Two-Point Local Optimality in $k$-Means via Boundary-Point Screening
Lloyd's algorithm and the discrete local (D-local) optimization method (Li et al., 2025) for $k$-means provide only weak local-optimality guarantees, and their solution quality remains sensitive to initialization. In this paper, we introduce $r$-point local optimality, under which no reassignment of at most $r$ samples decreases the objective function, and focus on $r=2$. The main computational obstacle is the $\mathcal{O}(n^2(k^2+d))$ cost of exhaustive two-point certification for $n$ samples in $d$ dimensions and $k$ clusters. To address this challenge, we prove that (i) every improving two-point move of a D-local optimum must involve a cluster shared by both reassignments, and (ii) only certificate-defined boundary points can participate in an improving pair. Exploiting this structure, we propose Boundary-Point-Screened Two-Point Local Search (BPS-2PLS), which terminates at a two-point local optimum. For fixed $k,d$ and nonvanishing cluster occupancy, the number $m$ of retained candidates satisfies $m=\mathcal{O}_{\mathbb{P}}(\log n)$ under i.i.d. sampling from a bounded-support distribution with bounded density or from a Gaussian mixture. Across twelve benchmarks, BPS-2PLS attains the lowest available mean WCSS on ten. In a subsampling study, screening retains 0.10% to 2.81% of samples on average at the largest tested sizes. The code is available at https://github.com/lwl-learning/BPS-2PLS.