Precision control for a binary classifier¶

In this example, we explain how to do risk control for binary classification with MAPIE.

# sphinx_gallery_thumbnail_number = 2

import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import make_circles
from sklearn.inspection import DecisionBoundaryDisplay
from sklearn.metrics import precision_score
from sklearn.model_selection import FixedThresholdClassifier
from sklearn.neural_network import MLPClassifier

from mapie.risk_control import BinaryClassificationController
from mapie.utils import train_conformalize_test_split

RANDOM_STATE = 42

First, load the dataset and then split it into training, calibration (for conformalization), and test sets.

X, y = make_circles(n_samples=5000, noise=0.3, factor=0.3, random_state=RANDOM_STATE)
(X_train, X_calib, X_test, y_train, y_calib, y_test) = train_conformalize_test_split(
    X,
    y,
    train_size=0.7,
    conformalize_size=0.1,
    test_size=0.2,
    random_state=RANDOM_STATE,
)

# Plot the three datasets to visualize the distribution of the two classes.
fig, axes = plt.subplots(1, 3, figsize=(18, 6))
titles = ["Training Data", "Calibration Data", "Test Data"]
datasets = [(X_train, y_train), (X_calib, y_calib), (X_test, y_test)]

for i, (ax, (X_data, y_data), title) in enumerate(zip(axes, datasets, titles)):
    ax.scatter(
        X_data[y_data == 0, 0],
        X_data[y_data == 0, 1],
        edgecolors="k",
        c="tab:blue",
        label='"negative" class',
        alpha=0.5,
    )
    ax.scatter(
        X_data[y_data == 1, 0],
        X_data[y_data == 1, 1],
        edgecolors="k",
        c="tab:red",
        label='"positive" class',
        alpha=0.5,
    )
    ax.set_title(title, fontsize=18)
    ax.set_xlabel("Feature 1", fontsize=16)
    ax.tick_params(labelsize=14)

    if i == 0:
        ax.set_ylabel("Feature 2", fontsize=16)
    else:
        ax.set_ylabel("")
        ax.set_yticks([])

handles, labels = axes[0].get_legend_handles_labels()
fig.legend(
    handles,
    labels,
    loc="lower center",
    bbox_to_anchor=(0.5, -0.01),
    ncol=2,
    fontsize=16,
)

plt.suptitle("Visualization of Train, Calibration, and Test Sets", fontsize=22)
plt.tight_layout(rect=[0, 0.05, 1, 0.95])
plt.show()

Visualization of Train, Calibration, and Test Sets, Training Data, Calibration Data, Test Data

Second, fit a Multi-layer Perceptron classifier on the training data.

clf = MLPClassifier(max_iter=150, random_state=RANDOM_STATE)
clf.fit(X_train, y_train)

MLPClassifier(max_iter=150, random_state=42)

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Next, we initialize a BinaryClassificationController using the probability estimation function from the fitted estimator: clf.predict_proba, a risk or performance metric (here, "precision"), a target risk level, and a confidence level. Then we use the calibration data to compute statistically guaranteed thresholds using a risk control method.

Different risks or performance metrics have been implemented, such as precision and recall, but you can also implement your own custom function using BinaryClassificationRisk and choose your own secondary objective.

target_precision = 0.8
confidence_level = 0.9
bcc = BinaryClassificationController(
    clf.predict_proba,
    "precision",
    target_level=target_precision,
    confidence_level=confidence_level,
)
bcc.calibrate(X_calib, y_calib)

print(
    f"{len(bcc.valid_predict_params)} thresholds found that guarantee a precision of "
    f"at least {target_precision} with a confidence of {confidence_level}.\n"
    "Among those, the one that maximizes the secondary objective (recall here) is: "
    f"{bcc.best_predict_param:.2f}."
)

Out:

49 thresholds found that guarantee a precision of at least 0.8 with a confidence of 0.9.
Among those, the one that maximizes the secondary objective (recall here) is: 0.50.

In the plot below, we visualize how the threshold values impact precision, and what thresholds have been computed as statistically guaranteed.

proba_positive_class = clf.predict_proba(X_calib)[:, 1]

tested_thresholds = bcc._predict_params
precisions = np.full(len(tested_thresholds), np.inf)
for i, threshold in enumerate(tested_thresholds):
    y_pred = (proba_positive_class >= threshold).astype(int)
    precisions[i] = precision_score(y_calib, y_pred)

naive_threshold_index = np.argmin(
    np.where(precisions >= target_precision, precisions - target_precision, np.inf)
)

valid_thresholds_indices = np.array(
    [t in bcc.valid_predict_params for t in tested_thresholds]
)
best_threshold_index = np.where(tested_thresholds == bcc.best_predict_param)[0][0]

plt.figure()
plt.scatter(
    tested_thresholds[valid_thresholds_indices],
    precisions[valid_thresholds_indices],
    c="tab:green",
    label="Valid thresholds",
)
plt.scatter(
    tested_thresholds[~valid_thresholds_indices],
    precisions[~valid_thresholds_indices],
    c="tab:red",
    label="Invalid thresholds",
)
plt.scatter(
    tested_thresholds[best_threshold_index],
    precisions[best_threshold_index],
    c="tab:green",
    label="Best threshold",
    marker="*",
    edgecolors="k",
    s=300,
)
plt.scatter(
    tested_thresholds[naive_threshold_index],
    precisions[naive_threshold_index],
    c="tab:red",
    label="Naive threshold",
    marker="*",
    edgecolors="k",
    s=300,
)
plt.axhline(target_precision, color="tab:gray", linestyle="--")
plt.text(
    0.7,
    target_precision + 0.02,
    "Target precision",
    color="tab:gray",
    fontstyle="italic",
)
plt.xlabel("Threshold")
plt.ylabel("Precision")
plt.legend()
plt.show()

proba_positive_class_test = clf.predict_proba(X_test)[:, 1]
y_pred_naive = (
    proba_positive_class_test >= tested_thresholds[naive_threshold_index]
).astype(int)
print(
    "With the naive threshold, the precision is:\n "
    f"- {precisions[naive_threshold_index]:.3f} on the calibration set\n "
    f"- {precision_score(y_test, y_pred_naive):.3f} on the test set."
)

print(
    "\n\nWith risk control, the precision is:\n "
    f"- {precisions[best_threshold_index]:.3f} on the calibration set\n "
    f"- {precision_score(y_test, bcc.predict(X_test)):.3f} on the test set."
)

plot risk control binary classification

Out:

With the naive threshold, the precision is:
 - 0.801 on the calibration set
 - 0.763 on the test set.


With risk control, the precision is:
 - 0.886 on the calibration set
 - 0.857 on the test set.

Contrary to the naive way of computing a threshold to satisfy a precision target on calibration data, risk control provides statistical guarantees on unseen data. In this example, the naive threshold results in a precision on the test set that is lower than the target precision while risk control takes a margin to guarantee the target precision on unseen data with high probability. In the plot above, we can see that not all thresholds corresponding to a precision higher than the target are valid. This is due to the uncertainty inherent to the finite size of the calibration set, which risk control takes into account.

In particular, the highest threshold values are considered invalid due to the small number of observations used to compute the precision, following the Learn Then Test procedure. In the most extreme case, no observation is available, which causes the precision value to be ill-defined and set to 0.

Besides computing a set of valid thresholds, BinaryClassificationController also outputs the "best" one, which is the valid threshold that maximizes a secondary objective (recall here).

After obtaining the best threshold, we can use the predict function of BinaryClassificationController for future predictions, or use scikit-learn's FixedThresholdClassifier as a wrapper to benefit from functionalities like easily plotting the decision boundary as seen below.

y_pred = bcc.predict(X_test)

clf_threshold = FixedThresholdClassifier(clf, threshold=bcc.best_predict_param)
clf_threshold.fit(X_train, y_train)
# .fit necessary for plotting, alternatively you can use sklearn.frozen.FrozenEstimator


disp = DecisionBoundaryDisplay.from_estimator(
    clf_threshold, X_test, response_method="predict", cmap=plt.cm.coolwarm
)

plt.scatter(
    X_test[y_test == 0, 0],
    X_test[y_test == 0, 1],
    edgecolors="k",
    c="tab:blue",
    alpha=0.3,
    label='"negative" class',
)
plt.scatter(
    X_test[y_test == 1, 0],
    X_test[y_test == 1, 1],
    edgecolors="k",
    c="tab:red",
    alpha=0.3,
    label='"positive" class',
)
plt.title("Decision Boundary of FixedThresholdClassifier")
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plt.legend()
plt.show()

	hidden_layer_sizes hidden_layer_sizes: array-like of shape(n_layers - 2,), default=(100,) The ith element represents the number of neurons in the ith hidden layer.	(100,)
	activation activation: {'identity', 'logistic', 'tanh', 'relu'}, default='relu' Activation function for the hidden layer. - 'identity', no-op activation, useful to implement linear bottleneck, returns f(x) = x - 'logistic', the logistic sigmoid function, returns f(x) = 1 / (1 + exp(-x)). - 'tanh', the hyperbolic tan function, returns f(x) = tanh(x). - 'relu', the rectified linear unit function, returns f(x) = max(0, x)	'relu'
	solver solver: {'lbfgs', 'sgd', 'adam'}, default='adam' The solver for weight optimization. - 'lbfgs' is an optimizer in the family of quasi-Newton methods. - 'sgd' refers to stochastic gradient descent. - 'adam' refers to a stochastic gradient-based optimizer proposed by Kingma, Diederik, and Jimmy Ba For a comparison between Adam optimizer and SGD, see :ref:`sphx_glr_auto_examples_neural_networks_plot_mlp_training_curves.py`. Note: The default solver 'adam' works pretty well on relatively large datasets (with thousands of training samples or more) in terms of both training time and validation score. For small datasets, however, 'lbfgs' can converge faster and perform better.	'adam'
	alpha alpha: float, default=0.0001 Strength of the L2 regularization term. The L2 regularization term is divided by the sample size when added to the loss. For an example usage and visualization of varying regularization, see :ref:`sphx_glr_auto_examples_neural_networks_plot_mlp_alpha.py`.	0.0001
	batch_size batch_size: int, default='auto' Size of minibatches for stochastic optimizers. If the solver is 'lbfgs', the classifier will not use minibatch. When set to "auto", `batch_size=min(200, n_samples)`.	'auto'
	learning_rate learning_rate: {'constant', 'invscaling', 'adaptive'}, default='constant' Learning rate schedule for weight updates. - 'constant' is a constant learning rate given by 'learning_rate_init'. - 'invscaling' gradually decreases the learning rate at each time step 't' using an inverse scaling exponent of 'power_t'. effective_learning_rate = learning_rate_init / pow(t, power_t) - 'adaptive' keeps the learning rate constant to 'learning_rate_init' as long as training loss keeps decreasing. Each time two consecutive epochs fail to decrease training loss by at least tol, or fail to increase validation score by at least tol if 'early_stopping' is on, the current learning rate is divided by 5. Only used when ``solver='sgd'``.	'constant'
	learning_rate_init learning_rate_init: float, default=0.001 The initial learning rate used. It controls the step-size in updating the weights. Only used when solver='sgd' or 'adam'.	0.001
	power_t power_t: float, default=0.5 The exponent for inverse scaling learning rate. It is used in updating effective learning rate when the learning_rate is set to 'invscaling'. Only used when solver='sgd'.	0.5
	max_iter max_iter: int, default=200 Maximum number of iterations. The solver iterates until convergence (determined by 'tol') or this number of iterations. For stochastic solvers ('sgd', 'adam'), note that this determines the number of epochs (how many times each data point will be used), not the number of gradient steps.	150
	shuffle shuffle: bool, default=True Whether to shuffle samples in each iteration. Only used when solver='sgd' or 'adam'.	True
	random_state random_state: int, RandomState instance, default=None Determines random number generation for weights and bias initialization, train-test split if early stopping is used, and batch sampling when solver='sgd' or 'adam'. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `.	42
	tol tol: float, default=1e-4 Tolerance for the optimization. When the loss or score is not improving by at least ``tol`` for ``n_iter_no_change`` consecutive iterations, unless ``learning_rate`` is set to 'adaptive', convergence is considered to be reached and training stops.	0.0001
	verbose verbose: bool, default=False Whether to print progress messages to stdout.	False
	warm_start warm_start: bool, default=False When set to True, reuse the solution of the previous call to fit as initialization, otherwise, just erase the previous solution. See :term:`the Glossary `.	False
	momentum momentum: float, default=0.9 Momentum for gradient descent update. Should be between 0 and 1. Only used when solver='sgd'.	0.9
	nesterovs_momentum nesterovs_momentum: bool, default=True Whether to use Nesterov's momentum. Only used when solver='sgd' and momentum > 0.	True
	early_stopping early_stopping: bool, default=False Whether to use early stopping to terminate training when validation score is not improving. If set to True, it will automatically set aside ``validation_fraction`` of training data as validation and terminate training when validation score is not improving by at least ``tol`` for ``n_iter_no_change`` consecutive epochs. The split is stratified, except in a multilabel setting. If early stopping is False, then the training stops when the training loss does not improve by more than ``tol`` for ``n_iter_no_change`` consecutive passes over the training set. Only effective when solver='sgd' or 'adam'.	False
	validation_fraction validation_fraction: float, default=0.1 The proportion of training data to set aside as validation set for early stopping. Must be between 0 and 1. Only used if early_stopping is True.	0.1
	beta_1 beta_1: float, default=0.9 Exponential decay rate for estimates of first moment vector in adam, should be in [0, 1). Only used when solver='adam'.	0.9
	beta_2 beta_2: float, default=0.999 Exponential decay rate for estimates of second moment vector in adam, should be in [0, 1). Only used when solver='adam'.	0.999
	epsilon epsilon: float, default=1e-8 Value for numerical stability in adam. Only used when solver='adam'.	1e-08
	n_iter_no_change n_iter_no_change: int, default=10 Maximum number of epochs to not meet ``tol`` improvement. Only effective when solver='sgd' or 'adam'. .. versionadded:: 0.20	10
	max_fun max_fun: int, default=15000 Only used when solver='lbfgs'. Maximum number of loss function calls. The solver iterates until convergence (determined by 'tol'), number of iterations reaches max_iter, or this number of loss function calls. Note that number of loss function calls will be greater than or equal to the number of iterations for the `MLPClassifier`. .. versionadded:: 0.22	15000