How to model changes in distribution shape?

I have a stochastic optimization algorithm (black box). The algorithm has (4 or 5) tweakable parameters. I want to model how the algorithm's performance changes on a specific optimization task, by setting different initial values for the parameters and recording the results.

Due to the stochastic nature of the algorithm, many "runs" are performed (re-initialization to random start state, and iterate until convergence on solution). The result is a small set of values, usually between 20 and 50 values total, one value per run, where the number of runs depend on how computationally expensive the optimization task is. This distribution of results represent the performance of the algorithm, tuned with this set of initial parameter values, on this particular problem.

If we assume the task is to minimize an error function (with global optimum at 0.0), then a well-tuned algorithm will have many values as close to 0.0 as possible (this is also true if it is an "easy" optimization task). Typically, plotting these recorded values reveal a distribution that (on visual inspection) looks like an exponential distribution.

If the objective function landscape is highly multi-modal (or the algorithm is poorly tuned), then the algorithm might struggle, and only a few "runs" (if any) may find the global optimum. Plotting these recorded values, reveal by visual inspection what resembles a normal distribution. This is confirmed by a test for normality.

Is it possible to build a model to predict how the distribution will change, from an exponential-like distribution, to a normal distribution, based only on a dataset of input parameters to recorded result values?

It doesn't really make sense to me to use a neural network to predict (20-50) values that were obtained through a guided random process. The "data set" of results often include outliers, when the algorithm simply "gets stuck" in local optima, or if the algorithm failed to converge and simply wandered off into bad regions of the search space.

Would it make sense to attempt and describe the distributions using some summary statistics as target "features"?

I have noticed some work on interpolating distributions, and also on quantile regression, but I'm not sure if there is an obvious technique/approach that I'm missing.

Any suggestions?