In this tutorial, we illustrate how to implement a constrained multi-objective (MO) Bayesian Optimization (BO) closed loop in BoTorch.
We use the parallel ParEGO ($q$ParEGO) [1] and parallel Noisy Expected Hypervolume Improvement ($q$NEHVI) [2] acquisition functions to optimize a synthetic C2-DTLZ2 test function with $M=2$ objectives, $V=1$ constraint, and $d=4$ parameters. The two objectives are $$f_1(\mathbf x) = (1+ g(\mathbf x_M))\cos\big(\frac{\pi}{2}x_1\big)$$ $$f_2(\mathbf x) = (1+ g(\mathbf x_M))\sin\big(\frac{\pi}{2}x_1\big)$$ where $g(\mathbf x) = \sum_{x_i \in \mathbf x_M} (x_i - 0.5)^2, \mathbf x \in [0,1]^d,$ and $\mathbf x_M$ represents the last $d - M +1$ elements of $\mathbf x$. Additionally, the C2-DTLZ2 problem uses the following constraint:
$$c(\mathbf x) = - \min \bigg[\min_{i=1}^M\bigg((f_i(\mathbf x) -1 )^2 + \sum_{j=1, j=i}^M (f_j^2 - r^2) \bigg), \bigg(\sum_{i=1}^M \big((f_i(\mathbf x) - \frac{1}{\sqrt{M}})^2 - r^2\big)\bigg)\bigg]\geq 0$$where $\mathbf x \in [0,1]^d$ and $r=0.2$.
The goal here is to minimize both objectives. Since BoTorch assumes maximization, we maximize the negative of each objective. Since there typically is no single best solution in multi-objective optimization problems, we seek to find the pareto frontier, the set of optimal trade-offs where improving one metric means deteriorating another.
For batch optimization (or in noisy settings), we strongly recommend using $q$NEHVI rather than $q$EHVI [1] because it is far more efficient than $q$EHVI and mathematically equivalent in the noiseless setting.
Note: $q$EHVI aggressively exploits parallel hardware and is much faster when run on a GPU. See [1] for details.
import os
import torch
tkwargs = {
"dtype": torch.double,
"device": torch.device("cuda" if torch.cuda.is_available() else "cpu"),
}
SMOKE_TEST = os.environ.get("SMOKE_TEST")
from botorch.test_functions.multi_objective import C2DTLZ2
d = 4
M = 2
problem = C2DTLZ2(dim=d, num_objectives=M, negate=True).to(**tkwargs)
We use a multi-output SingleTaskGP to model the two objectives with a homoskedastic Gaussian likelihood with an inferred noise level.
The models are initialized with $2(d+1)=10$ points drawn randomly from $[0,1]^{4}$.
from botorch.models.gp_regression import SingleTaskGP
from botorch.models.model_list_gp_regression import ModelListGP
from botorch.models.transforms.outcome import Standardize
from botorch.utils.sampling import draw_sobol_samples
from botorch.utils.transforms import normalize, unnormalize
from gpytorch.mlls.sum_marginal_log_likelihood import SumMarginalLogLikelihood
def generate_initial_data(n):
# generate training data
train_x = draw_sobol_samples(bounds=problem.bounds, n=n, q=1).squeeze(1)
train_obj = problem(train_x)
# negative values imply feasibility in botorch
train_con = -problem.evaluate_slack(train_x)
return train_x, train_obj, train_con
def initialize_model(train_x, train_obj, train_con):
# define models for objective and constraint
train_x = normalize(train_x, problem.bounds)
train_y = torch.cat([train_obj, train_con], dim=-1)
models = []
for i in range(train_y.shape[-1]):
models.append(SingleTaskGP(train_x, train_y[..., i : i + 1], outcome_transform=Standardize(m=1)))
model = ModelListGP(*models)
mll = SumMarginalLogLikelihood(model.likelihood, model)
return mll, model
The helper function below initializes the $q$NEHVI acquisition function, optimizes it, and returns the batch $\{x_1, x_2, \ldots x_q\}$ along with the observed function values.
For this example, we'll use a small batch of $q=2$. Passing the keyword argument sequential=True to the function optimize_acqfspecifies that candidates should be optimized in a sequential greedy fashion (see [1] for details why this is important). A simple initialization heuristic is used to select the 10 restart initial locations from a set of 512 random points. Multi-start optimization of the acquisition function is performed using LBFGS-B with exact gradients computed via auto-differentiation.
Reference Point
$q$NEHVI requires specifying a reference point, which is the lower bound on the objectives used for computing hypervolume. In this tutorial, we assume the reference point is known. In practice the reference point can be set 1) using domain knowledge to be slightly worse than the lower bound of objective values, where the lower bound is the minimum acceptable value of interest for each objective, or 2) using a dynamic reference point selection strategy.
Integrating over function values at in-sample designs
$q$NEHVI integrates over the unknown function values at the previously evaluated designs (see [2] for details). Therefore, we need to provide the previously evaluated designs (train_x, normalized to be within $[0,1]^d$) to the acquisition function.
Pruning baseline designs
To speed up integration over the function values at the previously evaluated designs, we prune the set of previously evaluated designs (by setting prune_baseline=True) to only include those which have positive probability of being on the current in-sample Pareto frontier.
from botorch.acquisition.multi_objective.monte_carlo import qNoisyExpectedHypervolumeImprovement
from botorch.acquisition.multi_objective.objective import IdentityMCMultiOutputObjective
from botorch.optim.optimize import optimize_acqf, optimize_acqf_list
from botorch.utils.multi_objective.scalarization import get_chebyshev_scalarization
from botorch.utils.sampling import sample_simplex
BATCH_SIZE = 2
NUM_RESTARTS = 10 if not SMOKE_TEST else 2
RAW_SAMPLES = 512 if not SMOKE_TEST else 4
standard_bounds = torch.zeros(2, problem.dim, **tkwargs)
standard_bounds[1] = 1
def optimize_qnehvi_and_get_observation(model, train_x, train_obj, train_con, sampler):
"""Optimizes the qNEHVI acquisition function, and returns a new candidate and observation."""
train_x = normalize(train_x, problem.bounds)
acq_func = qNoisyExpectedHypervolumeImprovement(
model=model,
ref_point=problem.ref_point.tolist(), # use known reference point
X_baseline=train_x,
sampler=sampler,
prune_baseline=True,
# define an objective that specifies which outcomes are the objectives
objective=IdentityMCMultiOutputObjective(outcomes=[0, 1]),
# specify that the constraint is on the last outcome
constraints=[lambda Z: Z[..., -1]],
)
# optimize
candidates, _ = optimize_acqf(
acq_function=acq_func,
bounds=standard_bounds,
q=BATCH_SIZE,
num_restarts=NUM_RESTARTS,
raw_samples=RAW_SAMPLES, # used for intialization heuristic
options={"batch_limit": 5, "maxiter": 200},
sequential=True,
)
# observe new values
new_x = unnormalize(candidates.detach(), bounds=problem.bounds)
new_obj = problem(new_x)
# negative values imply feasibility in botorch
new_con = -problem.evaluate_slack(new_x)
return new_x, new_obj, new_con
The helper function below initializes a ConstrainedMCObjective for $q$ParEGO. It creates the objective which fetches the outcomes required for the scalarized objective and applies the scalarization and the constraints, which are modeled outcomes.
from botorch.acquisition.objective import ConstrainedMCObjective
def get_constrained_mc_objective(train_obj, train_con, scalarization):
"""Initialize a ConstrainedMCObjective for qParEGO"""
n_obj = train_obj.shape[-1]
# assume first outcomes of the model are the objectives, the rest constraints
def objective(Z):
return scalarization(Z[..., :n_obj])
constrained_obj = ConstrainedMCObjective(
objective=objective,
constraints=[lambda Z: Z[..., -1]], # index the constraint
)
return constrained_obj
The helper function below similarly initializes $q$ParEGO, optimizes it, and returns the batch $\{x_1, x_2, \ldots x_q\}$ along with the observed function values.
$q$ParEGO uses random augmented chebyshev scalarization with the qExpectedImprovement acquisition function. In the parallel setting ($q>1$), each candidate is optimized in sequential greedy fashion using a different random scalarization (see [1] for details).
To do this, we create a list of qExpectedImprovement acquisition functions, each with different random scalarization weights. The optimize_acqf_list method sequentially generates one candidate per acquisition function and conditions the next candidate (and acquisition function) on the previously selected pending candidates.
from botorch.acquisition.monte_carlo import qExpectedImprovement
def optimize_qparego_and_get_observation(model, train_obj, train_con, sampler):
"""Samples a set of random weights for each candidate in the batch, performs sequential greedy optimization
of the qParEGO acquisition function, and returns a new candidate and observation."""
acq_func_list = []
for _ in range(BATCH_SIZE):
# sample random weights
weights = sample_simplex(problem.num_objectives, **tkwargs).squeeze()
# construct augmented Chebyshev scalarization
scalarization = get_chebyshev_scalarization(weights=weights, Y=train_obj)
# initialize ConstrainedMCObjective
constrained_objective = get_constrained_mc_objective(
train_obj=train_obj,
train_con=train_con,
scalarization=scalarization,
)
train_y = torch.cat([train_obj, train_con], dim=-1)
acq_func = qExpectedImprovement( # pyre-ignore: [28]
model=model,
objective=constrained_objective,
best_f=constrained_objective(train_y).max(),
sampler=sampler,
)
acq_func_list.append(acq_func)
# optimize
candidates, _ = optimize_acqf_list(
acq_function_list=acq_func_list,
bounds=standard_bounds,
num_restarts=NUM_RESTARTS,
raw_samples=RAW_SAMPLES, # used for intialization heuristic
options={"batch_limit": 5, "maxiter": 200},
)
# observe new values
new_x = unnormalize(candidates.detach(), bounds=problem.bounds)
new_obj = problem(new_x)
# negative values imply feasibility in botorch
new_con = -problem.evaluate_slack(new_x)
return new_x, new_obj, new_con
The Bayesian optimization "loop" for a batch size of $q$ simply iterates the following steps:
Just for illustration purposes, we run three trials each of which do N_BATCH=20 rounds of optimization. The acquisition function is approximated using MC_SAMPLES=128 samples.
Note: Running this may take a little while.
import time
import warnings
from botorch import fit_gpytorch_model
from botorch.exceptions import BadInitialCandidatesWarning
from botorch.sampling.samplers import SobolQMCNormalSampler
from botorch.utils.multi_objective.hypervolume import Hypervolume
from botorch.utils.multi_objective.pareto import is_non_dominated
warnings.filterwarnings("ignore", category=BadInitialCandidatesWarning)
warnings.filterwarnings("ignore", category=RuntimeWarning)
N_BATCH = 20 if not SMOKE_TEST else 5
MC_SAMPLES = 128 if not SMOKE_TEST else 16
verbose = True
hv = Hypervolume(ref_point=problem.ref_point)
hvs_qparego, hvs_qnehvi, hvs_random = [], [], []
# call helper functions to generate initial training data and initialize model
train_x_qparego, train_obj_qparego, train_con_qparego = generate_initial_data(n=2 * (d + 1))
mll_qparego, model_qparego = initialize_model(train_x_qparego, train_obj_qparego, train_con_qparego)
train_x_qnehvi, train_obj_qnehvi, train_con_qnehvi = train_x_qparego, train_obj_qparego, train_con_qparego
train_x_random, train_obj_random, train_con_random = train_x_qparego, train_obj_qparego, train_con_qparego
mll_qnehvi, model_qnehvi = initialize_model(train_x_qnehvi, train_obj_qnehvi, train_con_qnehvi)
# compute pareto front
is_feas = (train_con_qparego <= 0).all(dim=-1)
feas_train_obj = train_obj_qparego[is_feas]
if feas_train_obj.shape[0] > 0:
pareto_mask = is_non_dominated(feas_train_obj)
pareto_y = feas_train_obj[pareto_mask]
# compute hypervolume
volume = hv.compute(pareto_y)
else:
volume = 0.0
hvs_qparego.append(volume)
hvs_qnehvi.append(volume)
hvs_random.append(volume)
# run N_BATCH rounds of BayesOpt after the initial random batch
for iteration in range(1, N_BATCH + 1):
t0 = time.time()
# fit the models
fit_gpytorch_model(mll_qparego)
fit_gpytorch_model(mll_qnehvi)
# define the qParEGO and qNEHVI acquisition modules using a QMC sampler
qparego_sampler = SobolQMCNormalSampler(num_samples=MC_SAMPLES)
qnehvi_sampler = SobolQMCNormalSampler(num_samples=MC_SAMPLES)
# optimize acquisition functions and get new observations
new_x_qparego, new_obj_qparego, new_con_qparego = optimize_qparego_and_get_observation(
model_qparego, train_obj_qparego, train_con_qparego, qparego_sampler
)
new_x_qnehvi, new_obj_qnehvi, new_con_qnehvi = optimize_qnehvi_and_get_observation(
model_qnehvi, train_x_qnehvi, train_obj_qnehvi, train_con_qnehvi, qnehvi_sampler
)
new_x_random, new_obj_random, new_con_random = generate_initial_data(n=BATCH_SIZE)
# update training points
train_x_qparego = torch.cat([train_x_qparego, new_x_qparego])
train_obj_qparego = torch.cat([train_obj_qparego, new_obj_qparego])
train_con_qparego = torch.cat([train_con_qparego, new_con_qparego])
train_x_qnehvi = torch.cat([train_x_qnehvi, new_x_qnehvi])
train_obj_qnehvi = torch.cat([train_obj_qnehvi, new_obj_qnehvi])
train_con_qnehvi = torch.cat([train_con_qnehvi, new_con_qnehvi])
train_x_random = torch.cat([train_x_random, new_x_random])
train_obj_random = torch.cat([train_obj_random, new_obj_random])
train_con_random = torch.cat([train_con_random, new_con_random])
# update progress
for hvs_list, train_obj, train_con in zip(
(hvs_random, hvs_qparego, hvs_qnehvi),
(train_obj_random, train_obj_qparego, train_obj_qnehvi),
(train_con_random, train_con_qparego, train_con_qnehvi),
):
# compute pareto front
is_feas = (train_con <= 0).all(dim=-1)
feas_train_obj = train_obj[is_feas]
if feas_train_obj.shape[0] > 0:
pareto_mask = is_non_dominated(feas_train_obj)
pareto_y = feas_train_obj[pareto_mask]
# compute feasible hypervolume
volume = hv.compute(pareto_y)
else:
volume = 0.0
hvs_list.append(volume)
# reinitialize the models so they are ready for fitting on next iteration
# Note: we find improved performance from not warm starting the model hyperparameters
# using the hyperparameters from the previous iteration
mll_qparego, model_qparego = initialize_model(train_x_qparego, train_obj_qparego, train_con_qparego)
mll_qnehvi, model_qnehvi = initialize_model(train_x_qnehvi, train_obj_qnehvi, train_con_qnehvi)
t1 = time.time()
if verbose:
print(
f"\nBatch {iteration:>2}: Hypervolume (random, qParEGO, qNEHVI) = "
f"({hvs_random[-1]:>4.2f}, {hvs_qparego[-1]:>4.2f}, {hvs_qnehvi[-1]:>4.2f}), "
f"time = {t1-t0:>4.2f}.",
end="",
)
else:
print(".", end="")
The plot below shows the log feasible hypervolume difference: the log difference between the hypervolume of the true feasible pareto front and the hypervolume of the observed (feasible) pareto front identified by each algorithm. The log feasible hypervolume difference is plotted at each step of the optimization for each of the algorithms. The confidence intervals represent the variance at that step in the optimization across the trial runs. The variance across optimization runs is quite high, so in order to get a better estimate of the average performance one would have to run a much larger number of trials N_TRIALS (we avoid this here to limit the runtime of this tutorial).
The plot show that $q$NEHVI vastly outperforms the $q$ParEGO and Sobol baselines.
import numpy as np
from matplotlib import pyplot as plt
%matplotlib inline
iters = np.arange(N_BATCH + 1) * BATCH_SIZE
log_hv_difference_qparego = np.log10(problem.max_hv - np.asarray(hvs_qparego))
log_hv_difference_qnehvi = np.log10(problem.max_hv - np.asarray(hvs_qnehvi))
log_hv_difference_rnd = np.log10(problem.max_hv - np.asarray(hvs_random))
fig, ax = plt.subplots(1, 1, figsize=(8, 6))
ax.plot(
iters,
log_hv_difference_rnd,
label="Sobol",
linewidth=1.5,
color="gray",
)
ax.plot(
iters,
log_hv_difference_qparego,
label="qParEGO",
linewidth=1.5,
color="red",
)
ax.plot(
iters,
log_hv_difference_qnehvi,
label="qNEHVI",
linewidth=1.5,
color="blue",
)
ax.set(xlabel="number of observations (beyond initial points)", ylabel="Log Hypervolume Difference")
ax.legend(loc="lower right")
To examine optimization process from another perspective, we plot the collected observations under each algorithm where the color corresponds to the BO iteration at which the point was collected. The plot on the right for $q$NEHVI shows that the $q$NEHVI quickly identifies the pareto front and most of its evaluations are very close to the pareto front. $q$ParEGO also identifies has many observations close to the pareto front, but relies on optimizing random scalarizations, which is a less principled way of optimizing the pareto front compared to $q$NEHVI, which explicitly attempts focuses on improving the pareto front. Sobol generates random points and has few points close to the pareto front
from matplotlib.cm import ScalarMappable
fig, axes = plt.subplots(1, 3, figsize=(17, 5))
algos = ["Sobol", "qParEGO", "qNEHVI"]
cm = plt.cm.get_cmap('viridis')
batch_number = torch.cat(
[torch.zeros(2*(d+1)), torch.arange(1, N_BATCH+1).repeat(BATCH_SIZE, 1).t().reshape(-1)]
).numpy()
for i, train_obj in enumerate((train_obj_random, train_obj_qparego, train_obj_qnehvi)):
sc = axes[i].scatter(train_obj[:, 0].cpu().numpy(), train_obj[:,1].cpu().numpy(), c=batch_number, alpha=0.8)
axes[i].set_title(algos[i])
axes[i].set_xlabel("Objective 1")
axes[i].set_xlim(-2.5, 0)
axes[i].set_ylim(-2.5, 0)
axes[0].set_ylabel("Objective 2")
norm = plt.Normalize(batch_number.min(), batch_number.max())
sm = ScalarMappable(norm=norm, cmap=cm)
sm.set_array([])
fig.subplots_adjust(right=0.9)
cbar_ax = fig.add_axes([0.93, 0.15, 0.01, 0.7])
cbar = fig.colorbar(sm, cax=cbar_ax)
cbar.ax.set_title("Iteration")
