In this tutorial, we show how to implement Bayesian optimization with adaptively expanding subspaces (BAxUS) [1] in a closed loop in BoTorch. The tutorial is purposefully similar to the TuRBO tutorial to highlight the differences in the implementations.
This implementation supports either Expected Improvement (EI) or Thompson sampling (TS). We optimize the Branin2 function [2] with 498 dummy dimensions$ and show that BAxUS outperforms EI as well as Sobol.
Since BoTorch assumes a maximization problem, we will attempt to maximize $-f(x)$ to achieve $\max_{x\in \mathcal{X}} -f(x)=0$.
import math
import os
from dataclasses import dataclass
import botorch
import gpytorch
import matplotlib.pyplot as plt
import numpy as np
import torch
from gpytorch.constraints import Interval
from gpytorch.kernels import MaternKernel, ScaleKernel
from gpytorch.likelihoods import GaussianLikelihood
from gpytorch.mlls import ExactMarginalLogLikelihood
from torch.quasirandom import SobolEngine
from botorch.acquisition.analytic import ExpectedImprovement
from botorch.exceptions import ModelFittingError
from botorch.fit import fit_gpytorch_mll
from botorch.generation import MaxPosteriorSampling
from botorch.models import SingleTaskGP
from botorch.optim import optimize_acqf
from botorch.test_functions import Branin
device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
print(f"Running on {device}")
dtype = torch.double
SMOKE_TEST = os.environ.get("SMOKE_TEST")
The goal is to minimize the embedded Branin function
$f(x_1, x_2, \ldots, x_{20}) = \left (x_2-\frac{5.1}{4\pi^2}x_1^2+\frac{5}{\pi}x_1-6\right )^2+10\cdot \left (1-\frac{1}{8\pi}\right )\cos(x_1)+10$
with bounds [-5, 10] for $x_1$ and [0, 15] for $x_2$ (all other dimensions are ignored). The function has three minima with an optimal value of $0.397887$.
As mentioned above, since botorch assumes a maximization problem, we instead maximize $-f(x)$.
We first define a new function where we only pass the first two input dimensions to the actual Branin function.
branin = Branin(negate=True).to(device=device, dtype=dtype)
def branin_emb(x):
"""x is assumed to be in [-1, 1]^D"""
lb, ub = branin.bounds
return branin(lb + (ub - lb) * (x[..., :2] + 1) / 2)
fun = branin_emb
dim = 500
n_init = 10
max_cholesky_size = float("inf") # Always use Cholesky
BAxUS needs to maintain a state, which includes the length of the trust region, success and failure counters, success and failure tolerance, etc. In contrast to TuRBO, the failure tolerance depends on the target dimensionality.
In this tutorial we store the state in a dataclass and update the state of TuRBO after each batch evaluation.
Note: These settings assume that the domain has been scaled to $[-1, 1]^d$
@dataclass
class BaxusState:
dim: int
eval_budget: int
new_bins_on_split: int = 3
d_init: int = float("nan") # Note: post-initialized
target_dim: int = float("nan") # Note: post-initialized
n_splits: int = float("nan") # Note: post-initialized
length: float = 0.8
length_init: float = 0.8
length_min: float = 0.5**7
length_max: float = 1.6
failure_counter: int = 0
success_counter: int = 0
success_tolerance: int = 3
best_value: float = -float("inf")
restart_triggered: bool = False
def __post_init__(self):
n_splits = round(math.log(self.dim, self.new_bins_on_split + 1))
self.d_init = 1 + np.argmin(
np.abs(
(1 + np.arange(self.new_bins_on_split))
* (1 + self.new_bins_on_split) ** n_splits
- self.dim
)
)
self.target_dim = self.d_init
self.n_splits = n_splits
@property
def split_budget(self) -> int:
return round(
-1
* (self.new_bins_on_split * self.eval_budget * self.target_dim)
/ (self.d_init * (1 - (self.new_bins_on_split + 1) ** (self.n_splits + 1)))
)
@property
def failure_tolerance(self) -> int:
if self.target_dim == self.dim:
return self.target_dim
k = math.floor(math.log(self.length_min / self.length_init, 0.5))
split_budget = self.split_budget
return min(self.target_dim, max(1, math.floor(split_budget / k)))
def update_state(state, Y_next):
if max(Y_next) > state.best_value + 1e-3 * math.fabs(state.best_value):
state.success_counter += 1
state.failure_counter = 0
else:
state.success_counter = 0
state.failure_counter += 1
if state.success_counter == state.success_tolerance: # Expand trust region
state.length = min(2.0 * state.length, state.length_max)
state.success_counter = 0
elif state.failure_counter == state.failure_tolerance: # Shrink trust region
state.length /= 2.0
state.failure_counter = 0
state.best_value = max(state.best_value, max(Y_next).item())
if state.length < state.length_min:
state.restart_triggered = True
return state
We now show how to create the BAxUS embedding. The essential idea is to assign input dimensions to target dimensions and to assign a sign $\in \pm 1$ to each input dimension, similar to the HeSBO embedding. We create the embedding matrix that is used to project points from the target to the input space. The matrix is sparse, each column has precisely one non-zero entry that is either 1 or -1.
def embedding_matrix(input_dim: int, target_dim: int) -> torch.Tensor:
if (
target_dim >= input_dim
): # return identity matrix if target size greater than input size
return torch.eye(input_dim, device=device, dtype=dtype)
input_dims_perm = (
torch.randperm(input_dim, device=device) + 1
) # add 1 to indices for padding column in matrix
bins = torch.tensor_split(
input_dims_perm, target_dim
) # split dims into almost equally-sized bins
bins = torch.nn.utils.rnn.pad_sequence(
bins, batch_first=True
) # zero pad bins, the index 0 will be cut off later
mtrx = torch.zeros(
(target_dim, input_dim + 1), dtype=dtype, device=device
) # add one extra column for padding
mtrx = mtrx.scatter_(
1,
bins,
2 * torch.randint(2, (target_dim, input_dim), dtype=dtype, device=device) - 1,
) # fill mask with random +/- 1 at indices
return mtrx[:, 1:] # cut off index zero as this corresponds to zero padding
embedding_matrix(10, 3) # example for an embedding matrix
Next, we write a helper function to increase the embedding and to bring observations to the increased target space.
def increase_embedding_and_observations(
S: torch.Tensor, X: torch.Tensor, n_new_bins: int
) -> torch.Tensor:
assert X.size(1) == S.size(0), "Observations don't lie in row space of S"
S_update = S.clone()
X_update = X.clone()
for row_idx in range(len(S)):
row = S[row_idx]
idxs_non_zero = torch.nonzero(row)
idxs_non_zero = idxs_non_zero[torch.randperm(len(idxs_non_zero))].squeeze()
non_zero_elements = row[idxs_non_zero].squeeze()
n_row_bins = min(
n_new_bins, len(idxs_non_zero)
) # number of new bins is always less or equal than the contributing input dims in the row minus one
new_bins = torch.tensor_split(idxs_non_zero, n_row_bins)[
1:
] # the dims in the first bin won't be moved
elements_to_move = torch.tensor_split(non_zero_elements, n_row_bins)[1:]
new_bins_padded = torch.nn.utils.rnn.pad_sequence(
new_bins, batch_first=True
) # pad the tuples of bins with zeros to apply _scatter
els_to_move_padded = torch.nn.utils.rnn.pad_sequence(
elements_to_move, batch_first=True
)
S_stack = torch.zeros(
(n_row_bins - 1, len(row) + 1), device=device, dtype=dtype
) # submatrix to stack on S_update
S_stack = S_stack.scatter_(
1, new_bins_padded + 1, els_to_move_padded
) # fill with old values (add 1 to indices for padding column)
S_update[
row_idx, torch.hstack(new_bins)
] = 0 # set values that were move to zero in current row
X_update = torch.hstack(
(X_update, X[:, row_idx].reshape(-1, 1).repeat(1, len(new_bins)))
) # repeat observations for row at the end of X (column-wise)
S_update = torch.vstack(
(S_update, S_stack[:, 1:])
) # stack onto S_update except for padding column
return S_update, X_update
S = embedding_matrix(10, 2)
X = torch.randint(100, (7, 2))
print(f"S before increase\n{S}")
print(f"X before increase\n{X}")
S, X = increase_embedding_and_observations(S, X, 3)
print(f"S after increase\n{S}")
print(f"X after increase\n{X}")
state = BaxusState(dim=dim, eval_budget=500)
print(state)
This generates an initial set of Sobol points that we use to start of the BO loop.
def get_initial_points(dim, n_pts, seed=0):
sobol = SobolEngine(dimension=dim, scramble=True, seed=seed)
X_init = (
2 * sobol.draw(n=n_pts).to(dtype=dtype, device=device) - 1
) # points have to be in [-1, 1]^d
return X_init
Given the current state and a probabilistic (GP) model built from observations X and Y, we generate a new batch of points.
This method works on the domain $[-1, +1]^d$, so make sure to not pass in observations from the true domain. unnormalize is called before the true function is evaluated which will first map the points back to the original domain.
We support either TS and qEI which can be specified via the acqf argument.
def create_candidate(
state,
model, # GP model
X, # Evaluated points on the domain [-1, 1]^d
Y, # Function values
n_candidates=None, # Number of candidates for Thompson sampling
num_restarts=10,
raw_samples=512,
acqf="ts", # "ei" or "ts"
):
assert acqf in ("ts", "ei")
assert X.min() >= -1.0 and X.max() <= 1.0 and torch.all(torch.isfinite(Y))
if n_candidates is None:
n_candidates = min(5000, max(2000, 200 * X.shape[-1]))
# Scale the TR to be proportional to the lengthscales
x_center = X[Y.argmax(), :].clone()
weights = model.covar_module.base_kernel.lengthscale.detach().view(-1)
weights = weights / weights.mean()
weights = weights / torch.prod(weights.pow(1.0 / len(weights)))
tr_lb = torch.clamp(x_center - weights * state.length, -1.0, 1.0)
tr_ub = torch.clamp(x_center + weights * state.length, -1.0, 1.0)
if acqf == "ts":
dim = X.shape[-1]
sobol = SobolEngine(dim, scramble=True)
pert = sobol.draw(n_candidates).to(dtype=dtype, device=device)
pert = tr_lb + (tr_ub - tr_lb) * pert
# Create a perturbation mask
prob_perturb = min(20.0 / dim, 1.0)
mask = torch.rand(n_candidates, dim, dtype=dtype, device=device) <= prob_perturb
ind = torch.where(mask.sum(dim=1) == 0)[0]
mask[ind, torch.randint(0, dim, size=(len(ind),), device=device)] = 1
# Create candidate points from the perturbations and the mask
X_cand = x_center.expand(n_candidates, dim).clone()
X_cand[mask] = pert[mask]
# Sample on the candidate points
thompson_sampling = MaxPosteriorSampling(model=model, replacement=False)
with torch.no_grad(): # We don't need gradients when using TS
X_next = thompson_sampling(X_cand, num_samples=1)
elif acqf == "ei":
ei = ExpectedImprovement(model, train_Y.max(), maximize=True)
X_next, acq_value = optimize_acqf(
ei,
bounds=torch.stack([tr_lb, tr_ub]),
q=1,
num_restarts=num_restarts,
raw_samples=raw_samples,
)
return X_next
This simple loop runs one instance of BAxUS with Thompson sampling until convergence.
BAxUS works on a fixed evaluation budget and shrinks the trust region until the minimal trust region size is reached (state["restart_triggered"] is set to True).
Then, BAxUS increases the target space and carries over the observations to the updated space.
evaluation_budget = 100
state = BaxusState(dim=dim, eval_budget=evaluation_budget - n_init)
S = embedding_matrix(input_dim=state.dim, target_dim=state.d_init)
X_baxus_target = get_initial_points(state.d_init, n_init)
X_baxus_input = X_baxus_target @ S
Y_baxus = torch.tensor(
[branin_emb(x) for x in X_baxus_input], dtype=dtype, device=device
).unsqueeze(-1)
NUM_RESTARTS = 10 if not SMOKE_TEST else 2
RAW_SAMPLES = 512 if not SMOKE_TEST else 4
N_CANDIDATES = min(5000, max(2000, 200 * dim)) if not SMOKE_TEST else 4
# Disable input scaling checks as we normalize to [-1, 1]
with botorch.settings.validate_input_scaling(False):
for _ in range(evaluation_budget - n_init): # Run until evaluation budget depleted
# Fit a GP model
train_Y = (Y_baxus - Y_baxus.mean()) / Y_baxus.std()
likelihood = GaussianLikelihood(noise_constraint=Interval(1e-8, 1e-3))
covar_module = (
ScaleKernel( # Use the same lengthscale prior as in the TuRBO paper
MaternKernel(
nu=2.5,
ard_num_dims=state.target_dim,
lengthscale_constraint=Interval(0.005, 10),
),
outputscale_constraint=Interval(0.05, 10),
)
)
model = SingleTaskGP(
X_baxus_target, train_Y, covar_module=covar_module, likelihood=likelihood
)
mll = ExactMarginalLogLikelihood(model.likelihood, model)
# Do the fitting and acquisition function optimization inside the Cholesky context
with gpytorch.settings.max_cholesky_size(max_cholesky_size):
# Fit the model
try:
fit_gpytorch_mll(mll)
except ModelFittingError:
# Right after increasing the target dimensionality, the covariance matrix becomes indefinite
# In this case, the Cholesky decomposition might fail due to numerical instabilities
# In this case, we revert to Adam-based optimization
optimizer = torch.optim.Adam([{"params": model.parameters()}], lr=0.1)
for _ in range(100):
optimizer.zero_grad()
output = model(X_baxus_target)
loss = -mll(output, train_Y.flatten())
loss.backward()
optimizer.step()
# Create a batch
X_next_target = create_candidate(
state=state,
model=model,
X=X_baxus_target,
Y=train_Y,
n_candidates=N_CANDIDATES,
num_restarts=NUM_RESTARTS,
raw_samples=RAW_SAMPLES,
acqf="ts",
)
X_next_input = X_next_target @ S
Y_next = torch.tensor(
[branin_emb(x) for x in X_next_input], dtype=dtype, device=device
).unsqueeze(-1)
# Update state
state = update_state(state=state, Y_next=Y_next)
# Append data
X_baxus_input = torch.cat((X_baxus_input, X_next_input), dim=0)
X_baxus_target = torch.cat((X_baxus_target, X_next_target), dim=0)
Y_baxus = torch.cat((Y_baxus, Y_next), dim=0)
# Print current status
print(
f"iteration {len(X_baxus_input)}, d={len(X_baxus_target.T)}) Best value: {state.best_value:.3}, TR length: {state.length:.3}"
)
if state.restart_triggered:
state.restart_triggered = False
print("increasing target space")
S, X_baxus_target = increase_embedding_and_observations(
S, X_baxus_target, state.new_bins_on_split
)
print(f"new dimensionality: {len(S)}")
state.target_dim = len(S)
state.length = state.length_init
state.failure_counter = 0
state.success_counter = 0
As a baseline, we compare BAxUS to Expected Improvement (EI)
X_ei = get_initial_points(dim, n_init)
Y_ei = torch.tensor(
[branin_emb(x) for x in X_ei], dtype=dtype, device=device
).unsqueeze(-1)
# Disable input scaling checks as we normalize to [-1, 1]
with botorch.settings.validate_input_scaling(False):
while len(Y_ei) < len(Y_baxus):
train_Y = (Y_ei - Y_ei.mean()) / Y_ei.std()
likelihood = GaussianLikelihood(noise_constraint=Interval(1e-8, 1e-3))
model = SingleTaskGP(X_ei, train_Y, likelihood=likelihood)
mll = ExactMarginalLogLikelihood(model.likelihood, model)
optimizer = torch.optim.Adam([{"params": model.parameters()}], lr=0.1)
model.train()
model.likelihood.train()
for _ in range(50):
optimizer.zero_grad()
output = model(X_ei)
loss = -mll(output, train_Y.squeeze())
loss.backward()
optimizer.step()
# Create a batch
ei = ExpectedImprovement(model, train_Y.max(), maximize=True)
candidate, acq_value = optimize_acqf(
ei,
bounds=torch.stack(
[
-torch.ones(dim, dtype=dtype, device=device),
torch.ones(dim, dtype=dtype, device=device),
]
),
q=1,
num_restarts=NUM_RESTARTS,
raw_samples=RAW_SAMPLES,
)
Y_next = torch.tensor(
[branin_emb(x) for x in candidate], dtype=dtype, device=device
).unsqueeze(-1)
# Append data
X_ei = torch.cat((X_ei, candidate), axis=0)
Y_ei = torch.cat((Y_ei, Y_next), axis=0)
# Print current status
print(f"{len(X_ei)}) Best value: {Y_ei.max().item():.2e}")
X_Sobol = (
SobolEngine(dim, scramble=True, seed=0)
.draw(len(X_baxus_input))
.to(dtype=dtype, device=device)
* 2
- 1
)
Y_Sobol = torch.tensor(
[branin_emb(x) for x in X_Sobol], dtype=dtype, device=device
).unsqueeze(-1)
We show the regret of the different methods.
%matplotlib inline
names = ["BAxUS", "EI", "Sobol"]
runs = [Y_baxus, Y_ei, Y_Sobol]
fig, ax = plt.subplots(figsize=(8, 6))
for name, run in zip(names, runs):
fx = np.maximum.accumulate(run.cpu())
plt.plot(-fx + branin.optimal_value, marker="", lw=3)
plt.ylabel("Regret", fontsize=18)
plt.xlabel("Number of evaluations", fontsize=18)
plt.title(f"{dim}D Embedded Branin", fontsize=24)
plt.xlim([0, len(Y_baxus)])
plt.yscale("log")
plt.grid(True)
plt.tight_layout()
plt.legend(
names + ["Global optimal value"],
loc="lower center",
bbox_to_anchor=(0, -0.08, 1, 1),
bbox_transform=plt.gcf().transFigure,
ncol=4,
fontsize=16,
)
plt.show()
