Multi-fidelity Multi-objective Bayesian Optimization¶

Here we attempt to solve for the constrained Pareto front of the TNK multi-objective optimization problem using Multi-Fidelity Multi-Objective Bayesian optimization. For simplicity we assume that the objective and constraint functions at lower fidelities is exactly equal to the functions at higher fidelities (this is obviously not a requirement, although for the best results lower fidelity calculations should correlate with higher fidelity ones). The algorithm should learn this relationship and use information gathered at lower fidelities to gather samples to improve the hypervolume of the Pareto front at the maximum fidelity.

TNK function $n=2$ variables: $x_i \in [0, \pi], i=1,2$

Objectives:

• $f_i(x) = x_i$

Constraints:

• $g_1(x) = -x_1^2 -x_2^2 + 1 + 0.1 \cos\left(16 \arctan \frac{x_1}{x_2}\right) \le 0$
• $g_2(x) = (x_1 - 1/2)^2 + (x_2-1/2)^2 \le 0.5$

In [1]:

# set values if testing
import os
from copy import deepcopy

import pandas as pd
import numpy as np

from xopt import Xopt, Evaluator
from xopt.generators.bayesian import MultiFidelityGenerator
from xopt.resources.test_functions.tnk import evaluate_TNK, tnk_vocs
from xopt.vocs import get_feasibility_data

import matplotlib.pyplot as plt

# Ignore all warnings
import warnings

warnings.filterwarnings("ignore")

SMOKE_TEST = os.environ.get("SMOKE_TEST")
N_MC_SAMPLES = 1 if SMOKE_TEST else 128
NUM_RESTARTS = 1 if SMOKE_TEST else 20
BUDGET = 0.02 if SMOKE_TEST else 10

evaluator = Evaluator(function=evaluate_TNK)
print(tnk_vocs.dict())

# set values if testing import os from copy import deepcopy import pandas as pd import numpy as np from xopt import Xopt, Evaluator from xopt.generators.bayesian import MultiFidelityGenerator from xopt.resources.test_functions.tnk import evaluate_TNK, tnk_vocs from xopt.vocs import get_feasibility_data import matplotlib.pyplot as plt # Ignore all warnings import warnings warnings.filterwarnings("ignore") SMOKE_TEST = os.environ.get("SMOKE_TEST") N_MC_SAMPLES = 1 if SMOKE_TEST else 128 NUM_RESTARTS = 1 if SMOKE_TEST else 20 BUDGET = 0.02 if SMOKE_TEST else 10 evaluator = Evaluator(function=evaluate_TNK) print(tnk_vocs.dict())

/home/runner/work/Xopt/Xopt/.venv/lib/python3.12/site-packages/torch/jit/_script.py:1491: FutureWarning: `torch.jit.script` is deprecated. Please switch to `torch.compile` or `torch.export`.
  warnings.warn(

{'variables': {'x1': {'dtype': None, 'default_value': None, 'domain': [0.0, 3.14159], 'type': 'ContinuousVariable'}, 'x2': {'dtype': None, 'default_value': None, 'domain': [0.0, 3.14159], 'type': 'ContinuousVariable'}}, 'objectives': {'y1': {'dtype': None, 'type': 'MinimizeObjective'}, 'y2': {'dtype': None, 'type': 'MinimizeObjective'}}, 'constraints': {'c1': {'dtype': None, 'value': 0.0, 'type': 'GreaterThanConstraint'}, 'c2': {'dtype': None, 'value': 0.5, 'type': 'LessThanConstraint'}}, 'constants': {'a': {'dtype': None, 'value': 'dummy_constant', 'type': 'Constant'}}, 'observables': {}}

Set up the Multi-Fidelity Multi-objective optimization algorithm¶

Here we create the Multi-Fidelity generator object which can solve both single and multi-objective optimization problems depending on the number of objectives in VOCS. We specify a cost function as a function of fidelity parameter $s=[0,1]$ as $C(s) = s^{3.5} + 0.001$ as an example from a real life multi-fidelity simulation problem.

In [2]:

my_vocs = deepcopy(tnk_vocs)
generator = MultiFidelityGenerator(vocs=my_vocs, reference_point={"y1": 1.5, "y2": 1.5})

# set cost function according to approximate scaling of laser plasma accelerator
# problem, see https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.013063
generator.cost_function = lambda s: s**3.5 + 0.001
generator.numerical_optimizer.n_restarts = NUM_RESTARTS
generator.n_monte_carlo_samples = N_MC_SAMPLES
generator.gp_constructor.use_low_noise_prior = True

X = Xopt(generator=generator, evaluator=evaluator)

# evaluate at some explicit initial points
X.evaluate_data(pd.DataFrame({"x1": [1.0, 0.75], "x2": [0.75, 1.0], "s": [0.0, 0.1]}))

X

my_vocs = deepcopy(tnk_vocs) generator = MultiFidelityGenerator(vocs=my_vocs, reference_point={"y1": 1.5, "y2": 1.5}) # set cost function according to approximate scaling of laser plasma accelerator # problem, see https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.013063 generator.cost_function = lambda s: s**3.5 + 0.001 generator.numerical_optimizer.n_restarts = NUM_RESTARTS generator.n_monte_carlo_samples = N_MC_SAMPLES generator.gp_constructor.use_low_noise_prior = True X = Xopt(generator=generator, evaluator=evaluator) # evaluate at some explicit initial points X.evaluate_data(pd.DataFrame({"x1": [1.0, 0.75], "x2": [0.75, 1.0], "s": [0.0, 0.1]})) X

Out[2]:

            Xopt
________________________________
Version: 0.1.dev1+gd4c488f75
Data size: 2
Config as YAML:
data_dump_file: null
evaluator:
  function: xopt.resources.test_functions.tnk.evaluate_TNK
  function_kwargs:
    raise_probability: 0
    random_sleep: 0
    sleep: 0
  max_workers: 1
  vectorized: false
generator:
  computation_time: null
  custom_objective: null
  fixed_features: null
  gp_constructor:
    covar_modules: {}
    custom_noise_prior: null
    mean_modules: {}
    name: standard
    train_config: null
    train_kwargs: null
    train_method: lbfgs
    train_model: true
    trainable_mean_keys: []
    transform_inputs: true
    use_cached_hyperparameters: false
    use_low_noise_prior: true
  max_travel_distances: null
  model: null
  n_candidates: 1
  n_interpolate_points: null
  n_monte_carlo_samples: 128
  name: multi_fidelity
  numerical_optimizer:
    discrete_max_batch_size: 2048
    discrete_max_choices: 4096
    max_iter: 1000
    max_time: 5.0
    mixed_max_discrete_configurations: 512
    n_restarts: 20
    name: LBFGS
  reference_point:
    s: 0.0
    y1: 1.5
    y2: 1.5
  returns_id: false
  supports_batch_generation: true
  supports_constraints: true
  supports_contextual_variables: true
  supports_discrete_variables: true
  supports_multi_objective: true
  supports_no_objective: true
  turbo_controller: null
  use_cuda: false
  use_pf_as_initial_points: false
  vocs:
    constants:
      a:
        dtype: null
        type: Constant
        value: dummy_constant
    constraints:
      c1:
        dtype: null
        type: GreaterThanConstraint
        value: 0.0
      c2:
        dtype: null
        type: LessThanConstraint
        value: 0.5
    objectives:
      s:
        dtype: null
        type: MaximizeObjective
      y1:
        dtype: null
        type: MinimizeObjective
      y2:
        dtype: null
        type: MinimizeObjective
    observables: {}
    variables:
      s:
        default_value: null
        domain:
        - 0.0
        - 1.0
        dtype: null
        type: ContinuousVariable
      x1:
        default_value: null
        domain:
        - 0.0
        - 3.14159
        dtype: null
        type: ContinuousVariable
      x2:
        default_value: null
        domain:
        - 0.0
        - 3.14159
        dtype: null
        type: ContinuousVariable
serialize_inline: false
serialize_torch: false
stopping_condition: null
strict: true
xopt_dump_file: null

Run optimization routine¶

Instead of ending the optimization routine after an explict number of samples we end optimization once a given optimization budget has been exceeded. WARNING: This will slightly exceed the given budget

In [3]:

budget = BUDGET
while X.generator.calculate_total_cost() < budget:
    X.step()
    print(
        f"n_samples: {len(X.data)} "
        f"budget used: {X.generator.calculate_total_cost():.4} "
        f"hypervolume: {X.generator.get_pareto_front_and_hypervolume()[-1]:.4}"
    )

budget = BUDGET while X.generator.calculate_total_cost() < budget: X.step() print( f"n_samples: {len(X.data)} " f"budget used: {X.generator.calculate_total_cost():.4} " f"hypervolume: {X.generator.get_pareto_front_and_hypervolume()[-1]:.4}" )

n_samples: 3 budget used: 0.01473 hypervolume: 0.0375

n_samples: 4 budget used: 0.01732 hypervolume: 0.0375

n_samples: 5 budget used: 0.03505 hypervolume: 0.1127

n_samples: 6 budget used: 0.05911 hypervolume: 0.1127

n_samples: 7 budget used: 0.1461 hypervolume: 0.1127

n_samples: 8 budget used: 0.1547 hypervolume: 0.1841

n_samples: 9 budget used: 0.2066 hypervolume: 0.1841

n_samples: 10 budget used: 0.2275 hypervolume: 0.1841

n_samples: 11 budget used: 0.2409 hypervolume: 0.2846

n_samples: 12 budget used: 0.2764 hypervolume: 0.3496

n_samples: 13 budget used: 0.5328 hypervolume: 0.3496

n_samples: 14 budget used: 0.6121 hypervolume: 0.4418

n_samples: 15 budget used: 0.6632 hypervolume: 0.5061

n_samples: 16 budget used: 0.8175 hypervolume: 0.5886

n_samples: 17 budget used: 1.044 hypervolume: 0.5886

n_samples: 18 budget used: 1.398 hypervolume: 0.7353

n_samples: 19 budget used: 1.613 hypervolume: 0.7995

n_samples: 20 budget used: 2.179 hypervolume: 0.7995

n_samples: 21 budget used: 2.316 hypervolume: 0.7995

n_samples: 22 budget used: 3.317 hypervolume: 0.9872

n_samples: 23 budget used: 4.318 hypervolume: 1.104

n_samples: 24 budget used: 4.324 hypervolume: 1.104

n_samples: 25 budget used: 5.325 hypervolume: 1.147

n_samples: 26 budget used: 5.337 hypervolume: 1.147

n_samples: 27 budget used: 6.338 hypervolume: 1.194

n_samples: 28 budget used: 7.339 hypervolume: 1.194

n_samples: 29 budget used: 8.34 hypervolume: 1.194

n_samples: 30 budget used: 8.608 hypervolume: 1.194

n_samples: 31 budget used: 9.609 hypervolume: 1.216

n_samples: 32 budget used: 9.619 hypervolume: 1.22

n_samples: 33 budget used: 9.631 hypervolume: 1.223

n_samples: 34 budget used: 10.06 hypervolume: 1.223

Show results¶

In [4]:

X.data

X.data

Out[4]:

	x1	x2	s	a	y1	y2	c1	c2	xopt_runtime	xopt_error
0	1.000000	0.750000	0.000000	dummy_constant	1.000000	0.750000	0.626888	0.312500	0.005285	False
1	0.750000	1.000000	0.100000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000192	False
2	0.608686	1.342537	0.278628	dummy_constant	0.608686	1.342537	1.086494	0.721681	0.008890	False
3	0.510687	2.549458	0.158620	dummy_constant	0.510687	2.549458	5.860514	4.200392	0.005720	False
4	0.855723	0.963368	0.310762	dummy_constant	0.855723	0.963368	0.601822	0.341249	0.005516	False
5	1.784338	0.680038	0.340600	dummy_constant	1.784338	0.680038	2.556587	1.681938	0.006335	False
6	0.158465	0.038175	0.496135	dummy_constant	0.158465	0.038175	-0.893271	0.329928	0.002780	False
7	0.913674	0.472171	0.247661	dummy_constant	0.913674	0.472171	0.035703	0.171901	0.000152	False
8	0.156535	0.830466	0.427041	dummy_constant	0.156535	0.830466	-0.187112	0.227176	0.003155	False
9	0.088068	1.091102	0.326661	dummy_constant	0.088068	1.091102	0.170416	0.519089	0.005510	False
10	0.102631	1.007161	0.285070	dummy_constant	0.102631	1.007161	0.030305	0.415115	0.001514	False
11	1.019923	0.391467	0.382268	dummy_constant	1.019923	0.391467	0.102160	0.282100	0.000153	False
12	0.896526	0.002851	0.677052	dummy_constant	0.896526	0.002851	-0.296104	0.404390	0.000166	False
13	1.083308	0.165148	0.483084	dummy_constant	1.083308	0.165148	0.275939	0.452374	0.003063	False
14	0.070596	1.042019	0.425032	dummy_constant	0.070596	1.042019	0.043860	0.478171	0.000151	False
15	0.712138	0.854595	0.585217	dummy_constant	0.712138	0.854595	0.225506	0.170740	0.000160	False
16	1.021506	0.046121	0.653112	dummy_constant	1.021506	0.046121	-0.029453	0.477975	0.000161	False
17	1.062554	0.101368	0.742725	dummy_constant	1.062554	0.101368	0.134400	0.475375	0.000164	False
18	0.134089	1.064252	0.644080	dummy_constant	0.134089	1.064252	0.192711	0.452271	0.000144	False
19	0.060891	1.026159	0.849277	dummy_constant	0.060891	1.026159	-0.001596	0.469659	0.000176	False
20	0.820498	0.210605	0.565984	dummy_constant	0.820498	0.210605	-0.218596	0.186469	0.000168	False
21	0.133295	1.076737	1.000000	dummy_constant	0.133295	1.076737	0.216064	0.467098	0.000156	False
22	0.984489	0.351962	1.000000	dummy_constant	0.984489	0.351962	0.022685	0.256645	0.000160	False
23	0.450325	0.737921	0.220102	dummy_constant	0.450325	0.737921	-0.173551	0.059074	0.002794	False
24	0.781528	0.723950	1.000000	dummy_constant	0.781528	0.723950	0.053018	0.129411	0.000160	False
25	0.728125	0.389979	0.275521	dummy_constant	0.728125	0.389979	-0.316403	0.064145	0.000155	False
26	1.047410	0.067919	1.000000	dummy_constant	1.047410	0.067919	0.050719	0.486351	0.002189	False
27	0.019069	1.009053	1.000000	dummy_constant	0.019069	1.009053	-0.076913	0.490429	0.000161	False
28	1.074695	0.099807	1.000000	dummy_constant	1.074695	0.099807	0.156030	0.490429	0.000173	False
29	0.619968	0.555609	0.685429	dummy_constant	0.619968	0.555609	-0.371032	0.017485	0.000182	False
30	0.958902	0.195708	1.000000	dummy_constant	0.958902	0.195708	0.057478	0.303185	0.000161	False
31	0.800463	0.595784	0.266019	dummy_constant	0.800463	0.595784	0.064453	0.099453	0.002392	False
32	0.625300	0.877810	0.270549	dummy_constant	0.625300	0.877810	0.250315	0.158440	0.000157	False
33	0.616588	0.644239	0.782990	dummy_constant	0.616588	0.644239	-0.298683	0.034398	0.000158	False

Plot results¶

Here we plot the resulting observations in input space, colored by feasibility (neglecting the fact that these data points are at varying fidelities).

In [5]:

fig, ax = plt.subplots()

theta = np.linspace(0, np.pi / 2)
r = np.sqrt(1 + 0.1 * np.cos(16 * theta))
x_1 = r * np.sin(theta)
x_2_lower = r * np.cos(theta)
x_2_upper = (0.5 - (x_1 - 0.5) ** 2) ** 0.5 + 0.5

z = np.zeros_like(x_1)

# ax2.plot(x_1, x_2_lower,'r')
ax.fill_between(x_1, z, x_2_lower, fc="white")
circle = plt.Circle(
    (0.5, 0.5), 0.5**0.5, color="r", alpha=0.25, zorder=0, label="Valid Region"
)
ax.add_patch(circle)
history = pd.concat(
    [X.data, get_feasibility_data(tnk_vocs, X.data)], axis=1, ignore_index=False
)

ax.plot(*history[["x1", "x2"]][history["feasible"]].to_numpy().T, ".C1")
ax.plot(*history[["x1", "x2"]][~history["feasible"]].to_numpy().T, ".C2")

ax.set_xlim(0, 3.14)
ax.set_ylim(0, 3.14)
ax.set_xlabel("x1")
ax.set_ylabel("x2")
ax.set_aspect("equal")

fig, ax = plt.subplots() theta = np.linspace(0, np.pi / 2) r = np.sqrt(1 + 0.1 * np.cos(16 * theta)) x_1 = r * np.sin(theta) x_2_lower = r * np.cos(theta) x_2_upper = (0.5 - (x_1 - 0.5) ** 2) ** 0.5 + 0.5 z = np.zeros_like(x_1) # ax2.plot(x_1, x_2_lower,'r') ax.fill_between(x_1, z, x_2_lower, fc="white") circle = plt.Circle( (0.5, 0.5), 0.5**0.5, color="r", alpha=0.25, zorder=0, label="Valid Region" ) ax.add_patch(circle) history = pd.concat( [X.data, get_feasibility_data(tnk_vocs, X.data)], axis=1, ignore_index=False ) ax.plot(*history[["x1", "x2"]][history["feasible"]].to_numpy().T, ".C1") ax.plot(*history[["x1", "x2"]][~history["feasible"]].to_numpy().T, ".C2") ax.set_xlim(0, 3.14) ax.set_ylim(0, 3.14) ax.set_xlabel("x1") ax.set_ylabel("x2") ax.set_aspect("equal")

Plot path through input space¶

In [6]:

ax = history.hist(["x1", "x2", "s"], bins=20)

ax = history.hist(["x1", "x2", "s"], bins=20)

In [7]:

history.plot(y=["x1", "x2", "s"])

history.plot(y=["x1", "x2", "s"])

Out[7]:

<Axes: >

Plot the acquisition function¶

Here we plot the acquisition function at a small set of fidelities $[0, 0.5, 1.0]$.

In [8]:

fidelities = [0.0, 0.5, 1.0]
for fidelity in fidelities:
    X.generator.visualize_model(
        variable_names=["x1", "x2"],
        reference_point={"s": fidelity},
    )

fidelities = [0.0, 0.5, 1.0] for fidelity in fidelities: X.generator.visualize_model( variable_names=["x1", "x2"], reference_point={"s": fidelity}, )

In [9]:

# examine lengthscale of the first objective
list(X.generator.model.models[0].named_parameters())

# examine lengthscale of the first objective list(X.generator.model.models[0].named_parameters())

Out[9]:

[('likelihood.noise_covar.raw_noise',
  Parameter containing:
  tensor([-116.3282], requires_grad=True)),
 ('mean_module.raw_constant',
  Parameter containing:
  tensor(0.3566, requires_grad=True)),
 ('covar_module.raw_lengthscale',
  Parameter containing:
  tensor([[ 0.3126, 18.0883, 39.5529]], requires_grad=True))]