Bayesian-Optimization

This is the implementation of a new acquisition function for Batch Bayesian Optimization, named Optimistic Expected Improvement (OEI). For details, results and theoretical analysis, refer to the paper titled Distributionally Ambiguous Optimization Techniques for Batch Bayesian Optimization by Nikitas Rontsis, Michael A. Osborne, Paul J. Goulart.

This branch is a cleaned and updated implementation of OEI. The branch GPflow-based includes code for testing against the following batch acquisition functions:

• Multipoint expected improvement (qei.py)
• Multipoint expected improvement, Constant Liar Heuristic (qei_cl.py)
• Batch Lower Confidence Bound (blcb.py)
• Local Penalization of Expected Improvement (lp_ei.py)

Refer to the above mentioned paper for references and more detailed descriptions.

Please use this branch for OEI as the one in the GPflow-based branch is outdated and only intended for the other acquisition functions (QEI, QEI-Cl, BLCB and LP_EI).

Installation

This package was written in Python 3.6. It is recommended to use the Anaconda >=5.0.1 distribution, on a empty environment. The package is built on top of gpflow 0.5 which has to be installed from source. Then, proceed on installing the following

conda install numpy scipy matplotlib pyyaml

You should also install SCS, a Convex Solver. Do compile the package (--no-binary :all: flag below), as this can bring a significant speedup.

pip install 'scs>=2.0.2' --no-binary :all:

Moreover, you should install a Python interface to the Intel MKL Pardiso Linear Solver (freely distributed by the Anaconda distribution):

conda install -c haasad pypardiso

Finally, you have to download the second order non-linear solver KNITRO, install it and copy its contents in a folder named knitro inside this package. KNITRO is proprietary, but time-limited academic and commercial licenses are provided for free by Artelys.

Testing

Unit tests are included in the tests folder. You can invoke them using pytest. To do so, the following additional packages are required:

conda install pytest
pip install 'cvxpy<1' numdifftools
conda install -c mosek mosek

Usage example

Running Batch Bayesian Optimization (BO) for the Hartmann-6d function with the OEI acquisition function can be invoked as following:

python run.py --seed=123 --algorithm=OEI --function=hart6 --batch_size=20 --initial_size=10 --iterations=15 --noise=1e-6

The above will create an output saved in the folder named out. Detailed logging is saved in the log folder, the verbosity of which can be controller by the logging.yaml configuration file.

To compare against random search and plot the results run:

python run.py --seed=123 --algorithm=Random --function=hart6 --batch_size=20 --initial_size=10 --iterations=15 --noise=1e-6 --nl_solver=knitro --hessian=1

python plot.py hart6 out/hart6_OEI out/hart6_Random 

This will save a pdf plot in the results folder.

The user can define a different function for optimization by modifying benchmark_function.py, or run any of the ones presented in the paper, the definitions of which can be found here.

Basic files

File	Description
run.py	Script that runs BO on test functions defined in benchmark_functions.py.
plot.py	Plots results of run.py.
methods/oei.py	Definition of OEI, its gradient and hessian.
methods/bo.py	A parent class that implements a simple parametrizable BO setup.

Timing results

The main operations for the calculating value of OEI, its gradient and hessian are listed below:

• Value and gradient
  • Solving the Semidefinite Optimization Problem (relevant line).
• Hessian (given solution of SDP):
  • Calculating the derivatives of M (dominant operation) called from here.
  • Chain rules of tensorflow (relevant line #1, relevant line #2).

In the current implementation a considerable amount of time is spent in Python to create, reshape and move matrices, all of which could have been avoided in a more efficient version.