#!/usr/bin/env python3
# Copyright (c) Meta Platforms, Inc. and affiliates.
#
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
"""
Models and utilities for fully bayesian inference.
TODO: move some of this into botorch.
References
.. [Eriksson2021saasbo]
D. Eriksson, M. Jankowiak. High-Dimensional Bayesian Optimization
with Sparse Axis-Aligned Subspaces. Proceedings of the Thirty-
Seventh Conference on Uncertainty in Artificial Intelligence, 2021.
.. [Eriksson2021nas]
D. Eriksson, P. Chuang, S. Daulton, et al. Latency-Aware Neural
Architecture Search with Multi-Objective Bayesian Optimization.
ICML AutoML Workshop, 2021.
"""
import math
import sys
import time
import types
import warnings
from logging import Logger
from typing import Any, Callable, Dict, List, Optional, Tuple
import numpy as np
import pyro # @manual=//third-party-source/native/pyro:pyro
import torch
from ax.exceptions.core import AxError
from ax.models.torch.botorch import (
BotorchModel,
TAcqfConstructor,
TBestPointRecommender,
TModelConstructor,
TModelPredictor,
TOptimizer,
)
from ax.models.torch.botorch_defaults import (
get_NEI,
MIN_OBSERVED_NOISE_LEVEL,
recommend_best_observed_point,
scipy_optimizer,
)
from ax.models.torch.botorch_moo import MultiObjectiveBotorchModel
from ax.models.torch.botorch_moo_defaults import get_NEHVI, pareto_frontier_evaluator
from ax.models.torch.frontier_utils import TFrontierEvaluator
from ax.models.torch.fully_bayesian_model_utils import (
_get_single_task_gpytorch_model,
load_mcmc_samples_to_model,
pyro_sample_input_warping,
pyro_sample_mean,
pyro_sample_noise,
pyro_sample_outputscale,
pyro_sample_saas_lengthscales,
)
from ax.utils.common.docutils import copy_doc
from ax.utils.common.logger import get_logger
from botorch.acquisition import AcquisitionFunction
from botorch.models.gpytorch import GPyTorchModel
from botorch.models.model import Model
from botorch.models.model_list_gp_regression import ModelListGP
from torch import Tensor
logger: Logger = get_logger(__name__)
SAAS_DEPRECATION_MSG = (
"Passing `use_saas` is no longer supported and has no effect. "
"SAAS priors are used by default. "
"This will become an error in the future."
)
[docs]def predict_from_model_mcmc(model: Model, X: Tensor) -> Tuple[Tensor, Tensor]:
r"""Predicts outcomes given a model and input tensor.
This method integrates over the hyperparameter posterior.
Args:
model: A batched botorch Model where the batch dimension corresponds
to sampled hyperparameters.
X: A `n x d` tensor of input parameters.
Returns:
Tensor: The predicted posterior mean as an `n x o`-dim tensor.
Tensor: The predicted posterior covariance as a `n x o x o`-dim tensor.
"""
with torch.no_grad():
# compute the batch (independent posterior over the inputs)
posterior = model.posterior(X.unsqueeze(-3))
# the mean and variance both have shape: n x num_samples x m (after squeezing)
mean = posterior.mean.cpu().detach()
# TODO: Allow Posterior to (optionally) return the full covariance matrix
variance = posterior.variance.cpu().detach().clamp_min(0)
# marginalize over samples
t1 = variance.sum(dim=0) / variance.shape[0]
t2 = mean.pow(2).sum(dim=0) / variance.shape[0]
t3 = -(mean.sum(dim=0) / variance.shape[0]).pow(2)
variance = t1 + t2 + t3
mean = mean.mean(dim=0)
cov = torch.diag_embed(variance)
return mean, cov
[docs]def compute_dists(X: Tensor, Z: Tensor, lengthscale: Tensor) -> Tensor:
"""Compute kernel distances.
TODO: use gpytorch `Distance` module. This will require some care to make sure
jit compilation works as expected.
"""
mean = X.mean(dim=0)
X_ = (X - mean).div(lengthscale)
Z_ = (Z - mean).div(lengthscale)
x1 = X_
x2 = Z_
adjustment = x1.mean(-2, keepdim=True)
x1 = x1 - adjustment
# x1 and x2 should be identical in all dims except -2 at this point
x2 = x2 - adjustment
x1_eq_x2 = torch.equal(x1, x2)
# Compute squared distance matrix using quadratic expansion
x1_norm = x1.pow(2).sum(dim=-1, keepdim=True)
x1_pad = torch.ones_like(x1_norm)
if x1_eq_x2 and not x1.requires_grad and not x2.requires_grad:
x2_norm, x2_pad = x1_norm, x1_pad
else:
x2_norm = x2.pow(2).sum(dim=-1, keepdim=True)
x2_pad = torch.ones_like(x2_norm)
x2_norm = x2.pow(2).sum(dim=-1, keepdim=True)
x2_pad = torch.ones_like(x2_norm)
x1_ = torch.cat([-2.0 * x1, x1_norm, x1_pad], dim=-1)
x2_ = torch.cat([x2, x2_pad, x2_norm], dim=-1)
res = x1_.matmul(x2_.transpose(-2, -1))
if x1_eq_x2 and not x1.requires_grad and not x2.requires_grad:
res.diagonal(dim1=-2, dim2=-1).fill_(0)
# Zero out negative values
dist = res.clamp_min_(1e-30).sqrt()
return dist
[docs]def matern_kernel(X: Tensor, Z: Tensor, lengthscale: Tensor, nu: float = 2.5) -> Tensor:
"""Scaled Matern kernel."""
dist = compute_dists(X=X, Z=Z, lengthscale=lengthscale)
exp_component = torch.exp(-math.sqrt(nu * 2) * dist)
if nu == 0.5:
constant_component = 1
elif nu == 1.5:
constant_component = (math.sqrt(3) * dist).add(1)
elif nu == 2.5:
# pyre-fixme[58]: `**` is not supported for operand types `Tensor` and `int`.
constant_component = (math.sqrt(5) * dist).add(1).add(5.0 / 3.0 * (dist**2))
else:
raise AxError(f"Unsupported value of nu: {nu}")
return constant_component * exp_component
[docs]def rbf_kernel(X: Tensor, Z: Tensor, lengthscale: Tensor) -> Tensor:
"""Scaled RBF kernel."""
dist = compute_dists(X=X, Z=Z, lengthscale=lengthscale)
# pyre-fixme[6]: For 1st param expected `Tensor` but got `float`.
# pyre-fixme[58]: `**` is not supported for operand types `Tensor` and `int`.
return torch.exp(-0.5 * (dist**2))
[docs]def single_task_pyro_model(
X: Tensor,
Y: Tensor,
Yvar: Tensor,
use_input_warping: bool = False,
eps: float = 1e-7,
gp_kernel: str = "matern",
task_feature: Optional[int] = None,
rank: Optional[int] = None,
) -> None:
r"""Instantiates a single task pyro model for running fully bayesian inference.
Args:
X: A `n x d` tensor of input parameters.
Y: A `n x 1` tensor of output.
Yvar: A `n x 1` tensor of observed noise.
use_input_warping: A boolean indicating whether to use input warping
task_feature: Column index of task feature in X.
gp_kernel: kernel name. Currently only two kernels are supported: "matern" for
Matern Kernel and "rbf" for RBFKernel.
rank: num of latent task features to learn for task covariance.
"""
Y = Y.view(-1)
Yvar = Yvar.view(-1)
tkwargs = {"dtype": X.dtype, "device": X.device}
dim = X.shape[-1]
# TODO: test alternative outputscale priors
outputscale = pyro_sample_outputscale(concentration=2.0, rate=0.15, **tkwargs)
mean = pyro_sample_mean(**tkwargs)
if torch.isnan(Yvar).all():
# infer noise level
noise = pyro_sample_noise(**tkwargs)
else:
noise = Yvar.clamp_min(MIN_OBSERVED_NOISE_LEVEL)
# pyre-fixme[6]: For 2nd param expected `float` but got `Union[device, dtype]`.
lengthscale = pyro_sample_saas_lengthscales(dim=dim, **tkwargs)
# transform inputs through kumaraswamy cdf
if use_input_warping:
c0, c1 = pyro_sample_input_warping(dim=dim, **tkwargs)
# unnormalize X from [0, 1] to [eps, 1-eps]
X = (X * (1 - 2 * eps) + eps).clamp(eps, 1 - eps)
X_tf = 1 - torch.pow((1 - torch.pow(X, c1)), c0)
else:
X_tf = X
# compute kernel
if gp_kernel == "matern":
k = matern_kernel(X=X_tf, Z=X_tf, lengthscale=lengthscale)
elif gp_kernel == "rbf":
k = rbf_kernel(X=X_tf, Z=X_tf, lengthscale=lengthscale)
else:
raise ValueError(f"Expected kernel to be 'rbf' or 'matern', got {gp_kernel}")
# add noise
k = outputscale * k + noise * torch.eye(X.shape[0], dtype=X.dtype, device=X.device)
pyro.sample(
"Y",
pyro.distributions.MultivariateNormal( # pyre-ignore [16]
loc=mean.view(-1).expand(X.shape[0]),
covariance_matrix=k,
),
obs=Y,
)
def _get_model_mcmc_samples(
Xs: List[Tensor],
Ys: List[Tensor],
Yvars: List[Tensor],
task_features: List[int],
fidelity_features: List[int],
metric_names: List[str],
state_dict: Optional[Dict[str, Tensor]] = None,
refit_model: bool = True,
use_input_warping: bool = False,
use_loocv_pseudo_likelihood: bool = False,
num_samples: int = 256,
warmup_steps: int = 512,
thinning: int = 16,
max_tree_depth: int = 6,
disable_progbar: bool = False,
gp_kernel: str = "matern",
verbose: bool = False,
# pyre-fixme[24]: Generic type `Callable` expects 2 type parameters.
pyro_model: Callable = single_task_pyro_model,
# pyre-fixme[24]: Generic type `Callable` expects 2 type parameters.
get_gpytorch_model: Callable = _get_single_task_gpytorch_model,
rank: Optional[int] = 1,
**kwargs: Any,
# pyre-fixme[24]: Generic type `dict` expects 2 type parameters, use `typing.Dict`
# to avoid runtime subscripting errors.
) -> Tuple[ModelListGP, List[Dict]]:
r"""Instantiates a batched GPyTorchModel(ModelListGP) based on the given data and
fit the model based on MCMC in pyro.
Args:
pyro_model: callable to instantiate a pyro model for running MCMC
get_gpytorch_model: callable to instantiate a coupled GPyTorchModel to load the
returned MCMC samples.
"""
model = get_gpytorch_model(
Xs=Xs,
Ys=Ys,
Yvars=Yvars,
task_features=task_features,
fidelity_features=fidelity_features,
state_dict=state_dict,
num_samples=num_samples,
thinning=thinning,
use_input_warping=use_input_warping,
gp_kernel=gp_kernel,
**kwargs,
)
if state_dict is not None:
# Expected `OrderedDict[typing.Any, typing.Any]` for 1st
# param but got `Dict[str, Tensor]`.
model.load_state_dict(state_dict)
mcmc_samples_list = []
if len(task_features) > 0:
task_feature = task_features[0]
else:
task_feature = None
if state_dict is None or refit_model:
for X, Y, Yvar in zip(Xs, Ys, Yvars):
mcmc_samples = run_inference(
pyro_model=pyro_model,
X=X,
Y=Y,
Yvar=Yvar,
num_samples=num_samples,
warmup_steps=warmup_steps,
thinning=thinning,
use_input_warping=use_input_warping,
max_tree_depth=max_tree_depth,
disable_progbar=disable_progbar,
gp_kernel=gp_kernel,
verbose=verbose,
task_feature=task_feature,
rank=rank,
)
mcmc_samples_list.append(mcmc_samples)
return model, mcmc_samples_list
[docs]def get_and_fit_model_mcmc(
Xs: List[Tensor],
Ys: List[Tensor],
Yvars: List[Tensor],
task_features: List[int],
fidelity_features: List[int],
metric_names: List[str],
state_dict: Optional[Dict[str, Tensor]] = None,
refit_model: bool = True,
use_input_warping: bool = False,
use_loocv_pseudo_likelihood: bool = False,
num_samples: int = 256,
warmup_steps: int = 512,
thinning: int = 16,
max_tree_depth: int = 6,
disable_progbar: bool = False,
gp_kernel: str = "matern",
verbose: bool = False,
**kwargs: Any,
) -> GPyTorchModel:
r"""Instantiates a batched GPyTorchModel(ModelListGP) based on the given data and
fit the model based on MCMC in pyro. The batch dimension corresponds to sampled
hyperparameters from MCMC.
"""
model, mcmc_samples_list = _get_model_mcmc_samples(
Xs=Xs,
Ys=Ys,
Yvars=Yvars,
task_features=task_features,
fidelity_features=fidelity_features,
metric_names=metric_names,
state_dict=state_dict,
refit_model=refit_model,
use_input_warping=use_input_warping,
use_loocv_pseudo_likelihood=use_loocv_pseudo_likelihood,
num_samples=num_samples,
warmup_steps=warmup_steps,
thinning=thinning,
max_tree_depth=max_tree_depth,
disable_progbar=disable_progbar,
gp_kernel=gp_kernel,
verbose=verbose,
pyro_model=single_task_pyro_model,
get_gpytorch_model=_get_single_task_gpytorch_model,
)
for i, mcmc_samples in enumerate(mcmc_samples_list):
load_mcmc_samples_to_model(model=model.models[i], mcmc_samples=mcmc_samples)
return model
[docs]def run_inference(
# pyre-fixme[24]: Generic type `Callable` expects 2 type parameters.
pyro_model: Callable,
X: Tensor,
Y: Tensor,
Yvar: Tensor,
num_samples: int = 256,
warmup_steps: int = 512,
thinning: int = 16,
use_input_warping: bool = False,
max_tree_depth: int = 6,
disable_progbar: bool = False,
gp_kernel: str = "matern",
verbose: bool = False,
task_feature: Optional[int] = None,
rank: Optional[int] = None,
) -> Dict[str, Tensor]:
start = time.time()
try:
# @manual=//third-party-source/native/pyro:pyro
from pyro.infer.mcmc import MCMC, NUTS
# @manual=//third-party-source/native/pyro:pyro
from pyro.infer.mcmc.util import print_summary
except ImportError: # pragma: no cover
raise RuntimeError("Cannot call run_inference without pyro installed!")
kernel = NUTS(
pyro_model,
jit_compile=True,
full_mass=True,
ignore_jit_warnings=True,
max_tree_depth=max_tree_depth,
)
mcmc = MCMC(
kernel,
warmup_steps=warmup_steps,
num_samples=num_samples,
disable_progbar=disable_progbar,
)
mcmc.run(
X,
Y,
Yvar,
use_input_warping=use_input_warping,
gp_kernel=gp_kernel,
task_feature=task_feature,
rank=rank,
)
# compute the true lengthscales and get rid of the temporary variables
samples = mcmc.get_samples()
inv_length_sq = (
samples["kernel_tausq"].unsqueeze(-1) * samples["_kernel_inv_length_sq"]
)
samples["lengthscale"] = (1.0 / inv_length_sq).sqrt() # pyre-ignore [16]
del samples["kernel_tausq"], samples["_kernel_inv_length_sq"]
# this prints the summary
if verbose:
orig_std_out = sys.stdout.write
sys.stdout.write = logger.info # pyre-fixme[8]
print_summary(samples, prob=0.9, group_by_chain=False)
sys.stdout.write = orig_std_out
logger.info(f"MCMC elapsed time: {time.time() - start}")
# thin
for k, v in samples.items():
samples[k] = v[::thinning] # apply thinning
return samples
[docs]def get_fully_bayesian_acqf(
model: Model,
objective_weights: Tensor,
outcome_constraints: Optional[Tuple[Tensor, Tensor]] = None,
X_observed: Optional[Tensor] = None,
X_pending: Optional[Tensor] = None,
# pyre-fixme[9]: acqf_constructor has type `Callable[[Model, Tensor,
# Optional[Tuple[Tensor, Tensor]], Optional[Tensor], Optional[Tensor], Any],
# AcquisitionFunction]`; used as `Callable[[Model, Tensor,
# Optional[Tuple[Tensor, Tensor]], Optional[Tensor], Optional[Tensor],
# **(Any)], AcquisitionFunction]`.
acqf_constructor: TAcqfConstructor = get_NEI,
**kwargs: Any,
) -> AcquisitionFunction:
kwargs["marginalize_dim"] = -3
# pyre-ignore [28]
acqf = acqf_constructor(
model=model,
objective_weights=objective_weights,
outcome_constraints=outcome_constraints,
X_observed=X_observed,
X_pending=X_pending,
**kwargs,
)
base_forward = acqf.forward
# pyre-fixme[53]: Captured variable `base_forward` is not annotated.
# pyre-fixme[3]: Return type must be annotated.
# pyre-fixme[2]: Parameter must be annotated.
def forward(self, X):
# unsqueeze dim for GP hyperparameter samples
return base_forward(X.unsqueeze(-3)).mean(dim=-1)
acqf.forward = types.MethodType(forward, acqf) # pyre-ignore[8]
return acqf
[docs]def get_fully_bayesian_acqf_nehvi(
model: Model,
objective_weights: Tensor,
outcome_constraints: Optional[Tuple[Tensor, Tensor]] = None,
X_observed: Optional[Tensor] = None,
X_pending: Optional[Tensor] = None,
**kwargs: Any,
) -> AcquisitionFunction:
return get_fully_bayesian_acqf(
model=model,
objective_weights=objective_weights,
outcome_constraints=outcome_constraints,
X_observed=X_observed,
X_pending=X_pending,
acqf_constructor=get_NEHVI, # pyre-ignore [6]
**kwargs,
)
[docs]class FullyBayesianBotorchModelMixin:
model: Optional[Model] = None
[docs] def feature_importances(self) -> np.ndarray:
if self.model is None:
raise RuntimeError(
"Cannot calculate feature_importances without a fitted model"
)
elif isinstance(self.model, ModelListGP):
models = self.model.models
else:
models = [self.model]
lengthscales = []
for m in models:
ls = m.covar_module.base_kernel.lengthscale
lengthscales.append(ls)
lengthscales = torch.stack(lengthscales, dim=0)
# take mean over MCMC samples
lengthscales = torch.quantile(lengthscales, 0.5, dim=1)
# pyre-ignore [16]
# pyre-fixme[58]: `/` is not supported for operand types `int` and `Tensor`.
return (1 / lengthscales).detach().cpu().numpy()
[docs]class FullyBayesianBotorchModel(FullyBayesianBotorchModelMixin, BotorchModel):
r"""Fully Bayesian Model that uses NUTS to sample from hyperparameter posterior.
This includes support for using sparse axis-aligned subspace priors (SAAS). See
[Eriksson2021saasbo]_ for details.
"""
def __init__(
self,
model_constructor: TModelConstructor = get_and_fit_model_mcmc,
model_predictor: TModelPredictor = predict_from_model_mcmc,
acqf_constructor: TAcqfConstructor = get_fully_bayesian_acqf,
# pyre-fixme[9]: acqf_optimizer declared/used type mismatch
acqf_optimizer: TOptimizer = scipy_optimizer,
best_point_recommender: TBestPointRecommender = recommend_best_observed_point,
refit_on_cv: bool = False,
refit_on_update: bool = True,
warm_start_refitting: bool = True,
use_input_warping: bool = False,
# use_saas is deprecated. TODO: remove
use_saas: Optional[bool] = None,
num_samples: int = 256,
warmup_steps: int = 512,
thinning: int = 16,
max_tree_depth: int = 6,
disable_progbar: bool = False,
gp_kernel: str = "matern",
verbose: bool = False,
**kwargs: Any,
) -> None:
"""Initialize Fully Bayesian Botorch Model.
Args:
model_constructor: A callable that instantiates and fits a model on data,
with signature as described below.
model_predictor: A callable that predicts using the fitted model, with
signature as described below.
acqf_constructor: A callable that creates an acquisition function from a
fitted model, with signature as described below.
acqf_optimizer: A callable that optimizes the acquisition function, with
signature as described below.
best_point_recommender: A callable that recommends the best point, with
signature as described below.
refit_on_cv: If True, refit the model for each fold when performing
cross-validation.
refit_on_update: If True, refit the model after updating the training
data using the `update` method.
warm_start_refitting: If True, start model refitting from previous
model parameters in order to speed up the fitting process.
use_input_warping: A boolean indicating whether to use input warping
use_saas: [deprecated] A boolean indicating whether to use the SAAS model
num_samples: The number of MCMC samples. Note that with thinning,
num_samples/thinning samples are retained.
warmup_steps: The number of burn-in steps for NUTS.
thinning: The amount of thinning. Every nth sample is retained.
max_tree_depth: The max_tree_depth for NUTS.
disable_progbar: A boolean indicating whether to print the progress
bar and diagnostics during MCMC.
gp_kernel: The type of ARD base kernel. "matern" corresponds to a Matern-5/2
kernel and "rbf" corresponds to an RBF kernel.
verbose: A boolean indicating whether to print summary stats from MCMC.
"""
# use_saas is deprecated. TODO: remove
if use_saas is not None:
warnings.warn(SAAS_DEPRECATION_MSG, DeprecationWarning)
BotorchModel.__init__(
self,
model_constructor=model_constructor,
model_predictor=model_predictor,
acqf_constructor=acqf_constructor,
acqf_optimizer=acqf_optimizer,
best_point_recommender=best_point_recommender,
refit_on_cv=refit_on_cv,
refit_on_update=refit_on_update,
warm_start_refitting=warm_start_refitting,
use_input_warping=use_input_warping,
num_samples=num_samples,
warmup_steps=warmup_steps,
thinning=thinning,
max_tree_depth=max_tree_depth,
disable_progbar=disable_progbar,
gp_kernel=gp_kernel,
verbose=verbose,
)
[docs]class FullyBayesianMOOBotorchModel(
FullyBayesianBotorchModelMixin, MultiObjectiveBotorchModel
):
r"""Fully Bayesian Model that uses qNEHVI.
This includes support for using qNEHVI + SAASBO as in [Eriksson2021nas]_.
"""
@copy_doc(FullyBayesianBotorchModel.__init__)
def __init__(
self,
model_constructor: TModelConstructor = get_and_fit_model_mcmc,
model_predictor: TModelPredictor = predict_from_model_mcmc,
# pyre-fixme[9]: acqf_constructor has type `Callable[[Model, Tensor,
# Optional[Tuple[Tensor, Tensor]], Optional[Tensor], Optional[Tensor], Any],
# AcquisitionFunction]`; used as `Callable[[Model, Tensor,
# Optional[Tuple[Tensor, Tensor]], Optional[Tensor], Optional[Tensor],
# **(Any)], AcquisitionFunction]`.
acqf_constructor: TAcqfConstructor = get_fully_bayesian_acqf_nehvi,
# pyre-fixme[9]: acqf_optimizer has type `Callable[[AcquisitionFunction,
# Tensor, int, Optional[Dict[int, float]], Optional[Callable[[Tensor],
# Tensor]], Any], Tensor]`; used as `Callable[[AcquisitionFunction, Tensor,
# int, Optional[Dict[int, float]], Optional[Callable[[Tensor], Tensor]],
# **(Any)], Tensor]`.
acqf_optimizer: TOptimizer = scipy_optimizer,
# TODO: Remove best_point_recommender for botorch_moo. Used in modelbridge._gen.
best_point_recommender: TBestPointRecommender = recommend_best_observed_point,
frontier_evaluator: TFrontierEvaluator = pareto_frontier_evaluator,
refit_on_cv: bool = False,
refit_on_update: bool = True,
warm_start_refitting: bool = False,
use_input_warping: bool = False,
num_samples: int = 256,
warmup_steps: int = 512,
thinning: int = 16,
max_tree_depth: int = 6,
# use_saas is deprecated. TODO: remove
use_saas: Optional[bool] = None,
disable_progbar: bool = False,
gp_kernel: str = "matern",
verbose: bool = False,
**kwargs: Any,
) -> None:
# use_saas is deprecated. TODO: remove
if use_saas is not None:
warnings.warn(SAAS_DEPRECATION_MSG, DeprecationWarning)
MultiObjectiveBotorchModel.__init__(
self,
model_constructor=model_constructor,
model_predictor=model_predictor,
acqf_constructor=acqf_constructor,
acqf_optimizer=acqf_optimizer,
best_point_recommender=best_point_recommender,
frontier_evaluator=frontier_evaluator,
refit_on_cv=refit_on_cv,
refit_on_update=refit_on_update,
warm_start_refitting=warm_start_refitting,
use_input_warping=use_input_warping,
num_samples=num_samples,
warmup_steps=warmup_steps,
thinning=thinning,
max_tree_depth=max_tree_depth,
disable_progbar=disable_progbar,
gp_kernel=gp_kernel,
verbose=verbose,
)
