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Measuring the Runtime of Global Effect Plots

This notebook analyzes the runtime \(T(\cdot)\) of Global Effect plots, which depends on:

  • \(t_f\): The time required to evaluate the black-box function \(f\) on the entire dataset.
  • \(N\): The number of instances in \(X\).
  • \(D\): The number of features in \(X\).
  • \(K\): The number of points used for centering the feature effect plot.
  • \(M\): The number of evaluation points.

The key factors affecting runtime are \(t_f\), \(N\), and \(D\). Each method involves:

  1. Preparing the permuted/augmented dataset: This step depends only on \(N\) and is repeated independently for each feature, so it contributes \(D T_1(N)\) to the total runtime.
  2. Predicting on the permuted dataset: We here make the hypothesis, that \(f(X)\) runs in \(t_f\) independently of the number of instances. This is not generally true, however, it is a reasonable assumption as long as \(f(X)\) can be computed in a single pass or some batches. Additionally, the prediction must be repeated independently for each feature, contributing \(D T_2(t_f)\), except for RHALE, where all gradients are computed in a single pass, resulting in \(T_2(t_f)\).

Therefore, the runtime of each methods is: \(\(T(t_f, N, D) \approx D T_1(N) + T_2(t_f, D)\)\).

Now, let's see all these effects in practice!

import effector
import numpy as np
import timeit
import time
import matplotlib.pyplot as plt
np.random.seed(21)

def return_predict(t):
    def predict(x):
        time.sleep(t)
        model = effector.models.DoubleConditionalInteraction()
        return model.predict(x)
    return predict

def return_jacobian(t):
    def jacobian(x):
        time.sleep(t)
        model = effector.models.DoubleConditionalInteraction()
        return model.jacobian(x)
    return jacobian

def measure_time(method_name, features):
    fit_time_list, eval_time_list = [], []
    X = np.random.uniform(-1, 1, (N, D))
    xx = np.linspace(-1, 1, M)
    axis_limits = np.array([[-1] * D, [1] * D])

    method_map = {
        "pdp": effector.PDP,
        "d_pdp": effector.DerPDP,
        "ale": effector.ALE,
        "rhale": effector.RHALE,
        "shap_dp": effector.ShapDP
    }

    for _ in range(repetitions):
        # general kwargs
        method_kwargs = {"data": X, "model": model, "axis_limits": axis_limits, "nof_instances": "all"}
        fit_kwargs = {"features": features, "centering": True, "points_for_centering": K}

        # specialize kwargs per method
        if method_name in ["d_pdp", "rhale"]:
            method_kwargs["model_jac"] = model_jac
        if method_name in ["rhale", "ale"]:
            fit_kwargs["binning_method"] = effector.axis_partitioning.Fixed(nof_bins=20)

        # init
        method = method_map[method_name](**method_kwargs)

        # fit
        tic = time.time()
        method.fit(**fit_kwargs)
        fit_time_list.append(time.time() - tic)

        # eval
        tic = time.time()
        for feat in features:
            eval_kwargs = {"feature": feat, "xs": xx, "centering": True, "heterogeneity": True}
            method.eval(**eval_kwargs)
        eval_time_list.append(time.time() - tic)

    return {"fit": np.mean(fit_time_list), "eval": np.mean(eval_time_list), "total": (np.mean(fit_time_list) + np.mean(eval_time_list))}

import matplotlib.pyplot as plt

def bar_plot(xs, time_dict, methods, metric, title, xlabel, ylabel, bar_width=0.02):

    bar_width = (np.max(xs) - np.min(xs)) / 40
    method_to_label = {"ale": "ALE", "rhale": "RHALE", "pdp": "PDP", "d_pdp": "d-pdp", "shap_dp": "SHAP DP"}
    plt.figure()

    # Calculate the offsets for each bar group
    offsets = np.linspace(-2*bar_width, 2*bar_width, len(methods))

    for i, method in enumerate(methods):
        label = method_to_label[method]
        plt.bar(
            xs + offsets[i],
            [tt[metric] for tt in time_dict[method]],
            label=label,
            width=bar_width
        )

    plt.title(title)
    plt.xlabel(xlabel)
    plt.ylabel(ylabel)
    plt.xticks(xs)
    plt.legend()
    plt.show()

\(T_1\): runtime vs N

For one feature

t = 0.001
N = 10_000
D = 3
K = 100
M = 100
repetitions = 2
features=[0]

method_names = ["ale", "rhale", "pdp", "d_pdp"]
vec = np.array([10_000, 25_000, 50_000])
time_dict = {method_name: [] for method_name in method_names}
for N in vec:
    model = return_predict(t)
    model_jac = return_jacobian(t)
    for method_name in method_names:
        time_dict[method_name].append(measure_time(method_name, features))

for metric in ["total"]: # ["fit", "eval", "total"]:
    if metric in ["fit", "eval"]:
        title = "Runtime: ." + metric + "() -- single feature"
    else:
        title = "Runtime: .fit() + .eval() -- single feature"

    bar_plot(
        vec, 
        time_dict, 
        method_names,
        metric=metric,
        title=title,
        xlabel="N: number of instances",
        ylabel="time (sec)"
)

png

For all features

features=[i for i in range(D)]
method_names = ["ale", "rhale", "pdp", "d_pdp"]
vec = np.array([10_000, 25_000, 50_000])
time_dict = {method_name: [] for method_name in method_names}
for N in vec:
    model = return_predict(t)
    model_jac = return_jacobian(t)
    for method_name in method_names:
        time_dict[method_name].append(measure_time(method_name, features))

for metric in ["total"]: # ["fit", "eval", "total"]:
    if metric in ["fit", "eval"]:
        title = "Runtime: ." + metric + "() -- single feature"
    else:
        title = "Runtime: .fit() + .eval() -- single feature"

    bar_plot(
        vec, 
        time_dict, 
        method_names,
        metric=metric,
        title=title,
        xlabel="N: number of instances",
        ylabel="time (sec)"
)

png

Conclusion

Method .fit() .eval() \(T_1\) (single feature) \(T_1\) (all features)
PDP / d-PDP \(c_1 N\) \(c_2 N\) \((c_1 + c_2) N\) \(D (c_1 + c_2) N\)
ALE \(\epsilon\) Free \(\epsilon\) \(D \epsilon \approx 0\)
RHALE \(\epsilon\) Free \(\epsilon\) \(D \epsilon \approx 0\)

Here, \(c_1\) and \(c_2\) are small but nonzero, meaning the runtime scales linearly with \(N\) but remains low. In contrast, \(\epsilon\) is extremely small, making ALE and RHALE effectively free in practice.

\(T_2\): runtime vs. \(t_f\):

To isolate the impact of \(t_f\), we reduce \(N\) to a small value. This assumes that the execution time of \(f(X)\) remains constant regardless of the dataset size \(X\). While this is not always true in general, it is a reasonable assumption for many ML models with vectorized implementations, as long as \(f(X)\) can be computed in a single pass.

For a single feature

t = 0.001
N = 1_000
D = 3
K = 100
M = 100
repetitions = 2
features=[0]

method_names = ["ale", "rhale", "pdp", "d_pdp"]
vec = np.array([.1, .5, 1.])
time_dict = {method_name: [] for method_name in method_names}
for t in vec:
    model = return_predict(t)
    model_jac = return_jacobian(t)
    for method_name in method_names:
        time_dict[method_name].append(measure_time(method_name, features))

for metric in ["total"]: # ["fit", "eval", "total"]:
    if metric in ["fit", "eval"]:
        title = "Runtime: ." + metric + "() -- single feature"
    else:
        title = "Runtime: .fit() + .eval() -- single feature"

    bar_plot(
        vec, 
        time_dict, 
        method_names,
        metric=metric,
        title=title,
        xlabel="time (sec) to execute f(dataset)",
        ylabel="time (sec)"
)

png

For all features

t = 0.1
N = 10_000
D = 3
K = 100
M = 100
repetitions = 2

features=[i for i in range(D)]
method_names = ["ale", "rhale", "pdp", "d_pdp"]
vec = np.array([.1, .5, 1.])
time_dict = {method_name: [] for method_name in method_names}
for t in vec:
    model = return_predict(t)
    model_jac = return_jacobian(t)
    for method_name in method_names:
        time_dict[method_name].append(measure_time(method_name, features))

for metric in ["total"]: #["fit", "eval", "total"]:
    if metric in ["fit", "eval"]:
        title = "Runtime: ." + metric + "() -- all features"
    else:
        title = "Runtime: .fit() + .eval() -- all features"

    bar_plot(
        vec, 
        time_dict, 
        method_names,
        metric=metric,
        title=title,
        xlabel="time (sec) to execute f(dataset)",
        ylabel="time (sec)"
)

png

Conclusion

Method .fit() .eval() \(T_2\) (one feature) \(T_2\) (all features)
PDP / d-PDP \(t_f\) \(t_f\) \(2t_f\) \(2Dt_f\)
ALE \(2t_f\) Free \(2t_f\) \(2Dt_f\)
RHALE \(t_f\) Free \(t_f\) \(t_f\)

Total Runtime

Adding the two parts, we have the total runtime:

Method \(T = T_1 + T_2\) (one feature) \(T = T_1 + T_2\) (all features)
PDP / d-PDP \((c_1 + c_2) N + 2 t_f\) \(D (c_1 + c_2) N + 2 D t_f\)
ALE \(2 t_f\) \(2 D t_f\)
RHALE \(t_f\) \(t_f\)