Implemented algorithms

The core concept in adaptive is that of a learner. A learner samples a function at the best places in its parameter space to get maximum “information” about the function. As it evaluates the function at more and more points in the parameter space, it gets a better idea of where the best places are to sample next.

Of course, what qualifies as the “best places” will depend on your application domain! adaptive makes some reasonable default choices, but the details of the adaptive sampling are completely customizable.

The following learners are implemented:

• Learner1D, for 1D functions f: ℝ → ℝ^N,

• Learner2D, for 2D functions f: ℝ^2 → ℝ^N,

• LearnerND, for ND functions f: ℝ^N → ℝ^M,

• AverageLearner, for random variables where you want to average the result over many evaluations,

• AverageLearner1D, for stochastic 1D functions where you want to estimate the mean value of the function at each point,

• IntegratorLearner, for when you want to intergrate a 1D function f: ℝ → ℝ.

• BalancingLearner, for when you want to run several learners at once, selecting the “best” one each time you get more points.

Meta-learners (to be used with other learners):

• BalancingLearner, for when you want to run several learners at once, selecting the “best” one each time you get more points,

• DataSaver, for when your function doesn’t just return a scalar or a vector.

In addition to the learners, adaptive also provides primitives for running the sampling across several cores and even several machines, with built-in support for concurrent.futures, mpi4py, loky, ipyparallel and distributed.

Examples

Here are some examples of how Adaptive samples vs. homogeneous sampling. Click on the Play button or move the sliders.

import itertools
import adaptive
from adaptive.learner.learner1D import uniform_loss, default_loss
import holoviews as hv
import numpy as np

adaptive.notebook_extension()
hv.output(holomap="scrubber")

adaptive.Learner1D

Adaptively learning a 1D function (the plot below) and live-plotting the process in a Jupyter notebook is as easy as

from adaptive import notebook_extension, Runner, Learner1D
notebook_extension()  # enables notebook integration

def peak(x, a=0.01):  # function to "learn"
    return x + a**2 / (a**2 + x**2)

learner = Learner1D(peak, bounds=(-1, 1))

def goal(learner):
    return learner.loss() < 0.01  # continue until loss is small enough

runner = Runner(learner, goal)  # start calculation on all CPU cores
runner.live_info()  # shows a widget with status information
runner.live_plot()

def f(x, offset=0.07357338543088588):
    a = 0.01
    return x + a**2 / (a**2 + (x - offset)**2)

def plot_loss_interval(learner):
    if learner.npoints >= 2:
        x_0, x_1 = max(learner.losses, key=learner.losses.get)
        y_0, y_1 = learner.data[x_0], learner.data[x_1]
        x, y = [x_0, x_1], [y_0, y_1]
    else:
        x, y = [], []
    return hv.Scatter((x, y)).opts(style=dict(size=6, color="r"))

def plot(learner, npoints):
    adaptive.runner.simple(learner, lambda l: l.npoints == npoints)
    return (learner.plot() * plot_loss_interval(learner))[:, -1.1:1.1]

def get_hm(loss_per_interval, N=101):
    learner = adaptive.Learner1D(f, bounds=(-1, 1), loss_per_interval=loss_per_interval)
    plots = {n: plot(learner, n) for n in range(N)}
    return hv.HoloMap(plots, kdims=["npoints"])

layout = (
    get_hm(uniform_loss).relabel("homogeneous samping")
    + get_hm(default_loss).relabel("with adaptive")
)

layout.opts(plot=dict(toolbar=None))

adaptive.Learner2D

def ring(xy):
    import numpy as np
    x, y = xy
    a = 0.2
    return x + np.exp(-(x**2 + y**2 - 0.75**2)**2/a**4)

def plot(learner, npoints):
    adaptive.runner.simple(learner, lambda l: l.npoints == npoints)
    learner2 = adaptive.Learner2D(ring, bounds=learner.bounds)
    xs = ys = np.linspace(*learner.bounds[0], int(learner.npoints**0.5))
    xys = list(itertools.product(xs, ys))
    learner2.tell_many(xys, map(ring, xys))
    return (learner2.plot().relabel('homogeneous grid')
            + learner.plot().relabel('with adaptive')
            + learner2.plot(tri_alpha=0.5).relabel('homogeneous sampling')
            + learner.plot(tri_alpha=0.5).relabel('with adaptive')).cols(2)

learner = adaptive.Learner2D(ring, bounds=[(-1, 1), (-1, 1)])
plots = {n: plot(learner, n) for n in range(4, 1010, 20)}
hv.HoloMap(plots, kdims=['npoints']).collate()

adaptive.AverageLearner

def g(n):
    import random
    random.seed(n)
    val = random.gauss(0.5, 0.5)
    return val

learner = adaptive.AverageLearner(g, atol=None, rtol=0.01)

def plot(learner, npoints):
    adaptive.runner.simple(learner, lambda l: l.npoints == npoints)
    return learner.plot().relabel(f'loss={learner.loss():.2f}')

plots = {n: plot(learner, n) for n in range(10, 10000, 200)}
hv.HoloMap(plots, kdims=['npoints'])

adaptive.LearnerND

def sphere(xyz):
    import numpy as np
    x, y, z = xyz
    a = 0.4
    return np.exp(-(x**2 + y**2 + z**2 - 0.75**2)**2/a**4)

learner = adaptive.LearnerND(sphere, bounds=[(-1, 1), (-1, 1), (-1, 1)])
adaptive.runner.simple(learner, lambda l: l.npoints == 5000)

fig = learner.plot_3D(return_fig=True)

# Remove a slice from the plot to show the inside of the sphere
scatter = fig.data[0]
coords_col = [
    (x, y, z, color)
    for x, y, z, color in zip(
        scatter["x"], scatter["y"], scatter["z"], scatter.marker["color"]
    )
    if not (x > 0 and y > 0)
]
scatter["x"], scatter["y"], scatter["z"], scatter.marker["color"] = zip(*coords_col)

fig

see more in the Tutorial Adaptive.

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