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# Comparing initial point generation methods

Holger Nahrstaedt 2020

Bayesian optimization or sequential model-based optimization uses a surrogate model to model the expensive to evaluate function `func`. There are several choices for what kind of surrogate model to use. This notebook compares the performance of:

- Halton sequence,

- Hammersly sequence,

- Sobol’ sequence and

- Latin hypercube sampling

as initial points. The purely random point generation is used as a baseline.

    print(__doc__)
    import numpy as np

    np.random.seed(123)
    import matplotlib.pyplot as plt

    from skopt.benchmarks import hart6 as hart6_

## Toy model

We will use the `benchmarks.hart6` function as toy model for the expensive function. In a real world application this function would be unknown and expensive to evaluate.

    # redefined `hart6` to allow adding arbitrary "noise" dimensions
    def hart6(x, noise_level=0.0):
        return hart6_(x[:6]) + noise_level * np.random.randn()

    from skopt.benchmarks import branin as _branin

    def branin(x, noise_level=0.0):
        return _branin(x) + noise_level * np.random.randn()

    import time

    from matplotlib.pyplot import cm

    from skopt import gp_minimize

    def plot_convergence(
        result_list, true_minimum=None, yscale=None, title="Convergence plot"
    ):

        ax = plt.gca()
        ax.set_title(title)
        ax.set_xlabel("Number of calls $n$")
        ax.set_ylabel(r"$\min f(x)$ after $n$ calls")
        ax.grid()
        if yscale is not None:
            ax.set_yscale(yscale)
        colors = cm.hsv(np.linspace(0.25, 1.0, len(result_list)))

        for results, color in zip(result_list, colors):
            name, results = results
            n_calls = len(results[0].x_iters)
            iterations = range(1, n_calls + 1)
            mins = [[np.min(r.func_vals[:i]) for i in iterations] for r in results]
            ax.plot(iterations, np.mean(mins, axis=0), c=color, label=name)
            # ax.errorbar(iterations, np.mean(mins, axis=0),
            #             yerr=np.std(mins, axis=0), c=color, label=name)
        if true_minimum:
            ax.axhline(true_minimum, linestyle="--", color="r", lw=1, label="True minimum")
        ax.legend(loc="best")
        return ax

    def run(minimizer, initial_point_generator, n_initial_points=10, n_repeats=1):
        return [
            minimizer(
                func,
                bounds,
                n_initial_points=n_initial_points,
                initial_point_generator=initial_point_generator,
                n_calls=n_calls,
                random_state=n,
            )
            for n in range(n_repeats)
        ]

    def run_measure(initial_point_generator, n_initial_points=10):
        start = time.time()
        # n_repeats must set to a much higher value to obtain meaningful results.
        n_repeats = 1
        res = run(
            gp_minimize,
            initial_point_generator,
            n_initial_points=n_initial_points,
            n_repeats=n_repeats,
        )
        duration = time.time() - start
        print("%s: %.2f s" % (initial_point_generator, duration))
        return res

## Objective

The objective of this example is to find one of these minima in as few iterations as possible. One iteration is defined as one call to the `benchmarks.hart6` function.

We will evaluate each model several times using a different seed for the random number generator. Then compare the average performance of these models. This makes the comparison more robust against models that get “lucky”.

    from functools import partial

    example = "hart6"

    if example == "hart6":
        func = partial(hart6, noise_level=0.1)
        bounds = [
            (0.0, 1.0),
        ] * 6
        true_minimum = -3.32237
        n_calls = 30
        n_initial_points = 10
        yscale = None
        title = "Convergence plot - hart6"
    else:
        func = partial(branin, noise_level=2.0)
        bounds = [(-5.0, 10.0), (0.0, 15.0)]
        true_minimum = 0.397887
        n_calls = 30
        n_initial_points = 10
        yscale = "log"
        title = "Convergence plot - branin"

    from skopt.utils import cook_initial_point_generator

    # Random search
    dummy_res = run_measure("random", n_initial_points)
    lhs = cook_initial_point_generator("lhs", lhs_type="classic", criterion=None)
    lhs_res = run_measure(lhs, n_initial_points)
    lhs2 = cook_initial_point_generator("lhs", criterion="maximin")
    lhs2_res = run_measure(lhs2, n_initial_points)
    sobol = cook_initial_point_generator("sobol", randomize=False, min_skip=1, max_skip=100)
    sobol_res = run_measure(sobol, n_initial_points)
    halton_res = run_measure("halton", n_initial_points)
    hammersly_res = run_measure("hammersly", n_initial_points)
    grid_res = run_measure("grid", n_initial_points)

    random: 11.50 s
    <skopt.sampler.lhs.Lhs object at 0x0000020BCFA298D0>: 10.31 s
    <skopt.sampler.lhs.Lhs object at 0x0000020BCF824250>: 10.83 s
    D:\git\scikit-optimize\skopt\sampler\sobol.py:521: UserWarning: The balance properties of Sobol' points require n to be a power of 2. 0 points have been previously generated, then: n=0+10=10.
      warnings.warn(
    <skopt.sampler.sobol.Sobol object at 0x0000020BCFAF0210>: 8.19 s
    halton: 5.61 s
    hammersly: 6.14 s
    grid: 11.66 s

Note that this can take a few minutes.

    plot = plot_convergence(
        [
            ("random", dummy_res),
            ("lhs", lhs_res),
            ("lhs_maximin", lhs2_res),
            ("sobol'", sobol_res),
            ("halton", halton_res),
            ("hammersly", hammersly_res),
            ("grid", grid_res),
        ],
        true_minimum=true_minimum,
        yscale=yscale,
        title=title,
    )

    plt.show()

This plot shows the value of the minimum found (y axis) as a function of the number of iterations performed so far (x axis). The dashed red line indicates the true value of the minimum of the `benchmarks.hart6` function.

Test with different n_random_starts values

    lhs2 = cook_initial_point_generator("lhs", criterion="maximin")
    lhs2_15_res = run_measure(lhs2, 12)
    lhs2_20_res = run_measure(lhs2, 14)
    lhs2_25_res = run_measure(lhs2, 16)

    <skopt.sampler.lhs.Lhs object at 0x0000020BD8E162D0>: 5.83 s
    <skopt.sampler.lhs.Lhs object at 0x0000020BD8E162D0>: 5.66 s
    <skopt.sampler.lhs.Lhs object at 0x0000020BD8E162D0>: 4.54 s

n_random_starts = 10 produces the best results

    plot = plot_convergence(
        [
            ("random - 10", dummy_res),
            ("lhs_maximin - 10", lhs2_res),
            ("lhs_maximin - 12", lhs2_15_res),
            ("lhs_maximin - 14", lhs2_20_res),
            ("lhs_maximin - 16", lhs2_25_res),
        ],
        true_minimum=true_minimum,
        yscale=yscale,
        title=title,
    )

    plt.show()

Total running time of the script: (1 minutes 20.489 seconds)

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