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Local Search k-means++ with Foresight

Theo Conrads, Lukas Drexler, Joshua Könen, Daniel R. Schmidt, Melanie Schmidt

TL;DR

This work targets improving practical $k$-means clustering by enhancing initialization and local-search strategies. It introduces Foresight-LS++ (FLS++), a hybrid that interleaves one step of Lloyd's algorithm with LS++ center swaps, guided by d^2-sampling for candidate centers and foresight gained from a Lloyd update. The approach preserves the asymptotic runtime and approximation guarantees while delivering improved solution quality, demonstrated via extensive experiments showing GFLS++ often achieving the best costs on large datasets. The findings highlight the value of combining sampling, local search, and immediate refinement steps, and suggest greedy initialization can further boost performance across methods.

Abstract

Since its introduction in 1957, Lloyd's algorithm for $k$-means clustering has been extensively studied and has undergone several improvements. While in its original form it does not guarantee any approximation factor at all, Arthur and Vassilvitskii (SODA 2007) proposed $k$-means++ which enhances Lloyd's algorithm by a seeding method which guarantees a $\mathcal{O}(\log k)$-approximation in expectation. More recently, Lattanzi and Sohler (ICML 2019) proposed LS++ which further improves the solution quality of $k$-means++ by local search techniques to obtain a $\mathcal{O}(1)$-approximation. On the practical side, the greedy variant of $k$-means++ is often used although its worst-case behaviour is provably worse than for the standard $k$-means++ variant. We investigate how to improve LS++ further in practice. We study two options for improving the practical performance: (a) Combining LS++ with greedy $k$-means++ instead of $k$-means++, and (b) Improving LS++ by better entangling it with Lloyd's algorithm. Option (a) worsens the theoretical guarantees of $k$-means++ but improves the practical quality also in combination with LS++ as we confirm in our experiments. Option (b) is our new algorithm, Foresight LS++. We experimentally show that FLS++ improves upon the solution quality of LS++. It retains its asymptotic runtime and its worst-case approximation bounds.

Local Search k-means++ with Foresight

TL;DR

This work targets improving practical -means clustering by enhancing initialization and local-search strategies. It introduces Foresight-LS++ (FLS++), a hybrid that interleaves one step of Lloyd's algorithm with LS++ center swaps, guided by d^2-sampling for candidate centers and foresight gained from a Lloyd update. The approach preserves the asymptotic runtime and approximation guarantees while delivering improved solution quality, demonstrated via extensive experiments showing GFLS++ often achieving the best costs on large datasets. The findings highlight the value of combining sampling, local search, and immediate refinement steps, and suggest greedy initialization can further boost performance across methods.

Abstract

Since its introduction in 1957, Lloyd's algorithm for -means clustering has been extensively studied and has undergone several improvements. While in its original form it does not guarantee any approximation factor at all, Arthur and Vassilvitskii (SODA 2007) proposed -means++ which enhances Lloyd's algorithm by a seeding method which guarantees a -approximation in expectation. More recently, Lattanzi and Sohler (ICML 2019) proposed LS++ which further improves the solution quality of -means++ by local search techniques to obtain a -approximation. On the practical side, the greedy variant of -means++ is often used although its worst-case behaviour is provably worse than for the standard -means++ variant. We investigate how to improve LS++ further in practice. We study two options for improving the practical performance: (a) Combining LS++ with greedy -means++ instead of -means++, and (b) Improving LS++ by better entangling it with Lloyd's algorithm. Option (a) worsens the theoretical guarantees of -means++ but improves the practical quality also in combination with LS++ as we confirm in our experiments. Option (b) is our new algorithm, Foresight LS++. We experimentally show that FLS++ improves upon the solution quality of LS++. It retains its asymptotic runtime and its worst-case approximation bounds.
Paper Structure (8 sections, 2 theorems, 14 figures, 8 tables, 3 algorithms)

This paper contains 8 sections, 2 theorems, 14 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Algorithms
  3. Experimental Results
  4. Greedy vs. non-greedy $d^2$-sampling
  5. Conclusion
  6. Further datasets and cumulative costs
  7. Cost improvement through greedy initialization
  8. Local search remaining Figures

Key Result

Lemma 1

One iteration of the For-Loop in Lines $3$-$13$ of Algorithm FLS++ can be implemented such to run in time $\mathop{\mathrm{\mathcal{O}}}\nolimits(ndk)$.

Figures (14)

  • Figure 1: Compression of an image with $k=4$ centers (i.e. colors). Subfigure (a) shows the original image. Subfigure (b) shows a local optimum with a $k$-means cost of $55.18\cdot 10^8$. We found this local optimum in runs of Lloyd with uniform initialization and in single runs of $k$-means++. Subfigure (c) shows a solution with a $k$-means cost of $43.09 \cdot 10^8$ (for example found by FLS++).
  • Figure 2: This data set is by Fritzke fritzke2017boxes, the illustration by Conrads C21. The left side shows nine centers sampled by one run of $k$-means++ and the corresponding induced clusters are illustrated by colors. The right side shows how the clusters and centers look after running Lloyd's algorithm to convergence with the nine centers from the left as input.
  • Figure 3: Improving solutions with local search steps.
  • Figure 4: A example with eight optimal clusters (green). When swapping in a new center, the best center to delete is the one in the middle: without it, Lloyd's algorithm can repair the solution.
  • Figure 5: Comparison on two large datasets, for $R=50$ repetitions. GLS++ always performs 25 local search steps. For GFLS++, we display the results for performing 5, 10 and 15 such steps.
  • ...and 9 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Corollary 2