)

ax1.spines["left"].set_color(line1.get_color())
ax1.yaxis.label.set_color(line1.get_color())
ax1.tick_params(axis="y", colors=line1.get_color())

# second axes (latency)
ax2 = fig.add_subplot(111, sharex=ax1, frameon=False)
line2 = ax2.plot(complexities, prediction_times, c="tab:orange", ls="-")[0]
ax2.yaxis.tick_right()
ax2.yaxis.set_label_position("right")
y2_label = "Time (s)"
ax2.set_ylabel(y2_label)
ax1.spines["right"].set_color(line2.get_color())
ax2.yaxis.label.set_color(line2.get_color())
ax2.tick_params(axis="y", colors=line2.get_color())

plt.legend(
(line1, line2), ("prediction error", "prediction latency"), loc="upper center"
)

plt.title(
"Influence of varying '%s' on %s"
% (conf["changing_param"], conf["estimator"].__name__)
)

for conf in configurations:
prediction_performances, prediction_times, complexities = benchmark_influence(conf)
plot_influence(conf, prediction_performances, prediction_times, complexities)
plt.show()
```
```Benchmarking SGDClassifier(alpha=0.001, l1_ratio=0.25, loss='modified_huber',
n_iter_no_change=2, penalty='elasticnet', tol=0.1)
Complexity: 4948 | Hamming Loss (Misclassification Ratio): 0.2675 | Pred. Time: 0.060968s

Benchmarking SGDClassifier(alpha=0.001, l1_ratio=0.5, loss='modified_huber',
n_iter_no_change=2, penalty='elasticnet', tol=0.1)
Complexity: 1847 | Hamming Loss (Misclassification Ratio): 0.3264 | Pred. Time: 0.045105s

Benchmarking SGDClassifier(alpha=0.001, l1_ratio=0.75, loss='modified_huber',
n_iter_no_change=2, penalty='elasticnet', tol=0.1)
Complexity: 997 | Hamming Loss (Misclassification Ratio): 0.3383 | Pred. Time: 0.038429s

Benchmarking SGDClassifier(alpha=0.001, l1_ratio=0.9, loss='modified_huber',
n_iter_no_change=2, penalty='elasticnet', tol=0.1)
Complexity: 802 | Hamming Loss (Misclassification Ratio): 0.3582 | Pred. Time: 0.038288s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05, nu=0.05)
Complexity: 18 | MSE: 5558.7313 | Pred. Time: 0.000181s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05, nu=0.1)
Complexity: 36 | MSE: 5289.8022 | Pred. Time: 0.000250s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05, nu=0.2)
Complexity: 72 | MSE: 5193.8353 | Pred. Time: 0.000403s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05, nu=0.35)
Complexity: 124 | MSE: 5131.3279 | Pred. Time: 0.000667s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05)
Complexity: 178 | MSE: 5149.0779 | Pred. Time: 0.000897s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2, n_estimators=10)
Complexity: 10 | MSE: 4066.4812 | Pred. Time: 0.000162s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2, n_estimators=25)
Complexity: 25 | MSE: 3551.1723 | Pred. Time: 0.000185s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2, n_estimators=50)
Complexity: 50 | MSE: 3445.2171 | Pred. Time: 0.000226s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2, n_estimators=75)
Complexity: 75 | MSE: 3433.0358 | Pred. Time: 0.000262s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2)
Complexity: 100 | MSE: 3456.0602 | Pred. Time: 0.000295s
```

## Conclusion#

As a conclusion, we can deduce the following insights:

• a model which is more complex (or expressive) will require a larger training time;

• a more complex model does not guarantee to reduce the prediction error.

These aspects are related to model generalization and avoiding model under-fitting or over-fitting.

Total running time of the script: (0 minutes 5.297 seconds)

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