Visualizing the diversity of a Pareto front approximation using RadVis

During many-objective search we seek to find approximate Pareto fronts that are convergent, meaning they are close to the “true” Pareto front, and diverse, meaning they contain solutions that they uniformly cover the full range of the “true” front. This post will focus on examining the latter criteria for high dimensional problems using Radial Visualization (RadVis; Hoffman et al., 1997), a visualization technique that transforms a solution set into radial coordinates. I’ll discuss the theory behind RadVis, provide some example code, and discuss potential applications, limitations and extensions.

Background

When performing runtime diagnostics (a subject of two of my recent posts, which you can find here and here), we often asses MOEA performance using hypervolume because it captures both the convergence and diversity of the approximate Pareto front. Though tracking hypervolume provides us with an aggregate measure of diversity, it does not provide information on how the solutions are spread across the objective space. For low dimensional problems (two or three objectives), the diversity of the approximation set can easily be discerned through visualization (such as 2-D or 3-D scatter plots). For many-objective problems however, this becomes challenging. Tools such as parallel coordinate plots, bubble plots and glyph plots can allow us to visualize the solutions within an approximation set in their full dimensionality, but they are limited in their ability to convey information on diversity in high dimensional space. An alternative strategy for examining diversity within a solution set is to transform the objective space into a form coordinate system that highlights diversity. One common transformation is to view the set using radial coordinates.

RadVis

RadVis uses an analogy from physics to plot many dimensional data. Each of n-objectives are plotted uniformly around the circumference of a unit circle while, each solution is represented as a point that is attached to all objectives by a set of n hypothetical springs (one spring is attached to each objective-solution pair). A solution’s objective value for the ith objective defines the spring constant exerted by the ith spring. A point’s location thus represents the equilibrium point between all springs (Hoffman et al., 1997). For example, a point that has equal objective values for all n-objectives will lie directly in the center of the plot, while a point that maximizes one objective and minimizes all others will lie on the circumference of the circle at the location specified for the maximized objective.

Below, I’ve plotted RadVis plots for a 5 objective instance of DTLZ2 at runtime snapshots from 1,000 function evaluations (NFE), 5,000 NFE, 10,000 NFE and 50,000 NFE. Notice how the approximate front for 1,000 NFE is sparse and skewed toward one side, while the front for 50,000 NFE is fairly uniform and centered. DTLZ2 is a test problem with a known true Pareto front which is continuous and uniformly distributed, using this information, we can see that the algorithm has likely improved its approximation set over the course of the run.

RadVis representations of the approximate Pareto set of a 5 objective instance of the DTLZ2 test problem. Each subplot represents a different runtime snapshot.

I coded this example using the Yellowbrick machine learning visualization toolkit from sklearn. Code for this example has been added to my runtime diagnostics Github repository, which can be found here.

Caution! RadVis provides no information on convergence

It’s important to note that RadVis provides no information about solution convergence since the location of each point is dependent on equilibrium between all objectives. A set of solutions that performs poorly across all objectives will look similar to a set of solutions that performs well across all objectives. For this reason, RadVis should never be used alone to assess the quality of a Pareto front approximation. To include convergence information in RadVis, Ibrahim et al,. (2016) and Ibrahim et al., 2018 have proposed 3d-RadVis and 3D-RadVis antenna, extensions of RadVis methodology which augment Radvis by providing convergence information to decision makers. These methodologies may be subject to future posts.

References

Hoffman, P., Grinstein, G., Marx, K., Grosse, I., & Stanley, E. (1997, October). DNA visual and analytic data mining. In Proceedings. Visualization’97 (Cat. No. 97CB36155) (pp. 437-441). IEEE.

Ibrahim, A., Rahnamayan, S., Martin, M. V., & Deb, K. (2016, July). 3D-RadVis: Visualization of Pareto front in many-objective optimization. In 2016 IEEE Congress on Evolutionary Computation (CEC) (pp. 736-745). IEEE.

Ibrahim, A., Rahnamayan, S., Martin, M. V., & Deb, K. (2018). 3D-RadVis Antenna: Visualization and performance measure for many-objective optimization. Swarm and evolutionary computation, 39, 157-176.

Beyond Hypervolume: Dynamic visualization of MOEA runtime

Multi-objective evolutionary algorithms have become an essential tool for decision support in water resources systems. A core challenge we face when applying them to real world problems is that we don’t have analytic solutions to evaluate algorithmic performance, i.e. since we don’t know what solutions are possible before hand, we don’t have a point of reference to assess how well our algorithm is performing. One way we can gain insight into algorithmic performance is by examining runtime dynamics. To aid our understanding of the dynamics of the Borg MOEA, I’ve written a small Python library to read Borg runtime files and build a dynamic dashboard that visualizes algorithmic progress.

The Borg MOEA produces runtime files which track algorithmic parameters and the evolving Pareto approximate set over an optimization run. Often we use these data to calculate performance metrics, which provide information on the relative convergence of an approximation set and the diversity of solutions within it (for background on metrics see Joe’s post here). Commonly, generational distance, epsilon indicator and hypervolume are used to examine quality of the approximation set. An animation of these metrics for the 3 objective DTLZ2 test problem is shown in Figure 1 below.

Figure 1: Runtime metrics for the DTLZ2 test problem. The x-axis is number of function evaluations, the y-axis is the each individual metric

While these metrics provide a helpful picture of general algorithmic performance, they don’t provide insight into how the individual objectives are evolving or Borg’s operator dynamics.

Figure 2 shows a diagnostic dashboard of the same 3 objective DTLZ2 test problem run. I used the Celluloid python package to animate the figures. I like this package because it allows you to fully control each frame of the animation.

Figure 2: DTLZ2 runtime dynamics. The tree objectives are shown in a scatter plot and a parallel axis plot. The third figure plots hypervolume over the optimization run and the bottom figure shows Borg’s operator dynamics. (a higher resolution version of this file can be found here: https://www.dropbox.com/s/0nus5xhwqoqtghk/DTLZ2_runtime.gif?dl=0)

One thing we can learn from this dashboard is that though hypervolume starts to plateau around 3500 NFE, the algorithm is still working to to find solutions that represent an adequately diverse representation of the Pareto front. We can also observe that for this DTLZ2 run, the SPX and SBX operators were dominant. SBX is an operator tailored to problems with independent decision variables, like DTLZ2, so this results make sense.

I’m planning on building off this dashboard to include a broader suite of visualization tools, including pairwise scatter plots and radial plots. The library can be found here: https://github.com/dgoldri25/runtimeDiagnostics

If anyone has suggestions or would like to contribute, I would love to hear from you!