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/pyro/ops/stats.py:527: SyntaxWarning: invalid escape sequence '\g'
  we have :math:`ES^{*}(P,Q) \ge ES^{*}(Q,Q)` with equality holding if and only if :math:`P=Q`, i.e.

{'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+gb834d2348
Data size: 2
Config as YAML:
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:
    max_iter: 2000
    max_time: 5.0
    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_multi_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

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: 0.00332 hypervolume: 0.0375

n_samples: 4 budget used: 0.004845 hypervolume: 0.0375

n_samples: 5 budget used: 0.006602 hypervolume: 0.0375

n_samples: 6 budget used: 0.008348 hypervolume: 0.0375

n_samples: 7 budget used: 0.01114 hypervolume: 0.0375

n_samples: 8 budget used: 0.01496 hypervolume: 0.1091

n_samples: 9 budget used: 0.02475 hypervolume: 0.1633

n_samples: 10 budget used: 0.04521 hypervolume: 0.2166

n_samples: 11 budget used: 0.05336 hypervolume: 0.2714

n_samples: 12 budget used: 0.09079 hypervolume: 0.2714

n_samples: 13 budget used: 0.1307 hypervolume: 0.3366

n_samples: 14 budget used: 0.1936 hypervolume: 0.411

n_samples: 15 budget used: 0.1966 hypervolume: 0.4298

n_samples: 16 budget used: 0.2063 hypervolume: 0.4298

n_samples: 17 budget used: 0.7499 hypervolume: 0.4298

n_samples: 18 budget used: 0.9355 hypervolume: 0.546

n_samples: 19 budget used: 1.366 hypervolume: 0.7072

n_samples: 20 budget used: 2.047 hypervolume: 0.7072

n_samples: 21 budget used: 2.437 hypervolume: 0.7072

n_samples: 22 budget used: 2.459 hypervolume: 0.7553

n_samples: 23 budget used: 2.553 hypervolume: 0.8107

n_samples: 24 budget used: 3.174 hypervolume: 0.9745

n_samples: 25 budget used: 3.456 hypervolume: 0.9745

n_samples: 26 budget used: 4.457 hypervolume: 0.9745

n_samples: 27 budget used: 5.458 hypervolume: 1.084

n_samples: 28 budget used: 6.459 hypervolume: 1.123

n_samples: 29 budget used: 7.46 hypervolume: 1.153

n_samples: 30 budget used: 8.067 hypervolume: 1.153

n_samples: 31 budget used: 8.146 hypervolume: 1.153

n_samples: 32 budget used: 8.218 hypervolume: 1.153

n_samples: 33 budget used: 8.221 hypervolume: 1.153

n_samples: 34 budget used: 8.23 hypervolume: 1.153

n_samples: 35 budget used: 8.936 hypervolume: 1.153

n_samples: 36 budget used: 9.937 hypervolume: 1.23

n_samples: 37 budget used: 10.81 hypervolume: 1.23

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	0.004024	False
1	0.750000	1.000000	0.100000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000152	False
2	0.540887	1.314066	0.028709	dummy_constant	0.540887	1.314066	0.919392	0.664375	0.005181	False
3	0.403417	0.638485	0.115592	dummy_constant	0.403417	0.638485	-0.337830	0.028506	0.000163	False
4	0.000000	0.046606	0.128331	dummy_constant	0.000000	0.046606	-1.097828	0.455566	0.000160	False
5	3.066225	0.000000	0.127773	dummy_constant	3.066225	0.000000	8.301738	6.835512	0.000180	False
6	0.467859	0.847406	0.164242	dummy_constant	0.467859	0.847406	-0.041444	0.121724	0.000185	False
7	0.963973	0.517001	0.186729	dummy_constant	0.963973	0.517001	0.198786	0.215560	0.000190	False
8	1.070426	0.302125	0.258596	dummy_constant	1.070426	0.302125	0.267686	0.364541	0.000179	False
9	1.069543	0.164309	0.324465	dummy_constant	1.069543	0.164309	0.247234	0.437068	0.000182	False
10	0.467728	0.984127	0.243771	dummy_constant	0.467728	0.984127	0.118727	0.235420	0.000157	False
11	0.994552	0.020178	0.388130	dummy_constant	0.994552	0.020178	-0.105238	0.474811	0.000165	False
12	0.733148	0.823639	0.395430	dummy_constant	0.733148	0.823639	0.156021	0.159100	0.000156	False
13	1.041407	0.095960	0.451600	dummy_constant	1.041407	0.095960	0.083691	0.456369	0.000156	False
14	0.208640	1.072741	0.169582	dummy_constant	0.208640	1.072741	0.294073	0.412924	0.000153	False
15	0.023573	0.956861	0.257894	dummy_constant	0.023573	0.956861	-0.176195	0.435705	0.000153	False
16	0.032853	1.453496	0.839733	dummy_constant	0.032853	1.453496	1.020195	1.127380	0.000152	False
17	0.833777	0.645643	0.617143	dummy_constant	0.833777	0.645643	0.155806	0.132619	0.000152	False
18	1.013860	0.118878	0.785202	dummy_constant	1.013860	0.118878	0.071282	0.409306	0.000152	False
19	0.892034	0.005274	0.895870	dummy_constant	0.892034	0.005274	-0.303801	0.398444	0.000153	False
20	0.279911	0.610472	0.763585	dummy_constant	0.279911	0.610472	-0.631764	0.060643	0.000154	False
21	0.088601	1.049080	0.330691	dummy_constant	0.088601	1.049080	0.086333	0.470738	0.000152	False
22	0.098612	1.049903	0.506997	dummy_constant	0.098612	1.049903	0.104787	0.463506	0.000153	False
23	0.099518	1.044628	0.872458	dummy_constant	0.099518	1.044628	0.096043	0.457006	0.000153	False
24	0.037846	0.846545	0.695924	dummy_constant	0.037846	0.846545	-0.357450	0.333679	0.000153	False
25	0.929377	0.382861	1.000000	dummy_constant	0.929377	0.382861	-0.089628	0.198086	0.000153	False
26	0.659047	0.842830	1.000000	dummy_constant	0.659047	0.842830	0.181557	0.142828	0.000154	False
27	0.286208	0.991652	1.000000	dummy_constant	0.286208	0.991652	0.086788	0.287429	0.000157	False
28	0.058236	1.035700	1.000000	dummy_constant	0.058236	1.035700	0.013803	0.482130	0.000158	False
29	0.502402	0.438840	0.866582	dummy_constant	0.502402	0.438840	-0.602247	0.003746	0.000152	False
30	0.499333	0.456031	0.482440	dummy_constant	0.499333	0.456031	-0.617571	0.001934	0.000154	False
31	1.694445	0.484055	0.469401	dummy_constant	1.694445	0.484055	2.131183	1.426953	0.000153	False
32	0.815753	0.305133	0.168214	dummy_constant	0.815753	0.305133	-0.326370	0.137673	0.000153	False
33	0.034007	1.018247	0.254252	dummy_constant	0.034007	1.018247	-0.048085	0.485729	0.000152	False
34	0.076509	1.406007	0.904960	dummy_constant	0.076509	1.406007	0.918211	1.000193	0.000152	False
35	1.062487	0.088607	1.000000	dummy_constant	1.062487	0.088607	0.113004	0.485636	0.000155	False
36	0.379703	1.219809	0.962145	dummy_constant	0.379703	1.219809	0.620536	0.532597	0.000155	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([-74.0031], requires_grad=True)),
 ('mean_module.raw_constant',
  Parameter containing:
  tensor(1.1303, requires_grad=True)),
 ('covar_module.raw_lengthscale',
  Parameter containing:
  tensor([[ 0.4621, 15.8831, 36.9488]], requires_grad=True))]