In this tutorial, we illustrate how to perform robust multi-objective Bayesian optimization (BO) under input noise.

This is a simple tutorial; for support for constraints, batch sizes greater than 1, and many alternative methods, please see https://github.com/facebookresearch/robust_mobo.

We consider the problem of optimizing (maximizing) a vector-valued objective function $\mathbf f(\mathbf x)$ where at implementation time $\mathbf f(\mathbf x)$ is subject to input noise $\mathbf{f}(\mathbf{x} \diamond \mathbf{\xi})$ where $\mathbf{\xi} \sim P(\mathbf \xi | \mathbf x)$ is the random input noise and $\diamond$ denotes the perturbation function (e.g. addition, multiplication, or any arbitrary function).

We consider the scenario where:

- We have access to a simulator during optimization such that $\mathbf{f}$ can be queried at a given design $\mathbf x$ without input noise.
- Input noise is only present at implementation time. After optimization, the design that is chosen according to the decision maker's preferences will be subject to input noise.
- The perturbation function is known.
- We can sample from the generative process $P(\mathbf \xi | \mathbf x)$.

Quantifying risk is important to understand how the final selected design will perform under input noise.

To quantify risk in the multi-objective setting, the MVaR set is an appealing option. For a given design $\mathbf x$, MVaR is theis the set of points such that for every $\mathbf z$ in the MVaR set, $\mathbf z$ is Pareto dominated by the objectives under input noise $\mathbf f (\mathbf x \diamond \mathbf \xi)$ with probability $\alpha$. In other words, if $\mathbf x$ is the chosen final design, the objectives will be better than $\mathbf z$ with probability $\alpha$ for all $\mathbf z$ in the MVaR set.

However, during optimization we are interested in identifying the global MVaR set that is the optimal set of probabilistic lower bounds across all designs. The global MVaR set is the non-dominated set of points across the union of MVaR sets of all points in the design space. See [1] for a deeper discussion.

In this tutorial, we will optimize the 2 1-dimensional functions shown above to identify an approximate global MVaR set. See [1] for a description of these functions.

To do so, we will use Bayesian optimization with MARS (MVaR approximated via random scalarizations). MARS exploits the result in [1] that, under limited assumptions, there is a bijection between weights in the $M-1$-dimensional-simplex (where $M$ is the number of objectives) and points $\mathbf z$ in the MVaR set based on the value-at-risk (VaR) of a Chebyshev scalarization.