K-means random initialization

Course Q&A Machine Learning Specialization Unsupervised Learning, Recommenders, Reinforcement
1

While taking the quiz, I encountered a question in the image

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The question doesn’t clarify that the iterations are for selecting the correct number of centroids. If we have a fixed value of K and run the K-means algorithm for, say, 100 iterations, I’m confused about why the cost would never increase.

2

Because the centroids are updated on each iteration such that the centroid distance (which relates to the cost) will decrease.

3

In addition, if the centroids are already in an optimal position where moving them would increase the distance to their assigned points, the algorithm will leave the centroids unchanged and thus the cost will remain the same. It may plateau, indicating that the algorithm has reached a local minimum or a stable configuration of centroids where further movement would not reduce the cost, where the algorithm stops if a convergence criterion is met (e.g., cost changes below a certain threshold).