CS264: Beyond Worst-Case Analysis Lecture #6: Clustering in Approximation-Stable Instances

In some optimization problems, the objective function can be taken quite literally. If one wants to maximize profit or accomplish some goal at minimum cost, then the goal translates directly into a numerical objective function. In other applications, an objective function is only a means to an end. Consider, for example, the problem of clustering. Given a set of data points, the goal is to cluster them into “coherent groups,” with points in the same group being “similar” and those in different groups being “dissimilar.” There is not an obvious, unique way to translate this goal into a numerical objective function, and as a result many different objective functions have been studied (k-means, k-median, k-center, etc.) with the intent of making the fuzzy notion of a “good/meaningful clustering” into a concrete optimization problem. In this case, we do not care about the objective function value per se; rather, we want to discover interesting structure in the data. So we’re perfectly happy to compute a “meaningful clustering” with suboptimal objective function value, and would be highly dissatisfied with an “optimal solution” that fails to indicate any patterns in the data (which suggests that we were asking the wrong question, or expecting structure where none exists). The point is that if we are trying to cluster a data set, then we are implicitly assuming that interesting structure exists in the data. This perspective suggests that an explicit model of data could sharpen the insights provided by a traditional worst-case analysis framework (cf., modeling locality of reference in online paging). This lecture begins our exploration of the conjecture that clustering is hard only when it doesn’t matter. That is, clustering