Sunday, 15 November 2020 12:00 Written by

Bayesian Optimization (Part 2) Jupyter Notebook

I this second part of this series we introduce the main concept of Bayesian optimization.

Parameter optimization is a very common task in science and engineering. In photonics it is useful e.g. for the shape optimization of photonic structures or for reconstructing shape parameters from measured data. In the following, we consider the problem of minimizing an objective function (e.g. a loss). Of course, if the target is to maximize an objective, one can equivalently minimize the negative objective.

Definition: Minimization problem

Given a $d$-dimensional parameter space $\mathcal{X} \subset \mathbb{R}^d$ and an objective function $f:\mathcal{X}\rightarrow\mathbb{R}$, find parameter values that minimize the objective. I.e. find $$ \mathbf{x}_{\rm min} = {\rm argmin}_{\mathbf{x}\in\mathcal{X}}f(\mathbf{x})$$
Visualization of a minimization problem Visualization of a minimization problem for a function depending on two parameters.

Often, one cannot be sure to find the global minimum of an expensive objective function in a reasonable time. Instead, the practical goals of an optimization can be to:

  • find the lowest possible $f(\mathbf{x})$ in a given time budget.
  • find a value $f(\mathbf{x}) \leq f_{\rm target}$ as fast as possible.

For the sake of concreteness, let us consider a typical optimization task in nanophotonics.

Example: Metasurface

Metasurfaces use sub-wavelength structures in order to tailor the optical behavior of a material interface. One application of metasurfaces is to provide an anti-reflective coating for example in order to improve the performance of CCD or CMOS sensors. We consider the example of optimizing a dielectric metasurface in order to suppress the reflectivity for normal incident light between 500 nm and 800 nm wavelength (the visible spectrum from green to red). The parameterization of the design of the periodic surface is depicted in the figure below. The objective is to minimize the average reflectivity $f(\mathbf{s}) = \frac{1}{N}\sum_i^N r(\mathbf{s},\lambda_i)$, where $r(\mathbf{s},\lambda_i)$ is the reflectivity for normal incident light of wavelength $\lambda$ and for a parameter vector $\mathbf{s} = (p,h_{\rm lower},h_{\rm upper},w_{\rm bottom},w_{\rm middle},w_{\rm top})$.

Visualization of a minimization problem The metasurface consists of a square array of silicon bumps on a silicon substrate. The reflectivity of the structure is minimized with respect to the periodicity of the array ($p$), the heights of the lower and upper part of the bumps ($h_{\rm lower}$ and $h_{\rm upper}$), and the bottom, middle, and top width of the bumps ($w_{\rm bottom}$, $w_{\rm middle}$, and $w_{\rm top}$).

Bayesian Optimization

We can now employ the Gaussian process regression (GPR) introduced in the previous blog to perform an efficient global optimization. The GPR provides for every point $x\in\mathcal{X}$ a prediction in the form of a mean value $\overline{y}(x)$ and a standard deviation $\sigma(x)$. Based on this data, one has to define an acquisition function that determines a score for sampling the expensive black-box function at position $x$. The acquisition function has to make a trade-off between exploration and exploitation. For example, by sampling always at the point of the lowest prediction $\overline{y}(x)$, one can quickly converge into a local minimum (exploitation). However, one converged one would never jump out of the minimum again (exploration). A common choice for an acquisition function, which overcomes this issue, is the expected improvement.

Expected improvement

Suppose, we have drawn $t$ samples of the objective function $\mathcal{D}_t=\{(x_1,y_1),\dots,(x_t,y_t)\}$ and the lowest function value found so far is $y_{\rm min}=\min(y_1,\dots,y_t)$. Then, the improvement for a function value of $y$ is defined as $$I(y) = \left\{ \begin{matrix} 0 & \text{for}\;y>y_{\rm min} \\ y_{\rm min} - y &\text{for}\;y<y_{\rm min} \end{matrix}\right.$$

Now, the GPR provides us with a prediction in form of a probability distribution for $y$ such that one can determine the expected improvement $\alpha_{\rm EI} = \mathbb{E}[I(y)]$. Luckily, the expected improvement evaluates to an analytic expression. Suppose for $x^*$ the mean and standard deviation of the Gaussian process regression is given by $\overline{y}=\overline{y}(x^*|\mathcal{D}_t)$ and $\sigma=\sigma(x^*|\mathcal{D}_t)$. Then, $$ \begin{split} \alpha_{\rm EI}(x^*|\mathcal{D}_t) &= \mathbb{E}[I(y)] = \int I(y) P(y) {\rm d}y\\ &= \int_{-\infty}^{y_{\rm min}} \left(y_{\rm min} - y\right)\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(y-\overline{y})^2}{\sigma^2}\right) {\rm d}y\\ &= \frac{\sigma \exp\left(-u^2\right)}{\sqrt{2\pi}} + \frac{1}{2}\left(y_{\rm min}-\overline{y}\right)\left(1 + {\rm erf}(u)\right) \end{split} $$ with $ u = \frac{y_{\rm min}-\overline{y}}{\sqrt{2}\sigma}$

The Bayesian optimization algorithm consists of repeating the following steps:

Until some convergence criterion is met, repeat for $t=1,2,\dots$:
  1. Find the maximum of an acquisition function $x_t = {\rm argmax}_x \alpha_{\rm EI}(x|\mathcal{D}_{t-1})$.
  2. Compute the objective function value $y_t = f(x_t)$ and possibly also the gradient $g_t = \nabla f(x_t)$.
  3. Update the GPR for the new set of samples $\mathcal{D}_{t} = \mathcal{D}_{t-1} \cup \{(x_t,y_t,g_t)\}$.

The following animation examples show how Bayesian optimization works for a one dimensional objective function. The function is minimized the first time without and the second time with using derivative information.

In [1]:
%matplotlib notebook
from utils import BO_visualization 
In [2]:
%matplotlib notebook
from utils import BO_visualization