Control risk of a binary classifier with multiple prediction parameters¶

AI is a powerful tool for email sorting (for example between spam and urgent emails). However, because algorithms are not perfect, manual verification is sometimes required. Thus one would like to be able to control the amount of emails sent to human validation. One way to do so is to define a multi-parameter prediction function based on a classifier's predicted scores. This would allow defining a rule for email checking, which could be adapted by varying the prediction parameters.

In this example, we explain how to do risk control for binary classification relying on multiple prediction parameters with MAPIE.

# sphinx_gallery_thumbnail_number = 2

import matplotlib.patches as patches
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import LinearSegmentedColormap
from matplotlib.lines import Line2D
from matplotlib.patches import Patch
from sklearn.datasets import make_circles
from sklearn.neural_network import MLPClassifier

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

RANDOM_STATE = 1

First, load the dataset and then split it into training, calibration, 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=0.8,
    conformalize_size=0.1,
    test_size=0.1,
    random_state=RANDOM_STATE,
)

# Plot the three datasets to visualize the distribution of the two classes. We can
# assume that the feature space represents some embedding of emails.
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=1)

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Third define a multi-parameter prediction function. For an email to be sent to human verification, we want the predicted score of the positive class to be between two thresholds lambda_1 and lambda_2. High (respectively low) values of the score correspond to high confidence that the email is a spam (respectively not a spam). Therefore, emails with intermediate scores are the ones for which the classifier is the least certain, and we want these emails to be verified by a human.

def send_to_human(X, lambda_1, lambda_2):
    y_score = clf.predict_proba(X)[:, 1]
    return (lambda_1 <= y_score) & (y_score < lambda_2)

From the previous function, we know we have a constraint lambda_1 <= lambda_2. We can generate a set of values to explore respecting this constraint.

to_explore = []
for i in range(9):
    lambda_1 = i / 10
    for j in range(i + 1, 10):
        lambda_2 = j / 10
        if lambda_2 > 0.99:
            break
        to_explore.append((lambda_1, lambda_2))
to_explore = np.array(to_explore)

Because we want to control the proportion of emails to be verified by a human, we use predicted_positive_fraction, a specific BinaryClassificationRisk which represents the fraction of samples predicted as positive (i.e., sent to human verification).

Finally, we initialize a BinaryClassificationController using our custom function send_to_human, the chosen risk predicted_positive_fraction, a target risk level (0.2), and a confidence level (0.9). Then we use the calibration data to compute statistically guaranteed thresholds using a multi-parameter control method.

target_level = 0.2
confidence_level = 0.9

bcc = BinaryClassificationController(
    predict_function=send_to_human,
    risk="predicted_positive_fraction",
    target_level=target_level,
    confidence_level=confidence_level,
    best_predict_param_choice="recall",
    list_predict_params=to_explore,
)
bcc.calibrate(X_calib, y_calib)

print(
    f"{len(bcc.valid_predict_params)} multi-dimensional parameters "
    f"found that guarantee a proportion of emails sent\n"
    f"to verification of at most {target_level} with "
    f"a confidence of {confidence_level}."
)

Out:

19 multi-dimensional parameters found that guarantee a proportion of emails sent
to verification of at most 0.2 with a confidence of 0.9.

grid_size = 10
matrix = np.full((grid_size, grid_size), np.nan)

# Build p-values matrix
for i, (l1, l2) in enumerate(to_explore):
    row = int(l1 * grid_size)
    col = int(l2 * grid_size)
    matrix[row, col] = bcc.p_values[i, 0]

# Build valid thresholds mask
valid_matrix = np.zeros((grid_size, grid_size), dtype=int)
for l1, l2 in bcc.valid_predict_params:
    row = int(l1 * grid_size)
    col = int(l2 * grid_size)
    valid_matrix[row, col] = 1

# Plot p-value matrix
fig, ax = plt.subplots(figsize=(7.5, 7.5))

colors = ["#cde2f9", "#96bfd7", "#3765a9"]
cmap = LinearSegmentedColormap.from_list("custom_blue", colors, gamma=0.5)
masked_matrix = np.ma.masked_invalid(matrix)
im = ax.imshow(masked_matrix, cmap=cmap, interpolation="nearest")

for i in range(grid_size):
    for j in range(grid_size):
        if np.isnan(matrix[i, j]):
            rect = patches.Rectangle(
                (j - 0.5, i - 0.5),
                1,
                1,
                hatch="///",
                facecolor="none",
                edgecolor="grey",
                linewidth=0,
            )
            ax.add_patch(rect)

# Add valid parameters area shape
for i in range(grid_size):
    for j in range(grid_size):
        if valid_matrix[i, j] == 1:
            neighbors = [(i - 1, j), (i + 1, j), (i, j - 1), (i, j + 1)]
            if i - 1 < 0 or valid_matrix[i - 1, j] == 0:
                ax.plot([j - 0.5, j + 0.5], [i - 0.5, i - 0.5], color="#2ecc71", lw=3)
            if i + 1 >= grid_size or valid_matrix[i + 1, j] == 0:
                ax.plot([j - 0.5, j + 0.5], [i + 0.5, i + 0.5], color="#2ecc71", lw=3)
            if j - 1 < 0 or valid_matrix[i, j - 1] == 0:
                ax.plot([j - 0.5, j - 0.5], [i - 0.5, i + 0.5], color="#2ecc71", lw=3)
            if j + 1 >= grid_size or valid_matrix[i, j + 1] == 0:
                ax.plot([j + 0.5, j + 0.5], [i - 0.5, i + 0.5], color="#2ecc71", lw=3)

# Add best predict param as a star
best_l1, best_l2 = bcc.best_predict_param
ax.scatter(
    best_l2 * grid_size,
    best_l1 * grid_size,
    c="#2ecc71",
    marker="*",
    edgecolors="k",
    s=300,
    label="Best threshold pair",
)

# Set up axes, theme and legend
ax.set_xlabel(r"$\lambda_2$", fontsize=16)
ax.set_ylabel(r"$\lambda_1$", fontsize=16)
ax.set_title(
    "P-values per parameter pair\nwith valid parameter zone highlighted", fontsize=16
)
ax.set_xticks(range(grid_size))
ax.set_xticklabels(np.round(np.arange(grid_size) / grid_size, 2), fontsize=14)
ax.set_yticks(range(grid_size))
ax.set_yticklabels(np.round(np.arange(grid_size) / grid_size, 2), fontsize=14)

cbar = plt.colorbar(im, ax=ax, orientation="horizontal", pad=0.2, fraction=0.035)
cbar.set_label("P-value", fontsize=12)
legend_elements = [
    Patch(facecolor="none", edgecolor="grey", hatch="///", label="Non-explored zone"),
    Patch(facecolor="none", edgecolor="#2ecc71", label="Valid parameter zone", lw=2),
    Line2D(
        [0],
        [0],
        marker="*",
        color="w",
        label="Best threshold pair",
        markerfacecolor="#2ecc71",
        markeredgecolor="k",
        markersize=15,
    ),
]
ax.legend(
    handles=legend_elements,
    loc="lower center",
    bbox_to_anchor=(0.5, -0.25),
    ncol=2,
    fontsize=12,
    frameon=False,
)
plt.tight_layout()
plt.show()
ax.set_ylabel(r"$\lambda_1$", fontsize=16)
ax.set_title("Valid parameters")
fig.tight_layout()
plt.show()

Valid parameters

Total running time of the script: ( 0 minutes 1.186 seconds)

Download Python source code: plot_risk_control_multi_parameter_binary_classification.py

Download Jupyter notebook: plot_risk_control_multi_parameter_binary_classification.ipynb

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	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 `.	1
	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