Sampling Plans Many methods requires knowing a series of samples of the objective value to calculate local model or population methods, so… Full Factorial Grid it up. easy to implement good results bad: sample count grows exponentially with dimension Random Sampling Use a pseudorandom generator to pick points in your space. allows for any number of evaluations you specify statistically, the points clump when you do this! also need lots of samples to get good coverage Uniform Projection We take each point, and uniformly project it onto each dimension. To implement this, we grid up each dimension and shuffle the ordering of each dimension individually. Then, we read off the coordinates to create the points:

in d3... seq = range(axis_min, axis_max) d1 = random.shuffle(seq) d2 = random.shuffle(seq) d3 = random.shuffle(seq) sampling_points = zip(d1, d2, d3) Stratified Sampling perform Uniform Projection within each grid, make smaller grids and perform within them Uniform Projection again Space-Filling Metrics Pairwise Distances This requires each set to have the same number of points

figure the euclidian distance between every pair of points for each set of pairs, figure the closest together points, and call that the “pairwise distance” of the set Limitation: if there are just two points that are close together, this metric scores it worse. So maybe Morris-Mitchell. Morris-Mitchell We have a hype-parameter q, which checks all of the possible norms to use between points. Consider d_{i} to be the ith-pairwise distance between the points with the for your choice of p. Then, for:

\begin{equation} \Phi_{q}(X) = \left(\sum_{i}^{}d_{i}^{-q}\right)^{\frac{1}{q}} \end{equation}

and we try to solve for our set of points X such that:

\begin{equation} \min_{X} \max_{q \in \{1,2,5,10,20,50,100\}} \Phi_{q}(X) \end{equation}

“minimize the distance at the worst q possible norm” Space-Filling Subset A Space-Filling Subset is a subset S of a point set X which minimizes the maximum distance between a point in S and its closest point in X (i.e. making S a good representative of X).

\begin{equation} d_{\max}(X,S) = \max_{x \in X} \min_{s \in S} |s -x|_{p} \end{equation}

we can choose any p norm you’d like. greedy local search Choosing one best point to add to S which maximize d_{\max}, and then choose another point, and another one, … exchange algorithm randomly initialize S, and swap points within S and only in X