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())

{'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+g46cc86a6b
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:
    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

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: 0.003316 hypervolume: 0.03845

n_samples: 4 budget used: 0.007356 hypervolume: 0.03845

n_samples: 5 budget used: 0.01248 hypervolume: 0.03845

n_samples: 6 budget used: 0.01449 hypervolume: 0.03845

n_samples: 7 budget used: 0.04332 hypervolume: 0.03845

n_samples: 8 budget used: 0.05639 hypervolume: 0.1812

n_samples: 9 budget used: 0.1225 hypervolume: 0.2867

n_samples: 10 budget used: 0.1387 hypervolume: 0.2867

n_samples: 11 budget used: 0.1458 hypervolume: 0.2867

n_samples: 12 budget used: 0.1512 hypervolume: 0.3203

n_samples: 13 budget used: 0.1544 hypervolume: 0.3203

n_samples: 14 budget used: 0.1616 hypervolume: 0.3518

n_samples: 15 budget used: 0.1913 hypervolume: 0.4084

n_samples: 16 budget used: 0.3053 hypervolume: 0.4084

n_samples: 17 budget used: 0.4641 hypervolume: 0.5177

n_samples: 18 budget used: 0.792 hypervolume: 0.6305

n_samples: 19 budget used: 0.8108 hypervolume: 0.672

n_samples: 20 budget used: 0.9075 hypervolume: 0.7358

n_samples: 21 budget used: 1.093 hypervolume: 0.7358

n_samples: 22 budget used: 1.501 hypervolume: 0.8741

n_samples: 23 budget used: 2.196 hypervolume: 0.9903

n_samples: 24 budget used: 3.197 hypervolume: 1.114

n_samples: 25 budget used: 4.198 hypervolume: 1.166

n_samples: 26 budget used: 5.172 hypervolume: 1.166

n_samples: 27 budget used: 5.194 hypervolume: 1.166

n_samples: 28 budget used: 5.598 hypervolume: 1.166

n_samples: 29 budget used: 6.599 hypervolume: 1.166

n_samples: 30 budget used: 6.694 hypervolume: 1.166

n_samples: 31 budget used: 6.847 hypervolume: 1.166

n_samples: 32 budget used: 7.848 hypervolume: 1.256

n_samples: 33 budget used: 8.849 hypervolume: 1.256

n_samples: 34 budget used: 8.878 hypervolume: 1.256

n_samples: 35 budget used: 8.906 hypervolume: 1.256

n_samples: 36 budget used: 9.167 hypervolume: 1.256

n_samples: 37 budget used: 9.443 hypervolume: 1.256

n_samples: 38 budget used: 10.44 hypervolume: 1.267

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	0.005846	False
1	0.750000	1.000000	0.100000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.002394	False
2	0.534012	1.097060	0.010897	dummy_constant	0.534012	1.097060	0.431759	0.357638	0.005667	False
3	0.396694	0.583423	0.190902	dummy_constant	0.396694	0.583423	-0.403088	0.017632	0.008982	False
4	0.486198	2.270887	0.208335	dummy_constant	0.486198	2.270887	4.490611	3.136230	0.008348	False
5	0.654334	0.253531	0.139211	dummy_constant	0.654334	0.253531	-0.600847	0.084566	0.002434	False
6	0.169836	0.672649	0.359412	dummy_constant	0.169836	0.672649	-0.450150	0.138816	0.005766	False
7	0.596597	0.792467	0.283053	dummy_constant	0.596597	0.792467	0.046076	0.094868	0.002807	False
8	0.780085	0.742889	0.458165	dummy_constant	0.780085	0.742889	0.067951	0.137442	0.002234	False
9	0.035670	0.006462	0.302303	dummy_constant	0.035670	0.006462	-0.902421	0.459182	0.007374	False
10	2.610698	0.000000	0.233439	dummy_constant	2.610698	0.000000	5.715745	4.705047	0.002443	False
11	1.066680	0.377374	0.211615	dummy_constant	1.066680	0.377374	0.213659	0.336163	0.005067	False
12	0.000000	0.909100	0.174803	dummy_constant	0.000000	0.909100	-0.273537	0.417363	0.002992	False
13	0.271627	1.080229	0.233408	dummy_constant	0.271627	1.080229	0.310351	0.388820	0.000163	False
14	0.105696	1.041231	0.362772	dummy_constant	0.105696	1.041231	0.100114	0.448406	0.000178	False
15	0.045462	0.998748	0.536363	dummy_constant	0.045462	0.998748	-0.075100	0.455355	0.000159	False
16	0.129001	1.049812	0.590025	dummy_constant	0.129001	1.049812	0.156345	0.439934	0.000132	False
17	0.066898	1.029040	0.726516	dummy_constant	0.066898	1.029040	0.012666	0.467460	0.000167	False
18	1.101195	0.169803	0.316541	dummy_constant	1.101195	0.169803	0.318353	0.470466	0.000169	False
19	1.043747	0.188234	0.511497	dummy_constant	1.043747	0.188234	0.220756	0.392858	0.000178	False
20	1.003014	0.025076	0.617299	dummy_constant	1.003014	0.025076	-0.085444	0.478576	0.000155	False
21	1.047550	0.099088	0.773120	dummy_constant	1.047550	0.099088	0.100998	0.460541	0.002476	False
22	0.826608	0.640181	0.901179	dummy_constant	0.826608	0.640181	0.136785	0.126324	0.000139	False
23	0.086514	1.041212	1.000000	dummy_constant	0.086514	1.041212	0.067408	0.463881	0.000161	False
24	0.531085	0.852557	1.000000	dummy_constant	0.531085	0.852557	0.096130	0.125263	0.000160	False
25	0.893647	1.260931	0.992021	dummy_constant	0.893647	1.260931	1.479024	0.733974	0.000157	False
26	0.619758	0.645187	0.332073	dummy_constant	0.619758	0.645187	-0.294506	0.035421	0.000163	False
27	0.845821	0.076294	0.771597	dummy_constant	0.845821	0.076294	-0.291875	0.299119	0.000180	False
28	0.291013	0.954165	1.000000	dummy_constant	0.291013	0.954165	-0.007291	0.249941	0.000164	False
29	0.031432	0.933980	0.508126	dummy_constant	0.031432	0.933980	-0.212554	0.407894	0.000155	False
30	1.109761	0.017497	0.583931	dummy_constant	1.109761	0.017497	0.135041	0.604618	0.000152	False
31	1.035281	0.058399	1.000000	dummy_constant	1.035281	0.058399	0.013180	0.481538	0.000168	False
32	0.440970	0.919190	1.000000	dummy_constant	0.440970	0.919190	-0.024833	0.179205	0.000235	False
33	0.815236	0.154802	0.360436	dummy_constant	0.815236	0.154802	-0.212393	0.218535	0.000175	False
34	0.815378	0.369851	0.355056	dummy_constant	0.815378	0.369851	-0.284638	0.116402	0.000172	False
35	0.423354	1.243859	0.680955	dummy_constant	0.423354	1.243859	0.675299	0.559201	0.000163	False
36	0.315629	1.413824	0.691445	dummy_constant	0.315629	1.413824	1.191656	0.869068	0.000162	False
37	0.324810	0.980226	1.000000	dummy_constant	0.324810	0.980226	0.026739	0.261309	0.000163	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([-96.1147], requires_grad=True)),
 ('mean_module.raw_constant',
  Parameter containing:
  tensor(0.9341, requires_grad=True)),
 ('covar_module.raw_lengthscale',
  Parameter containing:
  tensor([[ 0.3680, 20.6058, 39.9552]], requires_grad=True))]