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. The objective is treated as unknown and are modeled with separate GPs. The objective is noisy.

Throughout the optimization we can make nosiy observations directly of the objective (an online observation), and we can make noisy observations of a biased version of the objective (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.

In [1]:

```
import os
import time
from copy import deepcopy
from typing import Optional
import numpy as np
import torch
from ax.core.data import Data
from ax.core.experiment import Experiment
from ax.core.generator_run import GeneratorRun
from ax.core.multi_type_experiment import MultiTypeExperiment
from ax.core.objective import Objective
from ax.core.observation import ObservationFeatures, observations_from_data
from ax.core.optimization_config import OptimizationConfig
from ax.core.parameter import ParameterType, RangeParameter
from ax.core.search_space import SearchSpace
from ax.metrics.hartmann6 import Hartmann6Metric
from ax.modelbridge.factory import get_sobol
from ax.modelbridge.registry import Models, MT_MTGP_trans, ST_MTGP_trans
from ax.modelbridge.torch import TorchModelBridge
from ax.modelbridge.transforms.convert_metric_names import tconfig_from_mt_experiment
from ax.plot.diagnostic import interact_batch_comparison
from ax.runners.synthetic import SyntheticRunner
from ax.utils.common.typeutils import checked_cast
from ax.utils.notebook.plotting import init_notebook_plotting, render
init_notebook_plotting()
```

[INFO 09-23 20:40:27] ax.utils.notebook.plotting: Injecting Plotly library into cell. Do not overwrite or delete cell.

[INFO 09-23 20:40:27] ax.utils.notebook.plotting: Please see (https://ax.dev/tutorials/visualizations.html#Fix-for-plots-that-are-not-rendering) if visualizations are not rendering.