This tutorial uses synthetic functions to illustrate Bayesian optimization using a multi-task Gaussian Process in Ax. A typical use case is optimizing an expensive-to-evaluate (online) system with supporting (offline) simulations of that system.
Bayesian optimization with a multi-task kernel (Multi-task Bayesian optimization) is described by Swersky et al. (2013). Letham and Bakshy (2019) describe using multi-task Bayesian optimization to tune a ranking system with a mix of online and offline (simulator) experiments.
This tutorial produces the results of Online Appendix 2 from that paper.
The synthetic problem used here is to maximize the Hartmann 6 function, a classic optimization test problem in 6 dimensions, subject to a constraint on the 2-norm of $x$. Both the objective and the constraint are treated as unknown and are modeled with separate GPs. Both objective and constraint are noisy.
Throughout the optimization we can make nosiy observations directly of the objective and constraint (an online observation), and we can make noisy observations of a biased version of the objective and constraint (offline observations). Bias is simulated by passing the function values through a piecewise linear function. Offline observations are much less time-consuming than online observations, so we wish to use them to improve our ability to optimize the online objective.
from copy import deepcopy
import numpy as np
import pandas as pd
from scipy.stats import norm
import time
from ax.core.data import Data
from ax.core.observation import ObservationFeatures, observations_from_data
from ax.core.optimization_config import OptimizationConfig
from ax.core.search_space import SearchSpace
from ax.core.objective import Objective
from ax.runners.synthetic import SyntheticRunner
from ax.modelbridge.random import RandomModelBridge
from ax.core.outcome_constraint import OutcomeConstraint
from ax.core.types import ComparisonOp
from ax.core.parameter import RangeParameter, ParameterType
from ax.core.multi_type_experiment import MultiTypeExperiment
from ax.metrics.hartmann6 import Hartmann6Metric
from ax.metrics.l2norm import L2NormMetric
from ax.modelbridge.factory import get_sobol, get_GPEI, get_MTGP
from ax.core.generator_run import GeneratorRun
from ax.plot.diagnostic import interact_batch_comparison
from ax.plot.trace import optimization_trace_all_methods
from ax.utils.notebook.plotting import init_notebook_plotting, render
init_notebook_plotting()
[INFO 04-17 18:41:02] ipy_plotting: Injecting Plotly library into cell. Do not overwrite or delete cell.
For this example, the online system is optimizing a Hartmann6 function subject to a 2-norm constraint. The Metric objects for these are directly imported above. We create analagous offline versions of these two metrics which are identical but have a transform applied (a piecewise linear function). We construct Metric objects for each of them.
# Create metrics with artificial offline bias, for both objective and constraint
# by passing the true values through a piecewise linear function.
class OfflineHartmann6Metric(Hartmann6Metric):
def f(self, x: np.ndarray) -> float:
raw_res = super().f(x)
m = -0.35
if raw_res < m:
return (1.5* (raw_res - m)) + m
else:
return (6.0 * (raw_res - m)) + m
class OfflineL2NormMetric(L2NormMetric):
def f(self, x: np.ndarray) -> float:
raw_res = super().f(x)
m = 1.25
if raw_res < m:
return (0.8 * (raw_res - m)) + m
else:
return (4 * (raw_res - m)) + m
A MultiTypeExperiment is used for managing online and offline trials together. It is constructed in several steps:
Finally, because this is a synthetic benchmark problem where the true function values are known, we will also register metrics with the true (noiseless) function values for plotting below.
def get_experiment(include_true_metric=True):
noise_sd = 0.1 # Observations will have this much Normal noise added to them
# 1. Create simple search space for [0,1]^d, d=6
param_names = [f"x{i}" for i in range(6)]
parameters=[
RangeParameter(
name=param_names[i], parameter_type=ParameterType.FLOAT, lower=0.0, upper=1.0
)
for i in range(6)
]
search_space=SearchSpace(parameters=parameters)
# 2. Specify optimization config
online_objective = Hartmann6Metric("objective", param_names=param_names, noise_sd=noise_sd)
online_constraint = L2NormMetric("constraint", param_names=param_names, noise_sd=noise_sd)
opt_config = OptimizationConfig(
objective=Objective(online_objective, minimize=True),
outcome_constraints=[OutcomeConstraint(online_constraint, op=ComparisonOp.LEQ, bound=1.25, relative=False)]
)
# 3. Init experiment
exp = MultiTypeExperiment(
name="mt_exp",
search_space=search_space,
default_trial_type="online",
default_runner=SyntheticRunner(),
optimization_config=opt_config,
)
# 4. Establish offline trial_type, and how those trials are deployed
exp.add_trial_type("offline", SyntheticRunner())
# 5. Add offline metrics that provide biased estimates of the online metrics
offline_objective = OfflineHartmann6Metric("offline_objective", param_names=param_names, noise_sd=noise_sd)
offline_constraint = OfflineL2NormMetric("offline_constraint", param_names=param_names, noise_sd=noise_sd)
# Associate each offline metric with corresponding online metric
exp.add_tracking_metric(metric=offline_objective, trial_type="offline", canonical_name="objective")
exp.add_tracking_metric(metric=offline_constraint, trial_type="offline", canonical_name="constraint")
# Add a noiseless equivalent for each metric, for tracking the true value of each observation
# for the purposes of benchmarking.
if include_true_metric:
exp.add_tracking_metric(Hartmann6Metric("objective_noiseless", param_names=param_names, noise_sd=0.0), "online")
exp.add_tracking_metric(L2NormMetric("constraint_noiseless", param_names=param_names, noise_sd=0.0), "online")
return exp
These figures compare the online measurements to the offline measurements on a random set of points, for both the objective metric and the constraint metric. You can see the offline measurements are biased but highly correlated. This produces Fig. S3 from the paper.
# Generate 50 points from a Sobol sequence
exp = get_experiment(include_true_metric=False)
s = get_sobol(exp.search_space, scramble=False)
gr = s.gen(50)
# Deploy them both online and offline
exp.new_batch_trial(trial_type="online", generator_run=gr).run()
exp.new_batch_trial(trial_type="offline", generator_run=gr).run()
# Fetch data
data = exp.fetch_data()
observations = observations_from_data(exp, data)
# Plot the arms in batch 0 (online) vs. batch 1 (offline)
render(interact_batch_comparison(observations, exp, 1, 0))
Here we construct a Bayesian optimization loop that interleaves online and offline batches. The loop defined here is described in Algorithm 1 of the paper. We compare multi-task Bayesian optimization to regular Bayesian optimization using only online observations.
Here we measure performance over 3 repetitions of the loop. Each one takes 1-2 hours so the whole benchmark run will take several hours to complete.
# Settings for the optimization benchmark.
# This should be changed to 50 to reproduce the results from the paper.
n_reps = 3 # Number of repeated experiments, each with independent observation noise
n_init_online = 5 # Size of the quasirandom initialization run online
n_init_offline = 20 # Size of the quasirandom initialization run offline
n_opt_online = 5 # Batch size for BO selected points to be run online
n_opt_offline = 20 # Batch size for BO selected to be run offline
n_batches = 3 # Number of optimized BO batches
For the online-only case, we run n_init_online sobol points followed by n_batches batches of n_opt_online points selected by the GP. This is a normal Bayesian optimization loop.
# This function runs a Bayesian optimization loop, making online observations only.
def run_online_only_bo():
t1 = time.time()
### Do BO with online only
## Quasi-random initialization
exp_online = get_experiment()
m = get_sobol(exp_online.search_space, scramble=False)
gr = m.gen(n=n_init_online)
exp_online.new_batch_trial(trial_type="online", generator_run=gr).run()
## Do BO
for b in range(n_batches):
print('Online-only batch', b, time.time() - t1)
# Fit the GP
m = get_GPEI(
experiment=exp_online,
data=exp_online.fetch_data(),
search_space=exp_online.search_space,
)
# Generate the new batch
gr = m.gen(
n=n_opt_online,
search_space=exp_online.search_space,
optimization_config=exp_online.optimization_config,
)
exp_online.new_batch_trial(trial_type="online", generator_run=gr).run()
## Extract true objective and constraint at each iteration
df = exp_online.fetch_data().df
obj = df[df['metric_name'] == 'objective_noiseless']['mean'].values
con = df[df['metric_name'] == 'constraint_noiseless']['mean'].values
return obj, con
Here we incorporate offline observations to accelerate the optimization, while using the same total number of online observations as in the loop above. The strategy here is that outlined in Algorithm 1 of the paper.
n_init_online Sobol points online, and n_init_offline Sobol points offline.n_opt_offline candidates using NEI.n_opt_offline candidates offline and observe their offline metrics.n_opt_online of the NEI candidates, after incorporating their offline observations, and run them online.# Online batches are constructed by selecting the maximum utility points from the offline
# batch, after updating the model with the offline results. This function selects the max utility points according
# to the MTGP predictions.
def max_utility_from_GP(n, m, experiment, search_space, gr):
obsf = []
for arm in gr.arms:
params = deepcopy(arm.parameters)
params['trial_type'] = 'online'
obsf.append(ObservationFeatures(parameters=params))
# Make predictions
f, cov = m.predict(obsf)
# Compute expected utility
mu_c = np.array(f['constraint'])
sigma_c = np.sqrt(cov['constraint']['constraint'])
pfeas = norm.cdf((1.25 - mu_c) / sigma_c)
u = -np.array(f['objective']) * pfeas
best_arm_indx = np.flip(np.argsort(u))[:n]
gr_new = GeneratorRun(
arms = [
gr.arms[i] for i in best_arm_indx
],
weights = [1.] * n,
)
return gr_new
# This function runs a multi-task Bayesian optimization loop, as outlined in Algorithm 1 and above.
def run_mtbo():
t1 = time.time()
online_trials = []
## 1. Quasi-random initialization, online and offline
exp_multitask = get_experiment()
# Online points
m = get_sobol(exp_multitask.search_space, scramble=False)
gr = m.gen(
n=n_init_online,
)
tr = exp_multitask.new_batch_trial(trial_type="online", generator_run=gr)
tr.run()
online_trials.append(tr.index)
# Offline points
m = get_sobol(exp_multitask.search_space, scramble=False)
gr = m.gen(
n=n_init_offline,
)
exp_multitask.new_batch_trial(trial_type="offline", generator_run=gr).run()
## Do BO
for b in range(n_batches):
print('Multi-task batch', b, time.time() - t1)
# (2 / 7). Fit the MTGP
m = get_MTGP(
experiment=exp_multitask,
data=exp_multitask.fetch_data(),
search_space=exp_multitask.search_space,
)
# 3. Finding the best points for the online task
gr = m.gen(
n=n_opt_offline,
optimization_config=exp_multitask.optimization_config,
fixed_features=ObservationFeatures(parameters={'trial_type': 'online'}),
)
# 4. But launch them offline
exp_multitask.new_batch_trial(trial_type="offline", generator_run=gr).run()
# 5. Update the model
m = get_MTGP(
experiment=exp_multitask,
data=exp_multitask.fetch_data(),
search_space=exp_multitask.search_space,
)
# 6. Select max-utility points from the offline batch to generate an online batch
gr = max_utility_from_GP(
n=n_opt_online,
m=m,
experiment=exp_multitask,
search_space=exp_multitask.search_space,
gr=gr,
)
tr = exp_multitask.new_batch_trial(trial_type="online", generator_run=gr)
tr.run()
online_trials.append(tr.index)
# Extract true objective at each online iteration for creating benchmark plot
obj = np.array([])
con = np.array([])
for tr in online_trials:
df_t = exp_multitask.trials[tr].fetch_data().df
df_tobj = df_t[df_t['metric_name'] == 'objective_noiseless']
obj = np.hstack((obj, df_tobj['mean'].values))
df_tcon = df_t[df_t['metric_name'] == 'constraint_noiseless']
con = np.hstack((con, df_tcon['mean'].values))
return obj, con
Run both Bayesian optimization loops and aggregate results.
runners = {
'GP, online only': run_online_only_bo,
'MTGP': run_mtbo,
}
iteration_objectives = {k: [] for k in runners}
iteration_constraints = {k: [] for k in runners}
for rep in range(n_reps):
print('Running rep', rep)
for k, r in runners.items():
obj, con = r()
iteration_objectives[k].append(obj)
iteration_constraints[k].append(con)
for k, v in iteration_objectives.items():
iteration_objectives[k] = np.array(v)
iteration_constraints[k] = np.array(iteration_constraints[k])
Running rep 0 Online-only batch 0 0.0031158924102783203 Online-only batch 1 37.21125078201294 Online-only batch 2 75.06295323371887 Multi-task batch 0 0.0073070526123046875
--------------------------------------------------------------------------- RuntimeError Traceback (most recent call last) <ipython-input-8-6b4facc37f42> in <module> 8 print('Running rep', rep) 9 for k, r in runners.items(): ---> 10 obj, con = r() 11 iteration_objectives[k].append(obj) 12 iteration_constraints[k].append(con) <ipython-input-7-88fb8b2e972e> in run_mtbo() 51 experiment=exp_multitask, 52 data=exp_multitask.fetch_data(), ---> 53 search_space=exp_multitask.search_space, 54 ) 55 ~/build/facebook/Ax/ax/modelbridge/factory.py in get_MTGP(experiment, data, search_space, trial_index) 262 torch_dtype=torch.double, 263 torch_device=DEFAULT_TORCH_DEVICE, --> 264 status_quo_features=status_quo_features, 265 ) 266 ~/build/facebook/Ax/ax/modelbridge/torch.py in __init__(self, experiment, search_space, data, model, transforms, transform_configs, torch_dtype, torch_device, status_quo_name, status_quo_features, optimization_config) 73 status_quo_name=status_quo_name, 74 status_quo_features=status_quo_features, ---> 75 optimization_config=optimization_config, 76 ) 77 ~/build/facebook/Ax/ax/modelbridge/base.py in __init__(self, search_space, model, transforms, experiment, data, transform_configs, status_quo_name, status_quo_features, optimization_config, fit_out_of_design) 154 search_space=search_space, 155 observation_features=obs_feats, --> 156 observation_data=obs_data, 157 ) 158 self.fit_time = time.time() - t_fit_start ~/build/facebook/Ax/ax/modelbridge/torch.py in _fit(self, model, search_space, observation_features, observation_data) 87 search_space=search_space, 88 observation_features=observation_features, ---> 89 observation_data=observation_data, 90 ) 91 ~/build/facebook/Ax/ax/modelbridge/array.py in _fit(self, model, search_space, observation_features, observation_data) 81 feature_names=self.parameters, 82 metric_names=self.outcomes, ---> 83 fidelity_features=list(target_fidelities.keys()), 84 ) 85 ~/build/facebook/Ax/ax/modelbridge/torch.py in _model_fit(self, model, Xs, Ys, Yvars, bounds, task_features, feature_names, metric_names, fidelity_features) 116 feature_names=feature_names, 117 metric_names=metric_names, --> 118 fidelity_features=fidelity_features, 119 ) 120 ~/build/facebook/Ax/ax/models/torch/botorch.py in fit(self, Xs, Ys, Yvars, bounds, task_features, feature_names, metric_names, fidelity_features) 291 fidelity_features=self.fidelity_features, 292 metric_names=self.metric_names, --> 293 **self._kwargs, 294 ) 295 ~/build/facebook/Ax/ax/models/torch/botorch_defaults.py in get_and_fit_model(Xs, Ys, Yvars, task_features, fidelity_features, metric_names, state_dict, refit_model, **kwargs) 129 # pyre-ignore: [16] 130 mll = ExactMarginalLogLikelihood(model.likelihood, model) --> 131 mll = fit_gpytorch_model(mll, bounds=bounds) 132 return model 133 ~/virtualenv/python3.7.1/lib/python3.7/site-packages/botorch/fit.py in fit_gpytorch_model(mll, optimizer, **kwargs) 65 for mll_ in mll.mlls: 66 fit_gpytorch_model( ---> 67 mll=mll_, optimizer=optimizer, max_retries=max_retries, **kwargs 68 ) 69 return mll ~/virtualenv/python3.7.1/lib/python3.7/site-packages/botorch/fit.py in fit_gpytorch_model(mll, optimizer, **kwargs) 102 mll.model.load_state_dict(original_state_dict) 103 sample_all_priors(mll.model) --> 104 mll, _ = optimizer(mll, track_iterations=False, **kwargs) 105 if not any(issubclass(w.category, OptimizationWarning) for w in ws): 106 mll.eval() ~/virtualenv/python3.7.1/lib/python3.7/site-packages/botorch/optim/fit.py in fit_gpytorch_scipy(mll, bounds, method, options, track_iterations, approx_mll, scipy_objective, module_to_array_func, module_from_array_func) 241 jac=True, 242 options=options, --> 243 callback=cb, 244 ) 245 iterations = [] ~/virtualenv/python3.7.1/lib/python3.7/site-packages/scipy/optimize/_minimize.py in minimize(fun, x0, args, method, jac, hess, hessp, bounds, constraints, tol, callback, options) 608 elif meth == 'l-bfgs-b': 609 return _minimize_lbfgsb(fun, x0, args, jac, bounds, --> 610 callback=callback, **options) 611 elif meth == 'tnc': 612 return _minimize_tnc(fun, x0, args, jac, bounds, callback=callback, ~/virtualenv/python3.7.1/lib/python3.7/site-packages/scipy/optimize/lbfgsb.py in _minimize_lbfgsb(fun, x0, args, jac, bounds, disp, maxcor, ftol, gtol, eps, maxfun, maxiter, iprint, callback, maxls, **unknown_options) 343 # until the completion of the current minimization iteration. 344 # Overwrite f and g: --> 345 f, g = func_and_grad(x) 346 elif task_str.startswith(b'NEW_X'): 347 # new iteration ~/virtualenv/python3.7.1/lib/python3.7/site-packages/scipy/optimize/lbfgsb.py in func_and_grad(x) 293 else: 294 def func_and_grad(x): --> 295 f = fun(x, *args) 296 g = jac(x, *args) 297 return f, g ~/virtualenv/python3.7.1/lib/python3.7/site-packages/scipy/optimize/optimize.py in function_wrapper(*wrapper_args) 325 def function_wrapper(*wrapper_args): 326 ncalls[0] += 1 --> 327 return function(*(wrapper_args + args)) 328 329 return ncalls, function_wrapper ~/virtualenv/python3.7.1/lib/python3.7/site-packages/scipy/optimize/optimize.py in __call__(self, x, *args) 63 def __call__(self, x, *args): 64 self.x = numpy.asarray(x).copy() ---> 65 fg = self.fun(x, *args) 66 self.jac = fg[1] 67 return fg[0] ~/virtualenv/python3.7.1/lib/python3.7/site-packages/botorch/optim/utils.py in _scipy_objective_and_grad(x, mll, property_dict) 274 return float("nan"), np.full_like(x, "nan") 275 else: --> 276 raise e # pragma: nocover 277 loss.backward() 278 param_dict = OrderedDict(mll.named_parameters()) ~/virtualenv/python3.7.1/lib/python3.7/site-packages/botorch/optim/utils.py in _scipy_objective_and_grad(x, mll, property_dict) 267 mll.zero_grad() 268 try: # catch linear algebra errors in gpytorch --> 269 output = mll.model(*train_inputs) 270 args = [output, train_targets] + _get_extra_mll_args(mll) 271 loss = -mll(*args).sum() ~/virtualenv/python3.7.1/lib/python3.7/site-packages/gpytorch/models/exact_gp.py in __call__(self, *args, **kwargs) 263 if not all(torch.equal(train_input, input) for train_input, input in zip(train_inputs, inputs)): 264 raise RuntimeError("You must train on the training inputs!") --> 265 res = super().__call__(*inputs, **kwargs) 266 return res 267 ~/virtualenv/python3.7.1/lib/python3.7/site-packages/gpytorch/module.py in __call__(self, *inputs, **kwargs) 22 23 def __call__(self, *inputs, **kwargs): ---> 24 outputs = self.forward(*inputs, **kwargs) 25 if isinstance(outputs, list): 26 return [_validate_module_outputs(output) for output in outputs] ~/virtualenv/python3.7.1/lib/python3.7/site-packages/botorch/models/multitask.py in forward(self, x) 153 covar_i = self.task_covar_module(task_idcs) 154 # Combine the two in an ICM fashion --> 155 covar = covar_x.mul(covar_i) 156 return MultivariateNormal(mean_x, covar) 157 ~/virtualenv/python3.7.1/lib/python3.7/site-packages/gpytorch/lazy/lazy_tensor.py in mul(self, other) 1145 return self._mul_constant(other.view(*other.shape[:-2])) 1146 -> 1147 return self._mul_matrix(lazify(other)) 1148 1149 def ndimension(self): ~/virtualenv/python3.7.1/lib/python3.7/site-packages/gpytorch/lazy/lazy_tensor.py in _mul_matrix(self, other) 504 other = other.evaluate_kernel() 505 if isinstance(self, NonLazyTensor) or isinstance(other, NonLazyTensor): --> 506 return NonLazyTensor(self.evaluate() * other.evaluate()) 507 else: 508 left_lazy_tensor = self if self._root_decomposition_size() < other._root_decomposition_size() else other ~/virtualenv/python3.7.1/lib/python3.7/site-packages/gpytorch/utils/memoize.py in g(self, *args, **kwargs) 32 cache_name = name if name is not None else method 33 if not is_in_cache(self, cache_name): ---> 34 add_to_cache(self, cache_name, method(self, *args, **kwargs)) 35 return get_from_cache(self, cache_name) 36 ~/virtualenv/python3.7.1/lib/python3.7/site-packages/gpytorch/lazy/lazy_tensor.py in evaluate(self) 880 eye = torch.eye(num_cols, dtype=self.dtype, device=self.device) 881 eye = eye.expand(*self.batch_shape, num_cols, num_cols) --> 882 res = self.matmul(eye) 883 return res 884 ~/virtualenv/python3.7.1/lib/python3.7/site-packages/gpytorch/lazy/interpolated_lazy_tensor.py in matmul(self, tensor) 400 # right_interp^T * tensor 401 base_size = self.base_lazy_tensor.size(-1) --> 402 right_interp_res = left_t_interp(self.right_interp_indices, self.right_interp_values, tensor, base_size) 403 404 # base_lazy_tensor * right_interp^T * tensor ~/virtualenv/python3.7.1/lib/python3.7/site-packages/gpytorch/utils/interpolation.py in left_t_interp(interp_indices, interp_values, rhs, output_dim) 228 else: 229 cls = getattr(torch.sparse, type_name) --> 230 summing_matrix = cls(summing_matrix_indices, summing_matrix_values, size) 231 232 # Sum up the values appropriately by performing sparse matrix multiplication RuntimeError: size is inconsistent with indices: for dim 1, size is 1 but found index 1
We plot the cumulative best point found by each online iteration of the optimization for each of the methods. We see that despite the bias in the offline observations, incorporating them into the optimziation with the multi-task model allows the optimization to converge to the optimal point much faster. With n_reps=50, this generates Fig. S4 of the paper.
best_objectives = {}
for m, obj in iteration_objectives.items():
x = obj.copy()
z = iteration_constraints[m].copy()
best_objectives[m] = np.array([np.minimum.accumulate(obj_i) for obj_i in x])
render(
optimization_trace_all_methods({k: best_objectives[k] for k in runners})
)
Benjamin Letham and Eytan Bakshy. Bayesian optimization for policy search via online-offline experimentation. arXiv preprint arXiv:1603.09326, 2019.
Kevin Swersky, Jasper Snoek, and Ryan P Adams. Multi-task Bayesian optimization. In Advances in Neural Information Processing Systems 26, NIPS, pages 2004–2012, 2013.