This tutorial will show two algorithms available in Ax for multi-objective optimization and visualize they compare to eachother and to quasirandom search.

Multi-objective optimization (MOO) covers the case where we care about multiple
outcomes in our experiment but we do not know before hand a specific weighting of those
objectives (covered by `ScalarizedObjective`

) or a specific constraint on one objective
(covered by `OutcomeConstraint`

s) that will produce the best result.

The solution in this case is to find a whole Pareto frontier, a surface in outcome-space containing points that can't be improved on in every outcome. This shows us the tradeoffs between objectives that we can choose to make.

Optimize a list of M objective functions $ \bigl(f^{(1)}( x),..., f^{(M)}( x) \bigr)$ over a bounded search space $\mathcal X \subset \mathbb R^d$.

We assume $f^{(i)}$ are expensive-to-evaluate black-box functions with no known analytical expression, and no observed gradients. For instance, a machine learning model where we're interested in maximizing accuracy and minimizing inference time, with $\mathcal X$ the set of possible configuration spaces

In a multi-objective optimization problem, there typically is no single best solution. Rather, the *goal* is to identify the set of Pareto optimal solutions such that any improvement in one objective means deteriorating another. Provided with the Pareto set, decision-makers can select an objective trade-off according to their preferences. In the plot below, the red dots are the Pareto optimal solutions (assuming both objectives are to be minimized).

Given a reference point $ r \in \mathbb R^M$, which we represent as a list of M `ObjectiveThreshold`

s, one for each coordinate, the hypervolume (HV) of a Pareto set $\mathcal P = \{ y_i\}_{i=1}^{|\mathcal P|}$ is the volume of the space dominated (superior in every one of our M objectives) by $\mathcal P$ and bounded from above by a point $ r$. The reference point should be set to be slightly worse (10% is reasonable) than the worst value of each objective that a decision maker would tolerate. In the figure below, the grey area is the hypervolume in this 2-objective problem.

The below plots show three different sets of points generated by the qEVHI algorithm with different objective thresholds (aka reference points). Note that here we use absolute thresholds, but thresholds can also be relative to a status_quo arm.

The first plot shows the points without the `ObjectiveThreshold`

s visible (they're set far below the origin of the graph).

The second shows the points generated with (-18, -6) as thresholds. The regions violating the thresholds are greyed out. Only the white region in the upper right exceeds both threshold, points in this region dominate the intersection of these thresholds (this intersection is the reference point). Only points in this region contribute to the hypervolume objective. A few exploration points are not in the valid region, but almost all the rest of the points are.

The third shows points generated with a very strict pair of thresholds, (-18, -2). Only the white region in the upper right exceeds both thresholds. Many points do not lie in the dominating region, but there are still more focused there than in the second examples.

A deeper explanation of our the qEHVI and qParEGO algorithms this notebook explores can be found at https://arxiv.org/abs/2006.05078, and the underlying BoTorch implementation has a researcher-oriented tutorial at https://botorch.org/tutorials/multi_objective_bo.

In [1]:

```
import pandas as pd
from ax import *
import torch
import numpy as np
from ax.metrics.noisy_function import NoisyFunctionMetric
from ax.plot.exp_utils import exp_to_df
from ax.runners.synthetic import SyntheticRunner
# Plotting imports and initialization
from ax.utils.notebook.plotting import render, init_notebook_plotting
from ax.plot.contour import plot_contour
from ax.plot.pareto_utils import compute_pareto_frontier
from ax.plot.pareto_frontier import plot_pareto_frontier
init_notebook_plotting()
# Factory methods for creating multi-objective optimization modesl.
from ax.modelbridge.factory import get_MOO_EHVI, get_MOO_PAREGO
# Analysis utilities, including a method to evaluate hypervolumes
from ax.modelbridge.modelbridge_utils import observed_hypervolume
```