BOSS.jl

BOSS stands for "Bayesian Optimization with Semiparametric Surrogate". BOSS.jl is a Julia package for Bayesian optimization. It provides a compact way to define an optimization problem and a surrogate model, and solve the problem. It allows changing the algorithms used for the subtasks of fitting the surrogate model and optimizing the acquisition function. Simple interfaces are defined for the use of custom surrogate models and/or algorithms for the subtasks.

Bayesian Optimization

Bayesian optimization is an abstract algorithm for blackbox optimization (or active learning) with expensive-to-evaluate objective functions. The general procedure is showcased by the pseudocode below. [1]

The real-valued acquisition function $\alpha(x)$ defines the utility of a potential point for the next evaluation. Different acquisition functions are suitable for optimization and active learning problems. We maximize the acquisition function to select the most useful point for the next evaluation.

Once the optimum $x_{n+1}$ of the acquisition function is obtained, we evaluate the blackbox objective function $y_{n+1} \gets f(x_{n+1})$ and we augment the dataset.

Finally, we update the surrogate model according to the augmented dataset.

See [1] for more information on Bayesian optimization.

Optimization Problem

The problem is defined using the BossProblem structure and it follows the formalization below.

We have some (noisy) blackbox objective function

\[y = f(x) = f_t(x) + \epsilon \;,\]

where $\epsilon \sim \mathcal{N}(0, \sigma_f^2)$ is a Gaussian noise. We are able to evaluate $f(x_i)$ and obtain a noisy realization

\[y_i \sim \mathcal{N}(f_t(x_i), \sigma_f^2) \;.\]

Our goal is to solve the following optimization problem

\[\begin{aligned} \text{max} \; & \text{fit}(y) \\ \text{s.t.} \; & y < y_\text{max} \\ & x \in \text{Domain} \;, \end{aligned}\]

where $\text{fit}(y)$ is a real-valued fitness function defined on the outputs, $y_\text{max}$ is a vector defining constraints on outputs, and $\text{Domain}$ defines constraints on inputs.

Active Learning Problem

The BOSS.jl package currently only supports optimization problems out-of-the-box. However, BOSS.jl can be adapted for active learning easily by defining a suitable acquisition function (such as information gain or Kullback-Leibler divergence) to use instead of the expected improvement (see AcquisitionFunction). An acquisition function for active learning will usually not require the fitness function to be defined, so the fitness function can be omitted during problem definition (see BossProblem).

Surrogate Model

The surrogate model approximates the objective function based on the available data. It is defined using the SurrogateModel type. The BOSS.jl package provides a Parametric model, a Nonparametric model, and a Semiparametric model combining the previous two.

The predictive distribution of the Parametric model

\[y \sim \mathcal{N}(m(x; \hat\theta), \hat\sigma_f^2)\]

is given by the parametric function $m(x; \theta)$, the parameter vector $\hat\theta$, and the estimated evaluation noise deviations $\hat\sigma_f$. The model is defined by the parametric function $m(x; \theta)$ together with parameter priors $\theta_i \sim p(\theta_i)$. The parameters $\hat\theta$ and the noise deviations $\hat\sigma_f$ are estimated based on the current dataset.

The Nonparametric model is just an alias for the GaussianProcess model. Gaussian process (GP) is a nonparametric model, so its predictive distribution is based on the whole dataset instead of some vector of parameters. The predictive distribution is given by equations 29, 30 in [1]. The model is defined by defining priors over all its hyperparameters (length scales, amplitudes).

The Semiparametric model combines the previous two models. It is a Gaussian process, but uses the parametric model as the prior mean of the GP (the $\mu_0(x)$ function in [1]). An alternative way of interpreting the semiparametric model is that it fits the data using a parametric model and uses a Gaussian process to model the residual errors of the parametric model. The model is defined by defining both the Parametric and Nonparametric models.

References

[1] Bobak Shahriari et al. “Taking the human out of the loop: A review of Bayesian optimization”. In: Proceedings of the IEEE 104.1 (2015), pp. 148–175