New Algorithms for the Learning-Augmented k-means Problem
Authors: Junyu Huang
Qilong Feng
Ziyun Huang
Zhen Zhang
Jinhui Xu
Jianxin Wang
Venue: ICLR 2025 Poster (2025)
Keywords: Learning-Augmented Clustering; Approximation Algorithm;
Primary area: optimization

Abstract: In this paper, we study the clustering problems in the learning-augmented setting, where predicted labels for a d-dimensional dataset with size m are given by an oracle to serve as auxiliary information to improve the clustering performance. Following the prior work, the given oracle is parameterized by some error rate α, which captures the accuracy of the oracle such that there are at most α fraction of false positives and false negatives in each predicted cluster. In this setting, the goal is to design fast and practical algorithms that can break the computational barriers of inapproximability. The current state-of-the-art learning-augmented k-means algorithm relies on sorting strategies to find good coordinates approximation, where a (1+O(α))-approximation can be achieved with near-linear running time in the data size. However, the computational demands for sorting may limit the scalability of the algorithm for handling large-scale datasets. To address this issue, in this paper, we propose new algorithms that can identify good coordinates approximation using sampling-based strategies, where (1+O(α))-approximation can be achieved with linear running time in the data size. To obtain a more practical algorithm for the problem with better clustering quality and running time, we propose a sampling-based heuristic which can directly find center approximations using sampling-based strategies. Empirical experiments show that our proposed methods are faster than the state-of-the-art learning-augmented k-means algorithms with comparable performances on clustering quality.

Published as a conference paper at ICLR 2025
NEW ALGORITHMS FOR THE LEARNING-AUGMENTER MEMANS PROBLEM
Junyu Huang
Qilong Feng
School Computer Science and Engineering,
School of Computer Science and Engineering,
Central South University
Central South University
Xiangjiang Laboratory
Xiangjiang Laboratory
Changsha, China
Changsha, China
junyuhuangcsuu foxmail com
csufeng@nne ssafeendmail..cu.euuu.rr cdu.
Ziyun Huang
Zhen Zhang
Department of Computer Science and Software Engineering, Penn State Erie, The Behrend College Erie, United States Exh201@psu.ed
School of Advanced Interdisciplinary Studies, Human University of Technology and Business
Xiangjiang Laboratory
Changsha, China
csuz2 foxmail.coomm com
Jinhui Xu
Jianxin Wang
School Information Science and Technology, University of Science and Technology of China Hefei, Anhui, China
School of Computer Science and Engineering, Central South University
Hunan Provincial Key Lab Bioinformatics Xiangjiang Laboratory
jbxu000 1exu000@mmill com
Changsha, China
jxwang@ma csu edu. cn
ABSTRACT
In this paper, we study the clustering problems in the learning-augmented setting, where predicted labels for d-dimensional dataset with size are given by an oracle to serve as auxiliary information to enhance the clustering performance. Following the prior work, the given oracle parameterized by some error rate a, which captures the accuracy the oracle such that there are at most a fraction of false positives and false negatives in each predicted cluster. In this setting, the goal is to design fast and practical algorithms that can break the computational barriers of inapproximability for clustering problems. The current state-of-the-art k-means algorithm relies on sorting strategies to find coor- dinates approximation, where a (1 ((a))--aproximmatio can achieved with near-linear running time in the data size. However, the sorting process may limit the scalability of the algorithm for handling large-scale datasets. To address this issue, in this paper, we propose new algorithms that can identify good coordinates approximation using sampling-bee can be achieved with linear running time in the data size. To obtain more practi- cal algorithm for the problem with better clustering quality and running time, we propose sampling- based heuristic which can directly find center approximations for each cluster. Empirical experiments show that our proposed methods are faster than the mate-of-the-ar learning augmented k-means algorithms with comparable performances on clustering quality.
strategies, where (1 + O(a) ((co)))appoxxiaatioon
"Corresponding Authors
The research of Jinhui Xu was partially done the State University of New York Buffalo.

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INTRODUCTION
Clustering is fundamental unsupervised learning task that has been extensively studied over the past decades. Among different clustering objectives, one of the most commonly used clustering formulations is the k-means clustering. In the k-means clustering, we are given set / of data points in d-dimensional Euclidean space, and the goal is to compute a set C' C Id of centers with size at most such that the sum of the squared distances between data points in to their closest centers in C is minimized. The k-means problem has received significant attention in the literature and is proven to NP-hard (Dasgupta 2008). Furthermore, Johen-Addat and Karthik (Cohen-Addad & Karthik 2019) showed that even finding a solution for k-means with approximation ratio smaller than 1.07 is NP-hard. The current best approximation ratio for the k-means problem is 5.912 (Cohen-) Addad et al. 2022). which is based on primal-dual and nested cuasi-independen set methods. For fixed dimensionality d or the number of clusters k, several (1 + c)-approximation algorithms were known @Jaswwll al. 2014, FFrigsstad et al. 2019). However, these algorithms with relatively tight approximation guarantees do not scale well for large-scale datasets. Thus, several practical methods with linear running time in the data size have been proposed, such as the Olog -ppprooxxnntiion k- means+++ method (Arthur & Vassil 2007) and the 0(1 (1)aapprxximation local search methods (Lattanzi& & Sohler 2019/Berrtta Beretta et 2023 Fan et al. 2023 Choo et al. 2020). Although these linear-time algorithms have been widely used in practice, their large approximation ratios could potentially deteriorate the clustering performance in scenarios that require high-quality solutions.
To overcome the barrier of inapproximability and develop more practical approximation algorithms, a series of studies has focused on algorithms augmented with predictions (Mitzenmacher Vassil- vitskii 2022; |Ashtiani et al., 2016;] Kraska et al.) 2018 Mitzenmacher, [2018). For the clustering problem, Gamlath et al. (Gamlath et al.] 2022) proposed clustering recovery with noisy labels, where predicted clustering labels are given additional information to aid the clustering process. The given predictor is parameterized by some error rate a E [0, 1), such that the size of the symmet- ric difference between the predicted cluster and its corresponding optimal cluster is bounded by times the size of the optimal cluster. Based on this model, Gamlath et al. developed a (1 + ((a))- approximation algorithm with polynomial running time, assuming fixed l: and d. Moreover, to guarantee the approximation, the error rate a is required to satisfy the condition that a < 1/4. Er- gun al. (Ergun et al., 2021) introduced another learning-augmented clustering model, aiming at designing fast and practical algorithms for clustering. In their setting, a given predictor also provides auxiliary information for each data point in the form of predicted clustering labels. However, the predictor's 'ss reliability is parameterized by an error rate a E [0, 1), ensuring that there are at most an a fraction of false positives and false negatives in each predicted cluster. Based on this model, an improved (1 + (ap))))approiimaaiii can be achieved with near-linear running time in the data size.
clustering problem proposed by Ergun clustering problem is as follows. From a theoretical perspective, reaming--augmente k-means can overcome the inapproximability barriers, enabling the development of algorithms that can achieve tight clustering quality guarantees with strong scalability. From a practical perspective, as pointed out by Ergun et al. (2021), reliable predictors are available for a wide range of natural datasets. For instance, in datasets with training labels, these labels can act as auxiliary information to enhance the clustering quality on testing datasets. Furthermore, when the clustering instances adhere to specific distributions, the efficient acquisition of a robust predictor is proved to be feasible (Ergun et al. (021). Even if the instances not generated from such distributions, empirical evidence (Ergun et al. 202 I Nguyen et al., 2022) suggests that labels produced by the existing clustering methods, such as k-means++ (Arthur & Vassilvitskii 2007) or heuristic approaches (Lloyd 1982). can serve as good predictors. However, as pointed out in Nguyen et al. (2022), even when the auxiliary labeling partition (or the predictor) nearly optimal, a single false positive located far from the true clustering centers can significantly influence the clustering structures. Therefore, a key challenge in the baaminng-uggmet problem is to design robust algorithms that can minimize the impact of false positives.
In this paper, we mainly focus on the
et al. (2021). The motivation for studying the
clustering
Based on statistical methods, Ergun et al. (Ergun et al. 2021 proposed a randomized algorithm that can achieve a (1 + 20a))=aaproximattoon in time O(md log m), where m and are the data size and dimension, respectively. However, to guarantee the approximation loss. it is required that the label error rate a should be bounded by a E [10 logm/ ym, 1/7], and each optimal cluster should have size ((k/a). To overcome the label error rate and cluster size constraints, Nguyen et al. 2022) pro-
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posed deterministic searching method, where improved (1+ ((a) approximation result with better approximation guarantees can be obtained in time Ommdlog m) for a E [0. 1/2). The current clustering algorithms (Ergun al. 2021 Nguyen et al., 2022) primarily rely on sorting strategies to approximate the optimal clustering centers. However, since sorting processes require near-linear running time in the data size, this can limit the scalability of the existing algo- rithms for handling large-scale datasets. Moreover, it is well known that comparison-basee sorting has lower bound running time of Oml l)) As result, achieving a time complexity better than O log m) using sorting- based strategies is a non-trivial task. It is also worth mentioning that the time complexity of Ergun al. (2021) and Nguyeen et al. (2022) cannot be further improved through dimensionality reduction techniques, such as the JL-method. As pointed out in Theorem 3.4 of Ergun et al. (2021). the JL-method embeds a clustering instance from Rd into space of dimension Ollog v.) and Ollog k)) in (md log m.) and polynomial time, respectívely. Thus, the total running time of using JL-method is still at least O(md log m.) A central challenge in design- ing faster algorithms for the learning-augmentee augmented k-mmans problem is to efficiently approximate the optimal clustering centers in each dimension without using sorting- -baieg strategies.
1.1 OUR CONTRIBUTION
In this paper, we aim to further enhance the efficiency of learning-augmentted clustering algorithms by proposing new sampling-based methods. A key challenge here is to identify high-quality coor- dinates without relying on sorting-based strategies. To overcome this challenge, we first propose a sampling-basee algorithm called Fast-Sampling. For each dimension of each predicted cluster, Fast-Sampling can identify intervals that contain coordinates sufficiently close to the coordinates of the optimal clustering centers. High-quality coordinates are then constructed within linear run- ning time in the data size through fine-grained divisions of the intervals. With these techniques, (1 + D(a))--pproximaae solution for the learning-augmentee k-means problem can obtained in time (md log(kd)). To further improve the running time and eliminate the additional O(log (kd)) term, we propose another algorithm called tast-Estimttoon approximation by designing estimators that provide accurate clustering cost estimates in sublinear time, assuming a bounded aspect ratio (the ratio of the maximum to minimum pairwise distances) in each dimension. Then, the constructed estimators can be used to guide the coordinates selection process, yielding (1 + ((aa)aaappooiimaaaaaaaaaaaaaaaaa approximation in time (md) + Ö(d//a) a
accelcrates coordinate
While the proposed cooplinn-basedd algorithms can achieve faster running time compared to other leamnig--uggmented k-means methods, the improvements come with slight compromises in cluster- ing quality guarantees. To benefit more from the sampling-base algorithm (called Fast-Filtering) to better preserve the clustering quality while maintaining the im- proved running time. Instead of enumerating all the dimensions of the predicted clusters for coordi- nates approximation, the Fast-Filterigg algorithm directly identifies approximate clustering centers by constructing estimators that can provide accurate clustering cost estimates in sub-linear running time. By filtering out data points that are far from the selected centers, the Fast-Filtering algorithm can obtain high-quality clustering centers while avoiding the loss in clustering quality that arises from integrating coordinate information across all the dimensions.
strategies, we propose a heuristic
Table I provides detailed comparison of the results for the learning- -ugginnted k-means problem. In the Appendix, we also give a plot (Figure in Appendix) of approximation ratios vs. the error rate It can be seen from the table that the current best result achieves (1 +1+22 ((x)) aaprrxxixxiiiiio with a E [0, 1/2) (Nguyen et al. 2022). Both of our proposed algorithms, Estimation, achieve linear running time in the data size. Compared to the state-of- ofhhh--artrtt results, the Fast-Sampling algorithm can achieve nearly the same theoretical guarantee on clustering quality (i.e., (1 + approxxoppoiiiaooo while providing faster running time when the number of clusters k and the dimensionality d are much smaller than data size m, which is natural in real-world ap- plications. Furthermore, our Estimaattnoo algorithm provides much faster running time while maintaining clustering quality guarantees comparable to other -aunenn- algorithms.
SaspliSa and Fast-
'Throughout this paper, use Ō()) notations to suppress polylog(m, (m, d) factors.
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Table 1: Comparison results of learning augmented k-means algorithms
Methods and References
Approximation Ratio
Label Error Range
Time Complexity
Partitioning and Sorting Ergun et al.
2021
1+ 20c
p 1/7]
Omndlog m)
Sorting (Nguyen et 2022
+rra
a
201l oe
0., 1/2)
O m)
Fast- Tast-Sampling (Ours)
1 1+ra a 1+1=a
2010
(0.1/2)
Fast Estimation (Ours)
(0. 1/3
((md] + Öle (od/a)))
PRELIMINARIES
We use P C IRd and k denote the given dataset and the number of clusters, respectively. Let m be the data size. For any two points p. q E Rd, let o(, q) and 6'(p.) q) denote their distance and squared distance, respectively. Given point p IR and a set C = {ci C2,..., Ck) of centers, we also use 0p, C) = mineec 8(p,c) c) to denote the distance from to its closest center in C. We use C* = [cT, ) and P(C*) = {Pi:... ...., to denote the set of the optimal clustering centers and the optimal clustering partition, respectively. For each of E C", Of can be represented by coordinates, i.c. of = -(5, Cgy,... G.a) We define the clustering cost of P with respect to C as (
o (x, Given collection C(P) {P1, P2. .... P&3 as the predictor, let Qi = P, be the set of data points in P, that belong to P.". Denote the projections of data points in P. and Q, onto the j-th dimension as P.y and Qıy, respectively. Let P be the projections of data points in P,' onto the j-th dimension. Let m. and m be the size of P. and P, respectively. For a set V C I of data points, let V be the geometric center for V. Denote P(j) as the projections of points in P onto the j-th dimension. Let A max be the maximum aspect ratio of the projected data points,
i... Amax
max IS) miDg..CP().. v
wEm() 6(2,8)
For a positive integer t, let [t] be the set [1,2,... t}.
dearning-Auumenterr k-means Problem. Given a dataset P C Rd of m points, let C* and P(C*) PPi. P?.....PP PF) be arbitrary optimal solution and its corresponding partition, spectively. In the Garning--augenterr setting, it is assumed that we have access to a predictor in the form of labeling partition (PP) [PI. P2. 2,... Pa] parameterized by some label error rate k-means clustering is to find a set C C Rd of data points as centers such that 8² C)) is minimized.
E [0, 1) such that P. PT > (1-a) ma. IP? The goal of
The following lemmas are folklore in k-means clustering.
Lemma Arthur & Vassilvitskii (007)) Given set X C Rdd with size and arbitrary data point c E R., it holds that (x(X,e) = (X..) + m. (c,
Lemma 2
Nguyen et al. (2022)) Given set J C R. let J1 S J be a subset of J with |J1] 2
(1 - () where (0 < Ç < 1). Then, it holds that (J, S (
Lemma 3 Nguyen et al. (002)) Given a set X C Ed and an a E (0,1), let X Then, it holds that (2 a (xxx)
arg XngniixxxxxX 0²
3 THE FAST-SAMPLING ALGORITHM
The general idea of our Fast-Sampling algorithm to efficiently approximate the optimal cluster- ing centers in each dimension by identifying high-quality coordinates without using sorting-based strategies. The primary technical challenge lies in handling the false negatives without significantly compromising the approximation guarantees. Although directly sampling a small subset of coor- dinates from each dimension for the predicted clusters can help identify points near the optimal clustering centers, the uniformly sampled coordinates may not accurately approximate the optimal centers, potentially leading to a constant-factor loss in the approximation guarantees. To address this issue, the Fast-Sampling algorithm first identifies candidate coordinates close to the coordinates of each optimal clustering center within linear running time in the data size. Then, the constructed can- didate coordinates are used define intervals that can precisely capture the positions of the optimal centers, enabling a better approximation through fine- ine-grained divisions of these intervals.
Note that it is common to assume that A Amx can be bounded polynomial function of the data size.
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The proposed Fast-Sampling algorithm (see Algorithm 1) mainly consists of the following two phases: (1) interval estimation (steps 3-6 of Algorithm 1); (2) candidate coordinates construction (step of Algorithm 1). In the interval estimation phase, for each dimension of the predicted clus- ters, the interval lengths are estimated through random sampling strategies. 'he samples are then symmetrically adjusted based on the interval length estimates construct intervals that enclose the coordinates of the optimal centers. In the second phase, the derived intervals are further divided into smaller ones, with each corresponding to new candidate coordinate, enabling a fine line-grained approximation of the optimal clustering centers. In the following, we provide a detailed analysis for the proposed algorithm. Due space limits, all the proofs are delivered to the Appendix.
Algorithm Fast-Sampling
Input: A k-means instance (P, k, d), set (P1.... Ps) of partitions with error rate a, and param- eter e E (0, I].
Output: A set C C IR of centers with CI k.
1: for E [k] do
2: for j E [d] do
3: Randomly and independently sample a set Uij from Py with size (log(kd)))
4: for u E Uij do
5:
Let Nef (u) be the set of the nearest (1 - -a)l cooridnates in Ps to u.
25 (w), (w)
6:
4i V
(1-a)
7:
su) E' A [- [] nz). where e' v
8:
U// = UnEVE s(u).
9:
arg min,eyy, 53 (NN,)(4).NN((1))
Ly Ni)(u)).
è [ij)aE(4- 12: return (c1, (2,,,,,..,
We first consider single dimension j E [d] of an arbitrary predicted cluster P: for some i E [M] Let Qú S. Qú be the set of the coordinates with size (1 a));; and minimum clustering cost, i.e., Qi = arg mingy, SQu, (01==)m)) (2 2²²)) Starting from step 3 of Algorithm 1. a set Un is constructed by randomly and independently drawing (log(kd)) samples from Paj. The goal here is to find coordinates close to the coordinates of the optimal centers. We will show that, with certain probability, there exists at least one coordinate u E Usj that can well approximate the geometric center of Q'y- To analyze the success probability, we define City = f= Qú (ar, (iy)) S (Q.j 1////((t as the set of coordinates close to Vy- where μ is a constant with a 1. The following lemma argues that C takes large fraction of Q'y-
Lemma 4 For any Qij P. n Paj, it holds that |G
2 M-1 11/hh
Let μ 2. According to Lemma dently sampling a set U2j with size
we have
G
2 1 11// (1-a)m, If randomly and indepen-
9
2 In( kd from P.j, the probability of sampling at least one
I-a
Ugyl
| 1- 107
13 1ly1
coordinate u E G2yy is 1
21-
21
21 2
kd
where the second inequality follows from In(1 x) < for E (0, By taking a union bound success probability over all the dimensions and the predicted clusters, we can argue that with constant probability, there exists at least one coordinate Uij such that u E G?, nu:.-
Corollary With constant probability, for each E [k] and j E [d]. there exists at least one coordinate u E U.j such that Giy-
Based on the sampled coordinates, in the remaining steps of the interval estimation phase (steps 4-6 of Algorithm 1), the Past-Sampling algorihtm estimates the interval lengths identify potential regions that can enclose the geometric center for Qiy According to Corollary [ we can assume that there always exists least coordinate Ug n B2 Then, in step of Algorithm 1, the
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algorithm identifies the set (u) of the nearest (1 a)m, coordinates in P.j to The following lemma shows that both lower and upper bounds for 8² (Qgg Viy) can be established using (u)(u).
Lemma Civen arbitrary coondinate NN)()) - (Qij. Qiy).
G Utj. it holds that 8 (Q Qi²) 1
Lemma 5 shows that, if the sampled point u E Ug is from Biy. by identifying the set (u)(u) of obtain both lower and upper bounds for 82 (Q)j. Qyy). According to the
coordinates in Paj.
definition of Cijy we have (1, 1i S (Q 0.y)/10gy Let 44g (N9 (w), (w)) be
(1-a)m
the interval length. Then, according to Lemma 5 we can get that Oij [u 4aj, u + 4y and the
(S R.y
interval has length at most 2v
(1-a)m; In the candidate coordinates construction phase (step
of Algorithm 1), the algorithm further divides the intervals into smaller blocks, where the length of
each block is parameterized by some Vig Then, holds trivially that there must exist least
one block [u e' e'aley, u c((2 + 1)4,y for some integer À Z, such that 8(u + Mij. V(y) S
(0, 2(1-a)m;
Consequently, in step 8 of Algorithm 1, get that the constructed candidate
55 (Q,
set Uly contains at least one coordinate u' E U// such that 6(u', Viy) S
2(1 2(1-a)m;
Corollary With constant probability, for each E [k] and j E [d]. there exists at least one
coordinate u' E U/ such that ō(u', Qiy) (R) 2(1-a)m.
Starting from step 9 in Algorithm the Fast-Sampling algorithm enumerates all the constructed candidate coordinates and their (1 - a m; nearest neighbors identify the set of the coordinates with the minimum clustering cost. Then, the geometric center for the set the coordinates with minimum clustering cost is selected serve the coordinate for the clustering center. Let Iış be the set the coordinates found in step 10 of Algorithm 1. The following lemma shows that the distance between Qy and Tif can be bounded using Ty (ig as bridge.
Lemma The following bound holds: 5² (T Q1y) <
(4o+ta))d"(1;,00)) (1-2(1)aa)
To this end, we can combine Lemma with Lemma 2 to bound the distance between Ii and P.
Lemma The following bound holds: 5² (T P I
I-o +
(1-20)(1-o)
a(1+c)
Putting all these together, we can get the following result for dearning-augmente k-means problem. Theorem There exists a learning-augmenntd k-means algorithm that can output a + (()))- approximate solution in time O log kd)) with constant probability, where E 1/2).
4 THE FAST-ESTIMATION ALGORITHM
Although the proposed Fast-Sampling algorithm achieves linear running time in the data size while maintaining the approximation guarantees, introduces an additional (log(kd)) factor loss when taking union bound success probability. The additional loss in the running time might influence practical performance of the algorithm when handling large-scale datasets. To address this is- sue, in this section, we propose faster sampling based algorithm called Fast-Estimation. The proposed Fast-Estimation algorithm can efficiently approximate the coordinates of each predicted cluster within linear running time, with only small trade-off in clustering quality guarantees.
The formal description for the Fast- Estiimattttion algorithm is given in Algorithm 2. The general idea behind the algorithm is to first generate candidate coordinates that closely approximate the coordinates of the optimal clustering centers. Then, in each dimension of each predicted cluster, estimator is constructed using uniform sampling strategy. The estimator is designed provide

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accurate clustering cost estimates for subsets of coordinates with sizes (1 a)m, In particular, for each dimension of each predicted cluster, the estimator is built by randomly selecting a set Syy from P.j. Each sampled coordinate is then assigned a uniform weight, enabling approximate clustering costs to be calculated using the weighted samples rather than the entire predicted cluster. With the constructed estimators, finding the set of coordinates with minimum clustering cost can be done in sub-linear time, removing the multiplicative O[log(kd) kd)) factor from the running time of the Fast- Sampling algorithm. Due the space limits, all the proofs are delivered the Appendix.
Algorithm 2 Fast-Estimation
Input: A k-means instance (P,k, d), a (P1, (2,,P...... Pk) of partitions with error rate a, and parameter () < < 0.5.
Output: A set C' C I2 of centers with [C| k.
1: for i E [k:] do
2:
for j [d] do
3: Randomly and independently sample set Utj from P.j with size (llog(kd)), then initial-
ize U/ = and E T26
4:
for q = 0 to Olog(mm max)) do 24-1
5:
lij = v (1-a)m
6:
for E Usg do
s(u) - uu egylyy AE (- 22 nzn. where 62 33
7:
8:
U// U/j su))..
9:
Randomly and independently sample a set Sij from Pi with size / log(m
log(m= ad log ( 2ma2
assign each point Siz a weight my
oct
|S:]]"
10:
Construct an estimator w such that Vu E U/S w()) = Epes, (ux)
m. 8 (p. u), where
18
F(u) is the set of the furthest (1 + 3c, )al s points from Sig
11:
Cjj arg minueuy w(u).
12:
Let Ly be the set of the nearest (1 2a ae)m; coordinates from Ps to Cij-
13:
c (Tt))
14: return {c1, 02,..., Cx).
In the following, we give the formal analysis for Algorithm 2. In each step 3 of Algorithm 2, for each dimension of the predicted cluster, the algorithm selects random sample Uij to approximate the co- ordinates the optimal clustering centers. According to Lemma 4 with constant probability, there
exists at least one sampled coordinate u E Ui such that 8(u, 2 S 22 0////// Then, starting from step 4 of Algorithm 2, the algorithm enumerates all the possible lengths of intervals for constructing the set of candidate coordinates. Without loss of generality, we can assume that the minimum pairwise distance between the coordinates in P.j is 1, and the maximum pairwise distance between the coordinates in P.y is Amax- Consequently, in step 5 of Algorithm 2, there exists at least
one guess for the interval length such that 25"
A8 R.y). (1-a)m;
Then, in steps 7-8 of Algorithm according to Lemma there also exists least one coordinate w' E U'g such that
<4 S
(I (I-a)m;
u' is close enough to the geometric center for Qiy. i.c., d(u', Ø'2)) S (1 (.)/)),)
For each u E U/ denote (u) as the set of the nearest (1 - a.mm coordinates from P.j to u. Let Ou) = W.yy) be the set of the furthest am; coordinates from P. to u. Before the construction of the estimator ws (steps 9-10 of Algorithm 2), start by dividing Nij(u) into
y (1-e1) mm blocks. Specifically, for each E U (Ny( is decomposed into y blocks BT) based on the distances from the coordinates in Niy Nij(ue (u) to u, where By =
f (denoted as B. B {a E (w) e1) < (2 u) < (1 (1++ +11). Then, we further divide the blocks into two cam,
groups based on the sizes of the blocks, where L(u) [ 1B
hl]
(1+e. log(m, A

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and S(u) (B ..,,,....11 (((u) are the groups of large and small blocks, respectively. Our goal is to well approximate each large block in L(u) while allowing to ignore the coordinates in small blocks. The following lemma shows that the estimator, constructed by randomly selecting a set Siz
from P.o with size clog(mm (m max 1/r)) (og(m
a large enough constant to be specified) can well approximate each large block and the set Ouu) of coordinates with high probability.
act
Lemma 8 With probability at least 1 md (m we have (1 <1 E n Styll - 1 n
Siyl S (1+e)) nStyl for each Bi E [(u), and (1-ci) S [O(u)SSsjl S
(1 el (()OO()) n S.yl-
For the small blocks in S(u), denote J(u) = Us, ES(u) By as the set of the coordinates in small blocks. Given that the number of coordinates in small blocks constitutes only a small fraction of the entire predicted cluster, we will show that the number of coordinates selected from these small blocks through random sampling can be approximately upper bounded by (5
Lemma' 9 With probability at least 1 lo ( it holds that | (u) n Sty S 2eja|Shy)-
Next, we prove that the estimator can give accurate estimations for the clustering cost induced by the set of the nearest (1 - a )m; coordinates from some sampled coordinate E U2y- For any coordinate u E U. let (u) be the set of the furthest (2 20c Jam; coordinates from Pij to u. The following lemma establishes the lower and upper bounds for the clustering cost given by the estimator w.
E U//-- with high probability, we have (P,) (w)) 76) $ (Ny)))
Lemma 10 Given arbitrary coordinate
According to Lemma 9 and Lemma 10 for an arbitrary sample u E U.j. with probability at least (m the constructed estimator w can give approximate clustering cost estima-
tions such that (P(P))FF (uu)= (1 + 6)) (N) u). Observe that there at most 147c1
O(E 1/2 (og(kd) log(mAmax)) constructed coordinates in each dimension of each predicted cluster. Then, by taking union bound success probability over all the dimensions of the predicted clusters,
we can get that with constant probability, (F), < w(u) < (1+ (NV (w), u) holds 1+7e1
for each u E U/ where E [k] and j E [d]. Based on the properties of the estimator, we will show that in each dimension of each predicted cluster, the Fast- dust--ssimation algorithm can find good coordinate approximation for the optimal clustering centers.
Lemma 11 The following bound holds: (Iij, Qiy) S
Finally, by using Lemma 7 Theorem 2 can be proved.
Theorem There exists a learning-augmenter k-means algorithm that can output a (1+0 approximate solution in time (md) + +Ok=-kdjjo -kd/a) ai) with constant probability for E (0, -).
5 THE FAST-FILTERING ALROTIHM
For the Fast-Sampling and Fast-Estimation algorithms, clustering centers are generated by finding coordinates approximation in each dimension. However, the sampling process may introduce cu- mulative errors, potentially leading to degradation in the overall clustering quality. In this section, based on our Fast-Sampling and Fast-Esse Easttmttiittion algorithms, we provide a more practical heuristic algorithm to better preserve the clustering quality while maintaining the improved running time.
The proposed algorithm presented in Algorithm 3, where the main idea is to directly find center approximations for each predicted cluster. In step 2 of Algorithm 3, set of samples is drawn ran- domly and independently from each predicted cluster to serve as candidate centers. Then in steps 3-4 of Algorithm 3, estimators are constructed using similar ideas from the Fast-Estimation algo- rithm. Based on the estimators, the candidate center with the minimum clustering cost is selected in step 5 of Algorithm to identify intervals containing the nearest (1 a)m; points. Finally, in step 7 of Algorithm 3, the geometric centers the identified intervals are selected as the final clustering centers. In Appendix A.4, we give theoretical analysis for the Fast-Filtering algorithm. We show
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that, with adjusted number of nearest neighbors (steps and 6 of Algorithm 3) and sample sizes R1 and R2, the Fist-riitgi algorithm can also give a (1 + ((ya))-approxiiii solution.
Algorithm 3 Fast-Filtering
Input: A k-means instance kP d), aet P2, Pt) of partitions with emror rate ax parameters
R > 0, Rg > and 0 < 1.
Output: A set C' C I2 of centers with [C| k.
1: for E [A:]do do
2: Randomly and independently sample set U2 from P; with size R1.
3: Randomly and independently sample set S; from P. with size R2 and assign each point in
ma S; weight [S.]
4: Construct an estimator w such that Vu E U, w()) EPESLF(W) my (p, u), where F(u)
is the set of the furthest (1 + e)a|S; points from S. to
5: C = arg min w(u).
6: Let be the set of the nearest (1 a)m; points from P. Cj.
7: è Tp.
8:
return {c1,
EXPERIMENTS
In this section, give empirical evaluations the performances of our proposed algorithms. All algorithms are implemented and executed in Python. The experiments were done on a machine with
17-12700KF processor and 256GB RAM. Following the prior work (Nguyen al. 2022, Ergun et al. 202T), we run each algorithm 10 times and report the average results with deviations.
Datasets. Following the work in Nguyen et al. (202)) and Ergun al. (202T), we test the algorithms
on datasets CIFARI0 (m = 10, 000, = 3, 072), PHY (m = 10. 000, d = 50) and MNIST 797, = 64) with varying error rate a and the number of clusters k. We also test the performances of the algorithms on other large datasets from UCI Machine I Learning Repository P 27), and one
including SUSY (m 5, 000,000000 000, = 18) and HIGGS (m 11, 000, 000, large-scale dataset SIFT (m 100, 000, 000, d 128) from Matsui et al. 2017
Algorithms. In our experiments, we mainly compare our proposed Fast- Dast-Sampling, Fast Estimation and Fast-Filtering algorithms (the version in Appendix A.4 with theoretical guarantees) with other learning-aut augmeenedd algorithms, including the algorithm in Ergun et al. (202T) (denoted as Ergun) and the algorithm in Nguyen al. (2022) (denoted Det). For the Fast- Sasst-Smmling algorithm, the sample size is set to 4, and we fix = 1. For the Fast-Filtering and Fast-Esstmmane 0.3, where m the size of the given clustering instance.
algorithms,
fix R₁ 10, R₂ m/20 and To further demonstrate the advantage of learning-augmentedd clustering model, we also give com- parisons between our algorithms and the k-means++ method (Arthur & Vassilvitskii 2007) without prediction information.
Predictor Description. Following the prior work (Nguyen et 2022). , the predictor is generated as follows. For each dataset, we first run the k- + (Arthur & Vassilvitskii (007) method as an initialization, and then run the Lloyd's algorithm (Lloyd 1982) until convergence, where the labels returned are regarded the optimal labeling partitions (denoted as (Pi,.., Pa)). To test the performance of the algorithms under different error rates of the predictor, following the previous work of Nguyen et al. (2022), we randomly change the labels of the am, points closest to e for each cluster P. to generate the corrupted labeling partitions {P'.... P.). as the predictors. For every dataset, we generate the set of corrupted labels for ranging from 0. to 0.5.
Detailed A Algorithm Implementations. As pointed out in Nguyen et al. (022). in most situations. will not have access the eиror rate and must try different guesses of to return the clustering with the best cost. Therefore, for each algorithm (including those in Ergun et al. (202]); Nguyen al. (002 well as ours), we iterate over 15 possible values of uniformly distributed in the interval 00.01, 0.5] as the inputs for the algorithms. The guessed a value that yields the best cluster-
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ing cost is selected the final result. For each algorithm, the runtime for each execution includes the cumulative time spent across 15 iterations of guessing error rates and solving the corresponding k-means instances based predictors. Additionally, we also compare the ARI amd NMI values for different algorithms to show the quality of clustering with respect to the ground truth labeling.
Table 2: Comparisons on dataset SIFT for varying a and fixed A = 20
(100,000000
Method
Ref
NMI
Timels)
k-moans++
166848-13::14 455++11
0.3285-00138
0.1550:0.0137
1000899
Ergun
3,97990--121110033
0.9263_00000
00000100.0001
16748.84L5776.22
Det
9.7791E-12-0..00222
2200000:4000
009-00-00000
(.9491+0.0000
15152
Fast--arma Fast-Sampling
L05/2E-12844.18,
9.7666E-122000 000000+000
0,9519
0,0000
0,9531 0.0000
15057.
1711
Fast- Fillering
9.7150E::12:22900 2.:00-0-0
0.9316=0.000
6933:00.0107
1006.31::43
Fast Estimation
9:800/E--1233,744x08
0,906520.0003
0,9466.0000012
8574.6661257..26 .6
1.6955--147.91955
0.3654-0.012
0.1940-0.0221
1017.12--1157
Ergun
LO210E-1312288
0,8901 10,000
17410.75161122
1:99199-122-00005--
0.011-000000
00060000000
Fast-Sampling
9,705E-121111.244
0.2
9.95766-12::00000 00:0000000
0.00300000000
0.8895 0.0000
1989 65 13270.53
Fast-Filiering
9.79149--12-9.500000
0.8690000116
0.815--0.016
1088,53-92 -2.992220
Fast Eatimttin
1.00MME-433..111--0
0,909300.0002
0.897900.0002
9567.52.1291. 24
K-meuns.4+
1.6561E-:3-99.4944
0,3531-0
0.1614-0.0296
927.07+31.33
Ergun
1.0526E=13.44.0EE.
0.8663.0.0000
0.8361 00611000000 0000
17586,
20 20518558. 30
0.8625=0.000
D.229 DIE0
15214.91 11214.11110 1914.66
Faut-Samplin
9:2478E-12(1500.9)
0.3
1.0288--13-00000
13-00000400
0,8743-
0.8896
26-+
Fast-Filtering
9.9.9E-11277.74...000
08180-00048
1095.48-66.17
Fast- Estimation
1.0300E--13333314088
0.8663.0.0002
0..0000000002
8618.38 1618.38.12378.00
k k-munn+
[.8811E-13L491E+11
0.3582.00111
0.175210.0126
991.0 48.
Ergun
1.9944E-1314:277-8888
0.027-000000
0,7801 0.000
16291 70.1 1506 28
Det
0.208=000000
D 0.7749
12999.81 :1:4441:2214440 99
Fast-Sampling
8:97805-1(1542.73)
0.4
1.06 1.06133.-13-0,000.00
-8333-00000
2.799000.000
13658 40++ 1766. 14
Fast-Filtrring
1.9125E-11:3314:.000
0.7879=00048
0.7522-0.0052
1091553-94466
Fast Fall-Eatiation
1.0687E-13:8.255 222.255
0.8260-00000
00,0005
8725 772.99412611.00
k-meam++
1.75:2E-131281E-11
0,3313-00073
0.150050.0065
972.59.160.40
Ergun
1.1414E-13::4922.. 922-00
0.7885-00000
871400000000
17256. 11 11+66660
Det
1. 1.1.500-13.000040000
0.7863_000000
0.710810.000
1122168811901.0
Fast-Sampling
8.75665-1214126114
1110099.11=00.0000 300000000000
0.7963-9.8000
0.7290-000000 000000
150422 91 1762
Fillerinn
0,7086 10000000000
0613330000097
1061.20 34.37
Fall-Falimation
1.1 11166F=-3++100 1.68.409
6.788--0.005
0.7153-0.012
8532, 2532.66+2152.11
Results. Table 2 compares our proposed algorithms with other augmented k-means meth- ods on the SIFT dataset for varying error rates and fixed clusters. "Ref" reports clustering costs of optimal labeling partitions and the running time for generating them using the Lloyd's algorithm. Due to space limit, the results for varying clusters and other datasets are given in Appendix A.6.
The results show that our Past-Sampling algorithm achieves clustering costs comparable to state-of- the-art methods, while Fast-Filtering consistently outperforms other with an average 1.5% reduction in clustering cost across all datasets. In terms of running time, Fast-Filtering is significantly faster than other algorithms, especially for large and high-dimensional datasets, achieving at least 3x speedup over current methods. On the SIFT dataset, is the only method faster than Lloyd's algorithm, where the running time is at least 10 times faster than other methods. For the NMI and ARI values, our algorithms consistently achieve NMI and ARI values above 0.80 across most datasets, with particularly better results MNIST and SIFT due their spa- tial coherence. Meanwhile, the Det algorithm performs better on high-dimensional datasets (SUSY, HIGGS, and PHY), while Ergun's 's algorithm excels on CIFARIO with its complex image features.
algorithms,
CONCLUSION
In this work, we present new sampling-basee based algorithms with linear running time in the data size for the augmente k-means problem. We show experimentally that our algorithm achieves bet- ter performances different datasets compared with other state-of-the-an algorithms. An interest- ing future direction is how to design algorithms with better approximation ratios while maintaining linear running time in leaming-auggmntted settings.
ACKNOWLEDGEMENTS
This work was supported by National Natural Science Foundation of China (62432016, 62172446). This work was also carried out in part using computing resources at the High Performance Comput- ing Center of Central South University.
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REFERENCES
David Arthur and Sergei Vassilvitskii. k-means++ the advantages of careful seeding. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1027-1035, 2007.
Hassan Ashtiani, Shrinu Kushagra, and Shai Ben-David. Clustering with same-cluster queries. In Proceedings of the 29th International Conference on Neural Information Processing Systems, pp- 3216-3224, 2016.
Lorenzo Beretta, Vincent Cohen-Addad, Silvio Lattanzi, and Nikos Parotsidis. Multi-swap k- means++. In Proceedings of the 37th International Conference on Neural Information Processing Systems, pp.
2023.
Manuel Blum, Robert W. Floyd, Vaughan R. Pratt, Ronald Rivest, Robert Endre Tarjan, et al. Time bounds for selection. Journal of Computer and System Science, 7(4):448 461, 1973.
Davin Choo, Christoph Grunau, Julian Portmann, and Václav Rozhon. k-means++: Few more steps yield constant approximation. In Proceedings of the 37th International Conference on Machine Learning, pp. 1909-1917, 2020.
Vincent Cohen-Adee Addad and CS Karthik. Inapproximability of clustering in LP metrics. In Proceed- ings of the 60th IEEE Annual Symposium on Foundations of Computer Science, pp. 519-539, 2019.
Vincent Cohen- Addand Hossein Esfandiari, Vahab Mirrokni, and Shyam Narayanan. Improved ap- proximations for Euclidean k-means and k-median, nested quasi-independent sets. In Pro- ceedings of the 54th Annual ACM SIGACT Symposium Theory of Computing, pp. 1621-1628, 2022.
Sanjoy Dasgupta. The hardness k-means clustering. 2008
Jon C Ergun, Zhili Feng. Sandeep Silwal, David Woodruff, and Samson Zhou. sarming-augmmntee k-means clustering. In Proceedings of the 9th International Conference on Learning Representa- tions, 2021.
Chenglin Fan, Ping Li, and Xiaoyun Li. LSDS+ Dual sampling for accelerated k-means+++ In Proceedings of the 40th International Conference on Machine Learning, pp. 9640-9649, 2023.
Zachary Friggstad, Mohsen Rezapour, and Mohammad R Salavatipour. Local search yields PTAS for k-means in doubling metrics. SIAM Journal on Computing, 48(2):452-480, 2019.
Buddhima Gamlath, Silvio Lattanzi, Ashkan Norouzi-Fard, and Ola Svensson. Approximate cluster recovery from noisy labels. In Proceedings of the 35th Conference on Learning Theory, pp. 1463-1509, 2022.
Ragesh Jaiswal, Amit Kumar, and Sandeep Sen. A simple D sa-slingng based PTAS for k-means and other clustering problems. Algorithmica, 70(1):22 46, 2014.
Tim Kraska, Alex Beutel, Ed H Chi, Jeffrey Dean, and Neoklis Polyzotis. The case for learned index structures. In Proceedings of the 2018 International Conference on Management of Data, pp. 489-504, 2018.
Silvio Lattanzi and Christian Sohler. A better k-means++ algorithm via local search. In Proceedings of the 36th International Conference on Machine Learning, pp. 3662-3671, 2019.
Stuart Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):
137, 1982.
Yusuke Matsui, Keisuke Ogaki, Toshihiko Yamasaki. and Kiyoharu Aizawa. Pyk-means: Billion- scale clustering for product-quantizer codes. Proceedings of the 25th ACM International Con- ference Multimedia, pp. 1725-1733 1733, 2017.
Michael Mitzenmacher. A model for learned bloom filters and optimizing by sandwiching. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp. 462-471, 2018.
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Michael Mitzenmacher and Sergei Vassilvitskii. Algorithms with predictions. Communications of the ACM, 65(7):33-355 10:33-5,,20 2022.
Thy Dinh Nguyen, Anamay Chaturvedi, and Huy Nguyen. Improved learning-augmentt algo- rithms for k-means and k-medians clustering. In Proceedings of the 10th International Confer- ence on Learning Representations, 2022.
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A APPENDIX
A.1
PLOT OF APPROXIMATION RATIO V.S. ERROR RATE
Plot of functions with respect to error rate a
Ergun
10a
Nguyen
Fast Sampling
Fast Estimation
10a
Approximation
0.0
0.1
0.2
0.3
Figure 1: Plot of approximation ratios v.s. the error rate a.
A.2
MISSING PROOFS IN SECTION
Lemma 4, For any Qi = P n Pij. it holds that G 2 p-1 14
Proof Observe that
8 (Q'y V1y) 82((Q') /
Vig)
|G
(0/ 2'1y)
2 14'y1
1
10
which implies that G - Vtyl
Lemma 5. Given an arbitrary coondinate u Giyy Uzj it holds that (Q Qiyy) < (N) Proof According to the definition of 0/ that Ø3 is the subset of Ps with size (1 a)m; and minimum clustering cost, it holds trivially that 8² (NNi (u) S Poj and [N)(u)] (1 a)m, On the other hand, we can establish an upper bound for
(0 Q,y)
s² Q,y)) since
as
8²
& ((NN)(u), 5 80 u)
= 8Q Vgy)
+ 10 12 (u, Vgy)
S (3 V,y)
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where the first step follows from the optimality of the geometric center, the second step follows from the definition of Naj (u) that Nay(u) contains the nearest (1 a)m; coordinates from Pij to u, the third step follows from Lemma and the last step follows from the definition of Gty.
Lemma 6, The following bound holds: (I Qay) &
(4a+a))6"(Q;) [Qs1(1 (vsl(1-0o)
Proof We first bound the distance between 15 n Niis and 1ig. Denote F. as the set of the nearest
19 Myy 2(1-a)m
(1 a)m; points from Pij to u' such that w' E U// and ò(u', 2iy) S that
Then, it holds
8 (4 - (I)
52 (Q(j.
1Q Viy) + 110
(1 ; 8 8Q(() V.y)
where the first step follows from the optimality of geometric center, the second step follows from the definition for 1/ that 1' contains the nearest (1 a)m; coordinates from Po to u', the third
<d' R,) Since I5 is 2(1 (11-a)m,
step follows from Lemmal and the last step follows from 5(u', Q,yy S the set of coordinates with minimum clustering cost found in step 10 of Algorithm 1, it holds that
(1 1ig) S (1. 1(g
$ (1 + ; 82 (Q T.y)
Let Ç ( =1 1-
Tby bg))) According to Lemma 2 by assigning Tij
and J1 = Qi O Tij. we can get that
1 1-5 (T,
(1 10g. 11) 5
1- 1-i n Lgyl/1as
(3(( Iiz)
10 0 [1
[Tg
[gg 12 n Lig
82 lig)
1Q: nig
[4g
where the first step follows from Lemma According to the definition of Qiy, since PpaiQisl am,, it holds that Q2g 454 2 [4l am, Then, we have
8 (Iij. lij)
(T m Neij 11p)
- 2a
[Ty]
(1+ 2)8 Vin)
1 2a
[T
0: + 0.5ac
10'yl
(Q Q.)
Qqyl [Tl
a + 0.5ce
8 82iqq Q4)
1- 2x
[Q1l
where the first inequality follows from Lemma the second inequality follows from (L Inr) < (1 + (Q'g. Q'y), the third inequality follows from Lemma and the last inequality follows from [41 [Tg = a.mm.
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Next, we give upper bound for the distance between I, n Qij and Qij- Let Ç (1 QQy (1)l//1s))) According to Lemma 2 by assigning J Qis and J1 = Qis n Izj, we can get that
(1Qt)) /.yl/1/1111111 Quij. Qi5)
8 (Tg Q.)
(1- 1Q 44) Qis Qu1
14 (Qsj. Q1j)
(1Qis] - 1Q:j Viz Qiyl
ad (Qij. Qis)
(1 -2a) 21pl
where the last inequality follows from 1Qi Mayl S am,, Putting all these together and using the triangle inequality, we have 8² (Iijn Qii) S (a(4- (1144-e))"((Quss)))
1Q, (112
Lemma 7. The following bound holds: 8 (I P.f) S - a((+e) (1-2a)(1-a)
8 (P,
1P:)
Proof Since Qúj Pijo we have
=
Qiy + 1Qu
Then, can get that 82 (PPj. P3 Q1z)
1P P 11
8 (P.j Quy. Define 7
[ \
We have ( by Pi)(0s = 1- 8² P Q1j). By decomposing the clustering cost of P with respect Pùj into 82 (P;) Qij. P.j and 8² P. Qij P.j), it holds that
82 (P. PTj ( Quy (Qi),, P.j))
Pi 662 (P.
PM (ij)) (1:))++0 6 1j
=(PPN P (7) (177)
8² Pr Qij)
+8 Q1) + +1- (Qij P.j)
1 1-
= = Nij, P7j Q1,) + 8 (Uj Qúj)
Pij08 82 (Qij. Pt)
2 (Qi Q1j 1 pij 82 Qij. PTy).
where the first step follows from Lemma and the last step follows from the definition of the predicted clusters that ? a. By Lemma we have
1- (1 (1-a) a) (Ly Qis) (1g
1-a
8 (P P
a(4 +c)
Then, by using Cauchy-Schwer Inequality, we have
(0(P Qis) + 8(Qi,, 2 <
8 6'(P PT)
a + - -1-2a)(1 a.)
a(4+
1-a
Finally, conclude the proof using the fact that 8 (P Iis) < (8(P7, Qτ) + 58 Q2y ) D
Theorem 1. There exists starning-augmentt k-means algorithm that can output a (1 + 0(a))- approximate solution in time mm log (kd)) with constant probability, where a 1/2).
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Proof Denote C' the set of centers returned by Algorithm 1, where C' { (2,,02,.... ..., Cx} and é is
consisting of d coordinates [C11, G22 ..,, Gd]. Then, we have
h
d
- (PP,O)
Σ
8
ci)
=1
8 (Pij
P.j
P
Pig
1 a +
(1-2a)( -a) (P.ii
a(4 +c)
=1j=1
c) 1 + 1 (1 (1-2a)(1-a) (P((,0").
Then, we give the runtime analysis for the proposed Fast-Sampling algorithm. In each step 3 of the Sampling algorithm, sampling coordinates from the predicted cluster takes ((1) time. In each step 5 of the Fast-Samplin Sampling algorithm, finding the nearest (1 a)m, coordinates in P., to given sample can be executed in linear time (m,) using linear selection methods (Blum et al. 1973)
In each step 7 of Algorithm 1, note that the interval length can be bounded by O(1) V (P(Q.).Q.)
(1-a)m;
Hence, by dividing the interval into smaller blocks with length c'Ls where e' / 45 we can get that the number of candidate coordinates constructed in each step of Algorithm can be bounded by O() According to the sample size of Uzj. the total number of coordinates constructed in step 8 of Algorithm can be bounded by O log(kd)). Then, finding the subset of coordinates with minimum clustering cost in steps 9-10 of Algorithm can be executed in time O(e log(kd)) using linear selection methods (Blum et al. 1973). Since the Fast-Sampling algorithm needs to enumerate all the dimensions of each predicted cluster, the overall running time can be bounded by Σ Σn O(e-,m, (kd)) = O(e mnd (k))kk))))).
A.3 MISSING PROOFS IN SECTION
Lemma 8. With probability least md log (ma we have (1 - =] S $ [8]
Syl (1+e1) (1+e)) [S;]]] holds for each Bt, E C(u), and (1-e))
(O())Ss S
(1 + (())) Stgll
Proof Observe that |S,yl clog(m² a log (mo2 1/2 log(m max! where is some large enough con-
max
act stant. Hence, for an arbitrary large block By L(u), have
[8.1- |
El|S% n B
Pull
2a
2log(m2dloo (m 3/E)) log( A
(1 + Am max) clog(m² log
ace
(1
Thus, by applying the Chernoff Bound, we can get that
P. (11 0 Syll= B Syyl s (1 E| Styl) 2 2c
3
& dog (m max
Ω
where the last inequality follows from the fact that e is a large enough constant.
Similarly, for the set O(u) of coordinates, we can get that with probability at least Ω midd log" (mA holds that (L ()TO)) < (O()) < (L+)))il)0(nn S.y]. By taking union bound success probability, Lemma can be proved.
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Lemma 9. With probability at least 1 m9 l (m=))))) it holds that [J Styl [Syl
Proof
According to the definition of small blocks, it holds that (T(x)] I 610m; Observe that
El|Stt [w)]]] ( that
Let X
2erem [3( By applying the Chernoff Bound, we can get
m4
R((J(w))s (1+ (1x ( ns,
18 Sm,
(2ejam: -1) (I(+))
|5( 8m,
(1am; ) [3(m))
7o 3J(1
2 0
mm (ma max
where the first step follows from Chernoff Bound, the second step follows from the definition of X, the third step follows from the fact that [J()] I ejam; and the last step follows from log(m)) Since (1 + (L))) = (1+
|Sty] 2 clog(m" log"
(m 1/3) (mm="
H
4a [ have [T()) S.yl S 201a|Sigh-
my
Lemma 10. Given an arbitrary coordinate u Ulg with high probability, we have s (1
(P (u) sw(u)
Proof According to Lemma g and Lemma by taking union bound over the success probability, we have [T(u)) n Siyl S 2eya and [Ou)) n Sigl S (1+ (1)a hold with probability at least
(1
Define F'(u) = (J(u) (O())) Then, it holds that (F( S
m=dlog=(ma
(1 + 3e For each large block B E L(u), by using Lemma & we have |B 5 S
(1 + 9) E = (1+a)) [By] Thus, the clustering cost of By, nSi can be upper bounded
by
(B n u)
8( u)
MEB n8:g
(1 y 13", 0 Sig
(1 2 B
(1 t
mi
(1+ 8² (3((3)), u).
mi
where the second step follows from the definition of B, E C(u) that 52 u) S (1+c)) holds for each æ E B and the fourth step follows from the definition of BY, E L(u) that 6x u) 2 (1 (1++1) holds for each x E By Consequently, by taking summation over all the coordinates sampled in large blocks, we can get that Σ (c) (BB SSg u) < (1+a)² | (Sylo u). Denote F(u) as the set of the furthest (1 + 3c1 S points from S.j to According to the definition of the constructed estimator w in step 10 Algorithm 2, we have
(w))
m,
82 (2(ss) F()),u)
[Sigl
m4
822 ( F' u)
[Sig]
m,
8²((), n S.y. u)
|Sig]
m;
(1 + (1+ Styl 82 (N).
<
S
m:
= (1 + (1)+))) (N)(
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where the second step follows from [F) 2 (1+ 3 = F(u)). Hence, upper bound for the clustering cost estimation can be obtained as w(u) S (1 + (1)))
Then, we show that the estimator can also give a lower bound for the clustering cost induced by the nearest (1 a m, coordinates from some sampled coordinate u E U/J- For each large block
By, f(u), define Z F(u) n By Let H' be an arbitrary subset of the coordinates in B with mi Denote F = (O(u) U (u)) U (UB) E(()) H.)) It holds that
size (1 + 32 [S,yl
(u)] Σ am; + 2Ejom, + (1+ 3m)) (m)))))))
am; + 2c am; + (1 + (m, ((1 + 3e, )
< (2 + 2am,
where the second step follows from Σs, EC(a) |241 S [F(w)] S (1 + 33) )al Styl. Define FT(u) as the set of the furthest (2 + 20e Jomn points from P.y to u. Then, we can get that
(PiP (u), - (Pú) (F) u)
Σ
(B H u)
By EC((()
Σ By BCC(())
(1 (1) ))) (lBa)) 17C,
[B8 S my
[m (1 (1+
Σ B(EE((u)
(1 + e1)
(1-
|Syl
(1 (1)(1 + 3m1)m
Σ
nsijl - -12)) (1+e))
By, ((()
< (1 (E)).(u).
where the first step follows from the definition of Ft(u) (u) that (w) S [F]),,, the third step follows from the definition of L(u) that (x u) S (1 + e1)(+1 holds for each æ E B the fourth step follows from Lemma & and the second to the last step follows from I-ei < + 361 for 0 < < 0.5.
Lemma 11. The following bound holds:
S O(a))* Qg)ll
Proof According to Lemma
5 with constant probability, there exists at least one coordinate in
UY such that 8² (u1, Qiy) =
10:/ Hence, we can get that (N (M1), un) ) z (Q,), u1) S
(1 + c)) (Q'y: Qgy) where the last step follows from Lemma I
Let Ciy be the coordinate chosen by the estimator in step 11 of Algorithm 2. According to the property of the estimator, we have
82 (2 ((Px)FF ((g))
S w((n)
1+ 761
< w(u))
I (1 + + (186
(1 + ) 882 (Q,, Vgy))
Denote Ity the set of coordinates found in step 12 of Algorithm 2. Then, we can give the bound between Tıj and Qij- Since e 125 according Lemma and the properties of the estimator, have
8² (2 Tog) < (T s (1+ (es)
Vyy)
< (1 + < (Q, V,).
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where the last step follows from 61 According to Lemma it holds that
126 Next, we bound the distance between Iúj and Qij-
(2a + ae) 0 (Q Piy) Qiy)
8 8 15 (is)
(1 3a [)]
(2a + ac) +() 10/1 8 (Qij, Q.j)
(1 3a -c) 11jllqss/
(2a + ac) c1 -a) a) 8² 8Q Q11)
3a e) 2a 2a-2a-26 e) e) 14
Similarly, according to Lemma 6 also have (Qi. Q1g n Tzg) 1
aat 5(Qi,,
1-3a 14u1
Putting all these together and using the triangle inequality, it holds that 82 81 Q43)
13a-150m s1(Q Qs5) where the last inequality follows from < 0.5. (1-3a- (1-3a-c)(1- 2a
14yl
Theorem 2. There exists a learning-augmentee k-means algorithm that can output a (1+ (a)))
approximate solution in time (md) + Ö(E (/kk-kdja *kdja) with constant probability for a E (0,1/3 - c).
Proof According to Lemma
we have 8² P.. I,y) S (1-a + (1-30-e)(1-2a-
13a-15et
5 PP,,,
IP: De-
note C' as the set of centers returned by Algorithm 2, where = {ê1, 62 ..., 2x and é, is consisting of d coordinates (c11; C22 ..., Ctal- Then, we have
8² (Pij. j.j 9) Σ
Σ
8 (Pi).
Σ
8 P Pij) = P2 (i). cit)
1=
+
a + (1 - 3a - -c)(c - 2a -2a
13a 15a
(PP.j.PP.)
1 -
1=1
(1 -1- 3a e) 20 - 8'(,,C)
13a 15a²
Then, we give the runtime analysis for the proposed East-Estimation algorithm. Similar to the analysis for Algorithm 2, in each step of the Fast-Estimation algorithm, sampling coordinates from the given predicted clusters takes O(1) time. Then, in step 7 of Algorithm 2, we can get that the constructed candidate set of coordinates has size
(ma(mmmax)) log) log(kd)). Note that in each step 10 of Algorithm 2, the constructed estimator has size Ö 0(2)2) assuming bounded aspect ratio Amax that Amax S (oly(m). Hence, in step 10 of Algorithm 2. the running time for estimating the clustering cost can be bounded by Ö (.3)) ). Finally, in each step 12 of Algorithm 2, finding the nearest (1 - 2a - aε) coordinates in Psj to the selected coordinate can be executed in time ((m)) using linear selection methods (Blum et al. 1973). Then, the overall running time can be bounded by Σt Σj= 0(m)) + Ō o = (md) + 1 (kd/aa)
A.4 THEORETICAL ANALYSIS FOR THE FAST-FILTERING ALGORITHM
In this section, we present a theoretical analysis for the proposed Fast-Filterig algorithm (Algo- rithm 3). The intuitive idea behind is as follows. In each step 2 of Algorithm 3, we show that by carefully adjusting the sample size R1, with constant probability, is possible to sample candidate data points from each predicted cluster such that the sampled data points are close enough to the optimal clustering centers. Then, in steps 3-4 of Algorithm 3, a random and independent sample is drawn from each predicted cluster to construct an estimator that can accurately approximate the clustering cost induced by the candidate centers. Finally, by using the constructed estimator to select the best candidate center for each predicted cluster (step 5 of Algorithm 3), we prove that the clus- tering cost of each optimal cluster can be well approximated by using the nearest neighbor searching process (step 6 of Algorithm 3). The modified Fast-Filtering algorithm is presented in Algorithm 4.
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Algorithm 4 Fast-Filtering (modified)
Input: A k-mmans instance (P, k. d), set (P1, ..., Pto) of partitions with error rate a, parameter
O << < 1/3.
Output: A set C TRd of centers with [C| k.
1: for E [A:] do
2: Randomly and independently sample set U3 from P. with size R1 = ( log
1
1-20
3: Randomly and independently sample a set S from P. with size R₂
=
[log(m" ima" ")/"") 1og((ma))) and assign each point in St a weight ma ac
O
m2 4: Construct an estimator w such that Vu E U2 (()) EpEs
O (p,u), u), where F(u)
[5
is the set of the furthest (1 + 361 )a[S: points from S; to where = c/126.
5: C = arg millge,, w(u).
6: Let I, be the set of the nearest (1 2ox 200xe) m, points from P; to
7: à = T,
8: return {c1, C2.,,...., C.
Theorem Let R3 O( lo 1-2a and R = 0 log(m feeemmdda log" (mm)))/e) ( log(ms??)
where is param-
ac
eter with 0 < 1/3 and A
is the aspect ratio of the whole dataset (i.e., A ma p 6(a.a)
min..a P.arty 6(a.y)
With constant probability, Algorithm 4 can return a (1 + OlVa)))a))))))) -aproxiiae solution in time (md) + ( (-11-110000 2aa1 for (0, 1/3 c).
kd
Recall that Qi = P.n is the set of data points in P. that belong the optimal cluster P. Define Gp(P))= = fz E Pr (x, ct S. u ()) as the set of data points in P. that are close to of (the distance is parameterized by some constant μ >> 1). According Lemma and the definition
of Qs, we have [G2(P)) 2 |P- and PPA.G2(PP) < 2
[P[L Then, it holds that IP, G2(()) Z
2
|PTNP: (P(()) 2 (1 a:) P7 - (3- (?))) where the last inequality follows from the definition of the learning- maggmntdd k-means model. Additionally, according to the definition
of Qi we have IP,1| 2 IQi1 2 (1 -1-an
(1-a)]P:) PPl. which indicates that Pa S PI.
I-a" Let G (Pi)Ga(PPI [Pa]
be the probability that a data point from G2 G2,P," can be sampled from P. through uniform sampling strategy. Then, it holds trivially that Si 2 (1 - a){ - a). Hence, according to Corollary by randomly and independently taking sample U; with size 0 log( )) from P,, with probability least R. there exists least one data point E U, such that u E (2())). By taking union bound over the success probability across all the predicted clusters, we can argue that with constant
probability, there exists at least one data point u E U. such that u E G2 (P") nu, for each predicted cluster E [k].
Corollary Let R1 e log (-1-2)
For each i [k], with constant probability, there exists at least one data point u E U; such that u E G2(P.).
Let T26 Similar to the analysis for the Fast-Estimation algorithm, in steps 3-4 of Algorithm 4, we can also construct a clustering cost estimator w by randomly and independently taking a small sample R2 from P. with size O log(mn llgg (m2²)mmm)))))) log(m
Given an arbitrary data point
oej
U, let zt (u) be the set of the furthest (2 + 20c )am, data points from P. to u. Denote (u) as the set of the nearest (1 - a)m;; points in P. to According to Lemma 10 the constructed estimator w in step 4 of Algorithm can well approximate the clustering cost using data points in U, as centers.
Corollary Let R2 = (log(m dd (m A3 3) l Given an arbitrary data point u E U, + 7c)) & w(u) $ +1 (H) u). with high probability, it holds that (P.)2
ast
According to Corollary 3 with constant probability, data point uy U, n (Pt) can be sam- pled from P. for each predicted cluster E [k]. Hence, we have 8 (u:) u;) (" u4) <
'It is naturally to assume that the aspect ratio of given dataset can he bounded by polynomial function of the data size, i.c., A poly(n).
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(p, I (P, c?), where the first inequality follows from the definition of Hu) that (u) contains the nearest (1 -1-00 a.mm, points from P, to u, the sec- ond inequality follows from Qi S Cr, the second to the last step follows from Lemma [ and the last step follows from u, E (.(P.). Then, by using Corollary 4 we can argue that with con- stant probability, the center G chosen by the estimator in step of Algorithm 4 satisfies that ((P)))(()))) s (1+e)0² (H,()) u4) S 46 (8 G), where the last inequality follows from e < 1/3.
((P = (P,,
For any predicted cluster P. recall that (c() is the set of the furthest (2 + 20e Jam, data points in P. to c₁. Let 1, = Ps (c)), Define A, = Q (c,). where data points in A, can be regarded as the set of the "false negatives" discarded by q. Let B, = (Q) U (c,)), where data points in B; can be regarded as the set of the "false positives" included by cz. The following lemma bounds the distance between T; and c.
Lemma 12 The following bound holds: 8 (T,,, (1-(3+e)a)m
(MP(P))))
Proof Observe that [4:] = (1 (22+22))aa)))) and |PA\PT| S. amm, Then, it holds that 140P
(1 (3 + 20c )a)m; Observe that the clustering cost of (2; Tt) can be upper bounded by
(T, S (T,, c4) s (P Hence, by using the relaxed triangle inequality and taking
summation over the distances between data points in 1, and ct we can get that
E Epennce (1 + 1/2)0² (T. + (1 + (p,)
(T,)))
P
(1 (1+11/3 112) I.) + 308 (8, cf)
(1 (1-(3 (3 20c )a)m;
952 (P7,))
(1 (3 + e)amm
where the last inequality follows from 20c S e.
Observe that the clustering cost of P with respect to T; can be decomposed into 82(PP, Tr) PP n I. + (PP) 4)) In the following, we first give an upper bound for the clustering cost of (PtP n P, T.).
Lemma 13 The following bound holds:
Olya) (P)) < 5 + 1-(3+eja
Proof According to the definition of Q,, we have Q = P,NP; = Hence, it holds that (,)
= -1 (0
We first give an upper bound for 52 B, ()) (B,, T,). Observe that
82 - (B, = Σ
(b, ( - (B, T
BEB,
s Σ (1+ (b. (1)1))00000000+01)10)011++111
(B..).
S V (B, T.) + (1 (1+ (1+1//a)B.10 (c) T.).
where the second inequality follows from the relaxed triangle inequality. Similarly, we can also establish an upper bound for (A, It) - 8'(A,, cf). We have
-(A,,))-((A,6))))
Σ - (A,,
(1, (1 -2 (A
aeA. < V (A, c + (1 (1+11/ (1+1/
where the second inequality follows from the relaxed triangle inequality.
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According the optimality of the geometric center, it holds trivially that ( I (I,,))) Hence, we can get that
/4t.:////4...0000000000400444440 IT)
- +.)
0(va)
- O(va)) + 1- (3+ (P):'
O(va) $ 1- (3+ (P,,C))
where the second inequality follows from B; S 4 A < (t(e)) - 3am; and [B:] s am,, and
the last inequality follows from Corollary 4 and Lemma 12
Finally, we can give an upper bound for the clustering cost of P. with respect T,.
Lemma The following bound holds: (2;; TT) S (1+ (I-a)]1(3+caa (P())
Olva)
Proof According to Lemma
13 we can get that
(P, T,) = (P,n Ta + (P
Olva)
s 1- (3 (3+ +
E (p..I.
DEPPIR
Olva) (PP
1 (3+
Σ (1+
PEPPAR
((va)
1
1 - (3 (3+c)a/
O(Va)
< | 1
(P(P),
(3 ()
(P)
(1 1/V0) 1-a
am.
(1-(3+
0(Va)
(P).)
(1- (3 c)a)
where the first incquality follows from Lemma 13 the second inequality follows from the relaxed
triangle inequality, and the second the last inequality follows from Lemma 12
Using similar ideas for the Fast-Sampling and Fast-Estimation algorithms, by taking summation over all the clustering cost of the optimal clusters, Theorem 3 can be proved. As for the running time, for each predicted cluster [k).. sampling U, and S; from P. takes O(1) time. Then, in each step 5 of Algorithm 4, estimating the clustering cost of each candidate center in Ug takes time O(R1 R2), which is (1-22)))) Finally, in each step 6 of Algorithm 4, finding the nearest neighbors for a given data point can be executed in time O(m,d) using lincar selection technique Blum et al. ) Thus, the overall running time for Algorithm 4 is ((md) + Ó (2]]- ). To conclude, the Fast-riig algorithm can return (1+0(v/)) -appoooxiaat solution with constant probability in time ((md) Ō ((2T1 (3[1-aaa
kd
A.5 EXTENSION TO THE K- MEDIAN OBJECTIVE
In this subsection, we show how to extend our proposed sampling- based methods to the learning- augmented k-median problem. The key challenge here arises from the difference in optimization
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objectives. In particular, for arbitrary set S R of coordinates, the geometric center of S can no longer serve as the optimal clustering center for under k-median objective, making difficult to identify high-quality candidate coordinates or centers. As result, existing boarning-uugmennt k-median algorithms often struggle to achieve high-quality approximation guarantees.
To overcome this challenge, our goal is to sampling-bas based strategies to construct centers that are close to the optimal clustering centers for each predicted cluster P.. Then, by grid discretization, can generate candidate centers that can well approximate the optimal clustering centers. Finally, by enumerating the constructed candidate centers, we prove that the clustering cost of each optimal cluster can be well approximated using the best center chosen from enumerations.
U/ of
Table 3: Comparison results of learning augmented k-means algorithms
Methods and Refereaces
Approximation Ratio
Label Error Range
Time Complexity
Paritioning and Sorting (Ergun 2021
O((ka) 1/4) 417- On 10a")
Small Constant
O'md log' i polyík, log m))
Sorting (guuye et al. 2022
1/2)
O millog" log (8/66)
1-3e
Fasp-Sanpling (Ours)
(0, 1/2)
and log bdd loy (m))
[1-0][1-22
1-20
Table 3 provides a detailed comparison the results for the aearning-augmented k-median problem. We also give plot (Figure 2) of approximation ratios vs. the error rate It can be seen from the table that the current best result achieves a (1 + Oaaax-approinamatoon with a E [0, 1/2) (Nguyen et al. 2002).. Compared to the offe--of-thear results, the Fast-Samse East-Sampling algorithm can achieve better clustering quality guarantees with slightly worse running time for fixed dimension d.
Approximation v.5.
Det
107
App-ocinaaton
201
387
0.0
2.5
Figure 2: Plot of approximation ratios v.s. the error for the k-median objective.
The formal description for the learning-augmenned i--eddia algorithm presented in Algorithm 5. The general idea behind the algorithm is to first generate candidate centers that can closely approxi- mate the optimal clustering centers for each predicted cluster. Then, by picking the best center with minimum k-median cost, we prove that the proposed algorithm can give better approximation guar- antees for the augunneed k-median problem. the following, we give a formal analysis for the proposed algorithm.
Without loss generality, we can assume that the minimum pairwise distance between data points in is while the maximum pairwise distance is A. Note that this can be done using standard scaling techniques. According to Lemma in each step 2 of Algorithm 5, with probability at least k
least one center u E U, can be found such that 5(u, ef) < 28(P; c S
25(P" .oi) (I-a)m. where
the last step follows from the fact that P 2 |Q 2 (1 a)m, Then, in step of Algorithm 5, since the algorithm enumerates the possible values between and log(mA), there exists at least one guess for the clustering radius (step 4 of Algorithm 5) such that ((Pr:c))/(
a)m; S I, S 28(Pr,c))/C (1 a)m, Hence, in step of Algorithm 5, the grid centered at u with side length 21; (G(u)) should contain the optimal clustering center cτ. Then, in step 7 of Algorithm 5,
by decomposing the grid G(u) into smaller subgrids with side length (1 a)a414/ Vd for some
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Algorithm 5 Fast-Sampling (k-median)
Input: A k-median instance (P, k, d), set (P1..., Pis) of partitions with enror rate and pa-
rameter E (0, I].
Output: A set C TR of centers with [C| k:
1: for E [k] do
2: Randomly and independently sample a set U, from P. with size
[og(ka) 1-2a
then initialize
U = 0.
3:
for 1- 0 Ollo(mAA) do
4:
4 - 1129-1/1-000
5:
for u E U4 do
6:
Let G(u) be the grid centered at u with side length 2(,.
7:
Decompose G(u) into smaller subgrids with side length (1 - aa)eeli//v and let s(u) be the set that contains the centers for the decomposed subgrids, where é1 e/4.
8:
U = U/ s(u).
9:
Set u, arg min ((N((u), u), where M(u) is the of the nearest (1 a)m; points in
Po E U.
10:
C U4
11:
return {er, 02, J.
G < < the optimal clustering center of must also belong to some of the subgrids. Since the subgrid has side length (1 a/)ee///VV Vd, there also exists least one u' U/ such that u' is close enough to c;, i.e. 8('', cf S (1 a)aE I aed(p, ( c)/m,, Let u, be the point chosen step 9 of Algorithm 5. For any data point UK, let (1) be the set of the nearest (1 - a)m; points in P. to u. Consequently, we have
o(Ni(u)) < 8(NG(u)),
< ((N((u)),
8(NG(u)) + (N)(()))
<8(N((u,), cf) + ae)(P;
where the first inequality follows from the fact that 5(NG(u)) uz) induces the minimum clustering cost, the second inequality follows from the definition of Ni(u²) that N₁()) contains the nearest (1 - aamm points in P: to w', the third inequality follows from the triangle inequality, and the last inequality follows from 6('',c) .c) I aed(p) m m. and [NC(u)) & m4.
Corollary 5 For some i E [k), with probability at least 1 1/k, it holds that d(Ne(us). u:) S 8(N((us; (4) + (e (7))
For any predicted cluster P., let 21(4) the set of the furthest om,, coordinates in P; to ui i.e., (u,) zt(u,) negatives" and "false positives", respectively. In the following lemma, we first give an upper bound for the distance between of and u,,
= P. (u.). Let A, Q, n zt (u:) and B, PAQQ,, U (u,)) be the set of the "false zt
((tae))(PPee)) (1 [1-2a)m
Lemma 15 The following bound holds: 8(u,, cf) <
Proof Since [V](u)] a1-a)m; and |P |PA.PP am, it holds trivially that (u,) (1
20x) m, According to the definition of u; that Nu(u;) induces the minimum clustering cost, we can get that (u,) u,) < ((NV(u'))) u²) S 5(Q,, u') S 8(Q2,, ((Qc.c)++2,,66(r <f) I (1+ac) ct), where the last inequality follows from 6(u', <t) I aed(P; ct c)/m; Then, by using the triangle inequality, we have
5(u,, c/) - EPENS(MDDPY 5(p, u:) + 8(v, c7)
(W((u)) P.
((Na(u)) + O(Pr, ())
1122am
(2 + (c))(Pp,,c))
(1 - 2a)
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where the first inequality follows from the triangle inequality.
The following lemma gives an upper bound for the clustering cost of Q, with respect u4-
Lemma 16 The following bound holds: 8(Qi, u4) - 6(Qi. ()) a(443e)
1-20 ,(P;'c))
Proof Since Qi = (N( U A, we have 5(Q; uz) - 8(Qi, <)) = (u:), us) 6( Bi, ui) + 6(A,, :4) (6.) ()) + 8(A A1, )) We consider bound these terms separately.
For the clustering cost of (,,, 14) ((4,,C)) C)), we can get that 6(4,, u4) -6(A,; (7) S 5(A ( [A,lo(c, u.) -((A, c - [A, u.) Similarly, the clustering cost of 6BB, of - u.) can
be bounded by 5(B, c) - 5 BB, 4;) S 8(B;, u;) + [B:j6(u, c)) - ö(B, u.) S [B,j0(u,
Next, we bound the size of |A.] and [B.) Observe that Q; (i U A, Consequently, we have [Q, = (u,) ||4 |B (1 (1-a)m- amm [4,] -BB 2 (1 (1-a)m;, where the last inequality follows from [Qal 2 (1 a)m, Hence, we can get that [B S |A, S am,.
Then, it holds that
8(Qu. (0Q..))-(((c))))
S ((.) u.) ((V(())). ((A.)
(AA (B.l)((,,)))
s 8(V.(u)). (.) + aeo(P, c)) ((N((u)).
(14,1 + [B.1)6(u,, 6i)
(2 ac oc)
s aeδ (ee((P?:c0 (P,,ct) 2am
(1 20
a14 + 3c)
20 5(P,
Finally, we can bound the clustering cost of P. with respect to u; for each E [k]]
Lemma 17 For each i E [k]] with probability at least 1 1/k, 5(P;, u:)
(I+ Aoc (1-a)(1 (1)
-3002
8(P7.c).
Proof For arbitrary predicted cluster Pr, the clustering cost of P with respect to u, can be decomposed into SPP u4) = 8(P; n P., us) + S(P.)P, u4). Hence, we can get that
,(P),u.) = 5(P; n P. u., + (PP))
S8(P, () + a(4 + 3c)
1-20 8(P, +P7.))+88PVVP...
a(4 +3)
1-20 8(P:: () + 8(P: P/. + &PP) () + IP.
am: (2 + ac)((P) 4)
a(4 3c)
< 6(Pr + 1-a a (1 - 2a)m;
((P).c)
1-2a 2a
1 6a + 4aE 4a² -
8(P;,c)).
(1- (1-a)(1- - 2aa)
for the Baaning-auggmene k-median problem. As for the running time, for each predicted cluster, the sampling process takes (11) time in step of Algorithm 5. Guessing the optimal clustering cost in step of Algorithm 5 induces an multiplicative (log(mA)) factor loss. Then, constructing the candidate coordinates (logtkd)
Putting all these together, we can obtain a (1 +
v oo()). Finally, finding the nearest coordinates for each candidate center takes time (m,d)) in step 9 of Algorithm 5 using linear selection method (Blum et al. 1973). Consequently, the overall running time of Algorithm 5 O md (kg) log(mA) (Vd)O(d) eec
takes time O I-2a
1-2x
A.6 COMPLEMENTARY EXPERIMENTS
Other Results with Varying Error Rate and Fixed Clusters. Tables 4-8 show the experimental re- sults datasets Mnist, PHY, CIFARIO, SUSY, and HIGGS for varying error rate with fixed clusters,
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where "Ref" reports the clustering cost of the optimal labeling partitions and the running time for constructing the labeling partitions using k-means++ and Lloyd's 's algorithms. On average, by calcu- lating the results over all the datasets used in the experiments, the Fast-Filtering algorithm achieves a 1.5% reduction in clustering cost compared to the current state-of-the-art means algorithm. As for the running time, it can be seen from the tables that our Fast-Filtering algorithm is significantly faster than other Gaarning-augmenter algorithms across all the datasets, with this trend becoming more pronounced the data sizes and dimensionality grow. On average, Fast- fast-Filtering is at least 3 times faster than the current state-of-the-an learning-augmente augmented algo- rithm. For the Fast- Esttmmssim,,,on, on average, the Fast- Eastmmssimttion algorithm at least times faster than previous learning-augmenter algorithms on large-scale datasets with sizes over 5 million.
k-
Results with Varying Clusters and Fixed Error Rate. Tables 9-14 present the comparison results for varying number of clusters with fixed error rate. For clustering cost, our Fast-Sampling algorithm comparable with other tently achieves better clustering quality for most cases. On average, by calculating the results over all the datasets used in the experiments, the Fast-Filtering algorithm achieves at least 1.6% reduction starning-augmentn
algorithms, while our Fast-Filtering algorithm consis-
in clustering cost compared to the current of-eeoo--theartt Regarding the running time, it can be seen from the table that the running time for all learning- augmented algorithms remains relatively stable the number of clusters increases. However, our Fast-Filtering algorithm significantly faster than other bearning-augmentie the datasets. with this trend becoming more pronounced as the data sizes and dimensionality grow. On average, our Fast-Filtering algorithm is at least 3 times faster than the current state-of-the-an Baaming-augmentt
k-means algorithm.
algorithms across all
algorithm. On the largest dataset SIFT, the running time of our Fast-Filtering algorithm is at least 10 times faster than other learning- augmented methods, which further demon- strates the effectiveness of our proposed sampling-basee -based strategies for handling large-scale datasets.
Comparisons with the Lloyd's and k-means++ Methods. When comparing with the Lloyd's method, it can seen from the tables that our Fast-Filtering algorithm at least 50.9% faster than the Lloyd's method across different datasets. However, other may not achieve much better running time than Lloyd's s method (such as on dataset SIFT). When comparing with the k-means++ method, average, our Fast-Filtering algorithm achieves at least a 20% reduction in clustering cost compared to k-means++. Furthermore, provides over 50% improvements in ARI and NMI values across most datasets. These results show the advantage of and sampling strategies for improving the clustering quality.
algorithms
incorporating
Discussions the Limitations for the Algorithms. In the following, give brief discussion on the bottlenecks for different algorithms. For sorting-based algorithms, such as those in Ergun et al. (202T); Nguyen et al. (2022), the primary bottlenecks are their running time due the sorting processes, where an additional Ólog m) term is included in the running time. The experiments demonstrate that the sorting-bast based algorithms do not scale well when handling large-scale datasets. On the other hand, one of the limitations for our sampling based algorithms is that they might achieve worse clustering quality guarantees compared with sorting- based methods. However, these guarantees are worst-case scenarios. Experiments show that our algorithms perform competitively on large-scale datasets like SUSY, HIGGS and SIFT.
A.6.1 EXPERIMENTAL RESULTS
26

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Published as a conference paper at ICLR 2025
Table 4: Comparisons on dataset MNIST for varying and fixed k = 20
Datssct MNIST (1,797 x 64)
Method
Ref
Cost
NMI
ARI
Timets)
k-means++
1.5448B406+22444
49
0.7087=0.013 39
0.5502=00.025
0.03=0.04
Ergun
9.9174E=05+1
0.968020.0036
0.9665=0.0053
0.34=0.68
Det
9:75666--055+00000::000
0.9819+00.000
0.9621=000000
0.13=0.17
Fast- Sas--Samping
0.60526405(3375)
9.6471EE-U5:::::33-000
0.984910.00025
0.9644-0.0027
0.11
Fast- Filtering
0.961410.0099
0.0793=000005
0.38
Fast-Estimann Estimation
9.7553E--05.3.799-00
0.980610.0023
0.0001=0.0025
1.82-0.37
k-means++
1.4755B=006-33770--40
0.6861+000170
0.422500.030
0.02-0.01
Ergun
1.000EE-06:-2255::000
0.9482+0.0055
0.9794-0.0077
0.12-0.01
Det
100366-006+00000000000
0.0000000000
0.9460=000000
0.09=0.02
Fast- Samp-ing
9.6177+5(1L49)
0.2
9.9901F=05++ 1.32:40
0.9581+0.0049
0.9484-6.0069
0.09-0.00
Fast-Filtcring
0.963220.0016
0.9553=0.0157
0.03=0.01
Fast- Estimation
L.199E--061
0.955611.00022
0.9487=000027
1.4_0.14
k-means++
L.551EE-0113411:44
0.707210.0191
0.542620.0494
0.01
Esgun
0.931620.0052
0.9254=000058
0.12=0.00
Det
1.441 1E-0610,00E-44
10,000
0.9300
0.08=0001
Fast. Sampling
9.5226F405(1.76)
0.3
1.073EE-06--33
0.9174+0.0040
0.8942-000065
Fast Fast-Filtering
9. 9.9369E-05544442
0.942210.0096
0.003
0.03
Fast- Esrimatio
1.995E-06+
0.9331+0.0065
0.9254-0.0094
1.78-0.13
k-means++
15541E=-0-+33300 0.4
0.6893+0002311
0.055
0.01-0.00
Engan
1. 18599=-06+5.2888 28E+03
0.8804+0,0059
0.359=000086
0.12=0.01
0.881++0.0000
0.8095-000000
0.08-0.00
Fast- Past-Samping
9.6728F:05(13325)
1. 13799-0022334444
0.8975=0.0009
0.84970000100
0.09=0.01
Fast- Filtering
10663E:-06:1 LoSE:04
0.897510.0022
0.017
0.03=0.00
Fast-Estiatiion
0.8851#0.0091
0.8433=0.0137
1.84=0.09
k-mcans++
0.7042=+0.0344
0.5263=0.0468
0.02-0.02
Ergun
0.848310.0084
0.7622-0.0130
0.12-0.01
Det
1.241 70-4117-00000000000000
0.8601 +00000
0.8040-0.0000
0.08-0.11
Fast Sampling
9.19928405(1229)
1.578E-00-15.77EEE
0.8622
0.8076
0.09-0.01
Filt-iittriin
16622R=06- 11-22:66+11111 1.11=044
+.0812
0.8000-0.0195
0
Fast- Past-Estimation
1.284:E-0062-4674460
0.847020.0049
0.7005=000080
181=0008
Table 5: Comparisons on dataset CIFARIO for varying and fixed A: = 20
Dataset CIFAR 10 (10,000 (10,000
3,072)
Method
Ref
a
Cost
NMI
ARI
Time(s)
means++
1.278IE+ 6626F++99
0.3995:-00011
0.1683=000207
1.90--0.08 0.08
Ergun
7.88 1
0.9096+0.0007
0.8879=0.0012
19.15=t1.00
Det
7,86 11:1119:1000:::00 0.00F+00
0.9009:0000000
0.8761-10.000
20.45:0.55
Fast- Sampling
7:83377+10337.115)
0.1
7.83355:-10::4666:::-:
0.897010.0131
0.867111.02255
19.91.1.49
Fast- Filtering
7.8611F=+0
1.59F:07
0.9106=0.0042
0.8941=0.0060
5.61
Fust- Estimation
7:86331E-10:16.566:-66
0.918610.0006
0.90741000000
185.51
k-meanse+
0.4043+0.0165
0.1601=000129
1.89-00.04
Ergun
2:9135E--10-33.77--00
0.8700:0.0010
0.8292.0.0013
18.67.10.32
Det
0.8515+0.0000
0.7999 0000
20.20+00.09
Fast Sampling
7.83066++10333199)
0.2
7.9257E=10:11622..
0.808710.0218
0.735410.0348
19.73
Fast- Filtering
7.91 101122+-114.777:447
0.831410.0100
0.768210.0172
5.1910.10
Fast- Esss-Estittton
7:9180E--10:339336600
0.856820.0008
0.806620.0013
187.99900.9
means++
1
0.3975
0.1643.10.0248
1.97
Ergun
8 1.0341E=11--6..0....00
0.7999
0.7190=0.0010
18.77++0.00
Det
8.0095E:10000 100.EE=10000000::0000
0.7727
0.671110.0000
10.33.10.18
Fast- Bast-Samliing
1,8202++10304424)
0.3
8.0278E=-1::22155::00
0.743510.0165
0.617320.0301
19.86:10.12
Fast- Filtering
1.0760E=+10.33001.:.
0..001...0014
0.607210.0231
5.06
Fast- Estimation
8.1302E=110:11.61::40
0.762610.0011
0.6857:20.0013
188.72110.22
k-meanse+
+ 1493.46F
0.4062+0.0104
0.1628+0.0075
1.99-00.06
Ergun
0.6316.100013.
18.94
Det
0.7347+0.0000
0.601 1+0000000
10.16-0..08
Fast. Sampling
7.8061E+1022696s)
0.4
1.19115=+10.1..88...
0.675110.0337
10.0600
19.10
Fast-Filtering
#.3463E--10::2224446
0.6767:0.0132
0.99662000174
5.03::0,08
Fust. Estimation
1..9000E=+0.13..6..22
0.703110..011
0.5772....0017
188.79 1.89
k-meanse+
+ 11:.5-++11122.1---000
0.411620.0221
0.1646=000024
1.95=20.04
Frgun
8.3372F 10-+ 1.82
0.7000=0.0010
1.561440.0016
18.76-40.23
Det
8.4653 111--100000000-000
0.6877100.0000
0,509010.0000
20.24.10.10
Fast-Sampling Fast- Filtering
7.8211E+10022.77))
0.5
8 4
1,2
0.6131-0.0207
+0.0306
19.85-0.115
8. 1.8030E+100.5.
0.005210.0170
4,95 0.04
Fast- Estimation
8.7858E==10::500355
0.6875:000012
0.484400.0014
190.8612.17
27

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Published as a conference paper at ICLR 2025
Table 6: Comparisons on dataset PHY for varying and fixed lt = 20
Dataset PHY (10,000 x 50)
Method
Ref
Cost
NM1
ARI
Timets)
k +-meanc++
11591E+111122115
0.8505.10.033
0..79510.0348
0.1910.08
Ergun
2.9413E+
+772
0.9921=00.018
0.9919+-00024
3.17:40.61
Det
2.9368E+1 111.++++1100000000 10000E+00
0.9937 0.0000
0.994210.0000
2.3910.117
Fast- Most-Sampling
2.9358-11(15333) 0.1
0.0658=000055
0.951411000188
2.37.10.13
Fast-Filleria Fillering
11171E++1115.222:/1
0.9869_0.0012
0.984210.0017
1.52
Fist- Estimation
2.9413E:11:2:.334-00
0.9888=0.0025
0.987010.00334
5,98:00.39
k-means++
0.8522-0.0134
0.679++0.0382
0.18--0.07
Ergun
3,1388E+11.122411
0.9815 0.0002
0.9767.0.0004
2.72.10.10
Det
0.9867-0.0000
0 0.9853+-000000
2.15+00.77
Fast. Sampling
3.0196E-11(99865)
0.2
3,157/1=+1118.000:
0.90122..0092
2.38..0.11 0.923110.0175
Bast-Fillering
0.9791=0.0041
0.972740000065
0,56
Fost-Estimation
11800E+11==300E5
0.9793=0.0024
0.9741#0.0033
6.00=0.133
k k-mmunn+++
4.1248E+11 1.......11...1..--11
0.8552.L0.0107
0.691510.0301
0.05
Ergun
3,2578F 09
0
0.9681 +00001
2.58+00110
Det
3,2548E+
10...00
0.9738_00.000
0,964410.000
2.2810.04
3.0331=-1(22.04) 0.3
3.5779E=+1::11114:11
0.0291=0.0165
0.874611000355
2.28:00066
Fast-Fillerne Fillering
111726E+1111..8EEA
0.9506_0..023
0.924810.00M9
0.50
Fast- satt-istimation
3.25:6E+11::1221::000
0.9508=0.0061
0.922810.0122
5,8020.13
k-means++
1377778+1++2..777
0.8514-0.0209
0.6700+0.0611
0.15--011
Ergun
3.150ME=1111113E-119
0.9793. 0.0033
0.9704100006
2.6310..10
Det
111356E+11++61000
0.9709-0.0000
0.9619+000000
2.34+-0.06
Fasi Sampling
3.0385--1120.17) 0.4
1.29077E:11144.32:--
0.0011
0.791410.0505
2.35.00.07
Fast-Filtering
3.3306E+11::..884411
0.9279=0.0077
0.893611.00157
0,53:20.04
Fast Estimatiin
3,3655E+11 11..644-111...4--1
0.9336_00.0101
0.888310.022
6.11.10.22
k-means+++
4.3389E+ 1:
0.8444000112
006601+0.0345
0.12=00.04
Ergun
3.2413F+
0.9796-0.000
0.9758
000004
2.68--0.09
Det
3,2330E+11 16..0000
0.9791-0.0000
0.975610.0000
2.4010.07
Fast- Sampling
2.93775-11166623)
0.5
6.4952F+ 117.75
0
0.721440.0346
2.36--0.08
Dill-riileggng
1.2278E411111.995
0.868410.0108
0.724910.0339
0,.08
Fast- Eess-Estimatin
3.4066E+1::93925
0.9157=0.0093
0.8503:0.0203
6.07=0.144
Table 7: Comparisons on dataset SUSY for varying and fixed k = 20
Dutaset SUSY (5,000,000 x 18)
Method
Ref
Cost
NMI
ARI
Time(s)
k-means++
1.4792840716617885
0.4743:0.0152
0.2597-10.0163
8.61:10,1
Ergunn
2. 2.5432E-077 4.5432E--7..5.5SEE
5.5SE++0
0.8778:000000
0.8672-10.0003
130.97.12.89
Det
2.522]E=07100000E4:
0.913820.0000
0.9178:10.0000
154.27122.44
Fast- Past-Sampling
2.4466+07(4246677)
0.1
2.5247E+0748.81E400
0.9166=0.0013
0.9209=:0.0017
119.50=22011
Fast- Filleril
2.4921E-4071667711000
0.8434.000003
0.820310.0104
36.77:1.115
Dast-Estimation
2.5246B-0742.43B400
0.9031=0.0008
0.9040=0.0010
51.30=0.76
k means++
1.7779E=07.11.777::
0.003310.0130
0.2604 0.0225
9.31
Ergun
2.75335-07:7.384440
0.847420.0001
0,8224
134.3511.44
Det
4.7162E=00.10.00EE..
0.8866 0.0000
0.0000 0,8801
155.83
Fast- Sampling
24551B-07(318.44))
0.2
2.72535-00:11.122:45
0.862720.0573
0.8447:10.051
121 201:291:.6 .664
Fast-Filtering
2.67699-407:22.06E---0
0.8207=0.0224
0.7920=0.034)
1.88
Fast- Eistimation
2.70015-0713.1254005
0.877020.0003
0.8674:10.0004
53.58:1 1.20
k k-means++
3.4445F
0.4908 +00012
0.2675
0.
Ergun
0.8395 0.0002
0.8000 0.0000
1..65
Det
1090033-0711000004000
0.8778
0.8593
157.23311.99
Fast-Sampling
2.446E--07257.188)
0.3
2.924555-0799.444--0
0.8489=0.0256
0.8281
122. 122.78222.88
Fast- Pist--iltering
7.84399--0022222002--5
0.823220.0153
0.78392000244
11.32:11.22 29
Dast-Estimation
2.905 1B+077772227
0.8669=0.0000
0.8443=20.005
54.033=0.94
k memmst+
1.46885E-07.15.34EE..
0.476410.0177
0.2616.10.028
9.37.10.41
Ergun
3.26236-0748.384-03
0.8523
0,8178
133.73+1.38
Del
1.26955E=07.10000E...
0.859010.0000
0.8595 0.0000
154.98 1.35
Fast- Sampling
2-4811E-07:303.399
0.4
3.24125-00::13322445
0.814520.0513
0.7796:10.075
124.0912.00
Fast-Filtering
3.2218E--07::388B--
0.8094=0.0172
0.7873=0.0236
0.959
Fast- Estimattin
1.770EE-77:1.0SE400
0.852520.0008
0.8508:000007
24.47
k- k-meanst++
0.4963+-0.000
9
Ergun
3.43165-07.L
0.8350 0.0004
0.8033
133.74
Dett
3.4516E-07200000--40
0.838720.0000
0000000000000
154.20=1172
Fust Sampling
2.4300--0(252.522)
0.5
0.7282.10.0611
0.6470 0.0994
123.75.1..04
Past-Filtering
3.3083E-40::44144--00
0.7705=0.0293
0.7183=0.0488
40.01:2.886
Past-Estimation
3.602E-07+1.308440
0.8467=0.0116
0.8225=0.0147
1..91=1.70
28

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Published as a conference paper at ICLR 2025
Table 8: Comparisons dataset HIGGS for varying and fixed k = 20
Datnset HIGGS (11,000,000 x 27)
Method
Ref
Cost
NMI
ARI
Time(s)
k meanset
2.0309E:08 1191E+06
0..749.0.00182
0.1813.0.0012
21.36.10.25
Ergun
1.4083F=08=+
0.9093+000006
0.9094=0.0009
432.424-2.72
Det
1.4019E=08 10.00E++0
0.9613.10.0000
0.9674.0.00M
481.06.12.76
Fast Dast-Sapliig
1.3924E-088661.322
0.1
1.41998E-UX:28.3335--4
0.9618:0000077
0.963920.0007
428.25.:3.12
Fast. Filtering
1. +.4 13.72E+004
0.8989100.0021
0.89371000002
136062.17.03
Tast-Estimation
0.9471110.0010
0.952420000008
171.1422.17
k-means++
2. 2.0562F=008-1 00877885.40
0. 0.3627+0.0206
0.1756+0.0166
20.66-+0.36
Ergun
1.4383E=08
0.906010.0010
0.905610.0013
4.5.37
Det
0.9146-0..000
0.9149-0.0000
3.1458+3.41
Fast Sampling
1.391E-000(22245)
0.2
+4 13.68EE+04
0.878010.0019
0.0021
1.2.06.11.99
Fast-Filtering
148122E-08=11344:5
0.8547::0.0114
0.83762000066
146.5315.221
Past-Estimation
1.4252E 1.300+0
0.9001=000013
0.8950=0.0018
171.63+3.22
k memanst
2.0574E=08 14.32E+006
0.367610.0145
0.179310.0095
20.81
Ergun
1.8947-0.000
0.8942-4 0.0023
0.8942-0.0023
418. 54--544-7
Det
1.4613E=08100 10.000++0
0.873910.000
0.8697
469.16 666.16.16.86
Fast- Dast-Saapling
1.3999E-08(79.155)
0.3
0.8198::0.0059
0.7849=0.0109
113.38:112.49
Fast Filtering
1.500SE:008.
11.37E+05
0.807510.0070
0.825610.0000
144.12.1..13
Fast- Nast-Estimation
0.8557:000008
0.8412=000006
167.43:1.117
k-means++
2. 2.0099F:00 1-3.888--5F00
0.3818+0.0094
0.1917+0.0150
20.00-+0.32
Ergun
1.5418E+08.11 11.05++0
0.8777 0.0002
0.870610.0035
413.61
Det
+8 08-0000F+000
2.8011+0.0000
0.7693+00.0000
459.55+ 1.29
Fast Sampling
1.3957E-00/707.88,
0.4
+0 12,42E+05
0.747910.0225
0.676110.0335
397. 4.55
Fast-Filtering
1. 1.8814 -.:-:0000055-::00
0.8054200.0082
0.771620.0092
13,727
Fast Estimattion
1.818E+00. 1.,222-6
0..065
0.7702.10.017
160.61 L2.36
k-means++
2.0108E 06
0. 0.3927+0.0080
0.1976=0.003
20.54=00.48
Ergun
0.6733+0.0072
0.5333-00.0119
431 5.45
Det
1.5447E=08-0000 10.0000+0
0.733210.0000
0.00000
478.20.18.2 28
Fast. Sat-SSmpling
L412TE-08(7L88)
0.5
08 +-2.188:+05
0.5783+-0 0.0020
0.3969 -00088
428.04 6.8.2
Fast- Filtering
1,5395E
1.17E
0.754310.0112
0.677110.0156
129.40.14.43
Pust-Estimation
1.543LE+08=6.900E
0.7498:0.0356
0.6550=0.0615
164.924:4.15
Table 9: Comparisons on dataset MNIST for varying k and fixed a 0.2
Dalasel MNIST (1,797 64)
Method
Cost
NMI
ART
Time(s)
k-means++
1.9717E-06-22633E-
0.639210.0615
0.521310.0737
0.01:10.01
Ergun
1.2304E--06
8888
1000039
0.9490 0.0009
0.33
Dei
1.2186E=06112.33:
0.945610.0000
0.0499000000
0.141:.117
Past-Sampling
1. 1.16944062.01s)) 10
1.2201E=00=1.50EE-
0.933010.0059
0.9256=2.0077
0.12-0.11
Fast-Filtering Filtering
0.948910.0123
0.952010.0130
0.4911.41
Fast- Estimation
1.23099+0:21.85544
0.9425=0.0055
0.9426=000057
1.32#0.41
means++
1.5699E=06..00 5669E=006.4.622444 14.62E:+04
0.85510.0015
0.406110.0117
0.01
Eigun
1.01199406:2..22400
0.951910.000
0.039110.0061
0.11110.00
Dei
9 10000E
0.9591 10.0000
0,9479 1447910.000
0.07.1..00
Fast- Samp-iag
9.5547+05(1.99)
20
9.991 11.44
0.958410.000
0.9493
0.08:00.00
Fast-Filtering
9.6684E--5511111B--
0.9619
0.9498=0.0099
0,02-10.00
Fast- Estimation
1.00115-0629.514400
0.9619:1.0032
0.0007
1.6020.00
k-meanse+
+
0.7127+0.012
0.4844-0.0313
0.02+00011
Ergun
11.79EE00
L9.002
0.9283 000006
0.1..0..22
Det
1.88972--05:111166-11
0.959210.0000
0.9482-000000
0.0910.01
Past-Sampling
8.5000B+05(1.679)
30
4:97995+0599,56E-000
0.9488:0.000 0049
0.9266 0.098
0.0000.22
Fast- Filtering
8.646682--05
0.9622:000000
0.9524:0.0091
0,03:10.00
Fast- Estimation
8.9725E-05=9.9444000
0.9503 00001
0.9345=0.0026
2.45=0.24
means++
0.7373100.077
0.459910.0197
0.03.1.01
Ergun
8.3789E405
0.950810.000
0.9207
0.17200.2
Dei
0.9640 10.000
0.9451 0.0000
0.11
Fast- Dast-Samplin
7.2018B+05(1444) 40
8.2700E+05=11001:4:0
0.9573
0.9328
0.12-00.2
Fast-Filtering
7.844588-05=11141E---
0.9595=0.008
0.9360=0.0145
0.04=0001
Fast- Estimation
8.23222-05=992224442
0.962210.003
0.9427=000066
3.08
E-means++
1. 1.1248T=06 -52299--0
0.7384+00.066
0.4296-0.0179
0.03+-0.01
Ergun
7.9776E+05 12246E+03
1943440.0003
1.8944_1.0007
0.19.0..1
Det
7.6429E-05=0000E-4-0
0.9601 10.0000
0.9274=0.0000
0.12-0.01
Fast Sass-Saallingg
7:22738405(1.539)
50
7.7443E=00 11.
0.953310.000
0.0033
0.9224 0.0057
0.13100.00
Fast-Filtering
7-41
0.6611-0.0108
0.9257=0.0237
0.05:10.1
Fast- Estimation
7.7657E-05=1.15E44
0.9567+0.0030
0.9200=0.0077
3.62+0.06
29

---

Published as a conference paper at ICLR 2025
Table 10: Comparisons dataset CIFAR10 for varying & and fixed a 0.2
Dataset CIFARIO (10,000 3,072)
Mothod
Ref
Cost
NMI
ARI
Timo(s)
means++
1,4120E-11 1.97550099
0.387310.0302
0.2036..0.04
1.02.0.09
Ergun
0.8480+0.0046
0.8335-0.0055
19.59-1.09 1.09
Det
8.5355E: 1.53E-405
0.803510.0000
0.8160 0.0000
0.57
Hast-Sampling
1.6673E+10(10.44)
10
117771E-10::44444440
0.7855:10.3303
0.7499
20.76=0,48
Fast. Filtering
8.5735E:-10-110000--
0..18610.0116
0.7920
6,1511441
Fast- Estimation
8.5537E-102:40044-00
0.846910.0011
0.8527
125.27::2.99
k-means++
1.2415F= 147.00000
0.4005+0.0294
0.1541
1885-0005
Ergun
1.9972EE-1114511B.
0.8700
0.8281
0.0011
18.55
Det
7.8750F: 10000000000
0.8516+0.0000
0.090000.0000
10.19-0.26
Fast Sampling
7.8808E+10.36.115)
20
119133EE-10....311-00
0.803310.0157
0.7266
1916
Bast-iiteriin
7.8920EE--0:8.799:-00
00,00140.0084
0.7750=0.0157
4.92:00,88
Fast- Estimation
7.9032E-10=3.84B44
0.8602#0.0010
0.8140=0.0016
182.33:11.99
mean+++
1.1677E-1113.6224
0.430410.0109
0.1612L0.0131
3,4410.12
Ergun
0.8618+0.0009
0,7931 0.0016
18282-0.22
Det
0.847210.0000
0.7746.000000
19.12-.0.11
Sist-Sampiing
7:5010E+10:29.54)
30
7.61099--10-78855-411
0.81474:0.1167
0.0203
19.49=0.22
Fast Filtering
7.559915-100...29--77
0.829210.0061
0.0471
4.7410.11
Fast- Estimation
7.6431E--1422.0044000
0.851020.0008
0.7906=0.0013
238.71:14.0
k-means++
1.1570F= 1103F---11-444455 1444488--9
0.4279+0.0152
0.1268-0.0233 0.0233
4.67--0.0
Ergun
7.3929E+10.1. 10..:0E--10121111-0000000000
0.8851 10.0008
0.8288
0.0013
19.35
Det
7. 10-0,00F=00
0.8667+0.0000
0,8000 0.0000
19.74-0.16
Fast- Sampling
7:2836E+10:37.33)
40
7.3839E-10....990 1.39E+0
0.824510.0126
0.7198
19.49_0.22
Fast-Filtering
1:5691E--10:8:40--007
0.8272=0.0101
0.7275=0.0185
4,45:0.09
Fast Estimation
199833E-1012.5900
0.862210.0009
0.7878L0.0012
279.6910.81
k-means++
1.4455= 1+:4 80E+09
0.4276+0.0137
0,1235-0.0186 86
5.77-0012
Ergun
7.29022:-10--22 2.1
0.8693:0.0006
0,7979 0.0011
19.46 0.31
Det
0.848110.0000
0..000
19.98-0.18
Fast- Sampling
7.1331E+10:99244)
50
-
10-
0..10
0.6888 0.0365
19.67-0.14
Fust Filtering
7.2331E=-1-7.119--00
0.849810.0064
0.7603
1.,7
Fast- Estimation
7.304E--0==3001--06
0.8555=0000066
0.7715=0.0013
334.36:1..25
Table 11: Comparisons on dataset PHY for varying i and fixed a 0.2
Dataset PITY (10,000 x 50)
Method
Cost
NMI
ARI
Timo(s)
k-meamst+
1-450E+11211.21-111
0.8077
0.667811.0766
0.0810001
Frggnn
1.033EE
1.55500
0.9772- 0.0043
0.9052.10.0051
2.75.10.66
Det
1.0236E+12:1.226-00
0.9001
0.980610.00000
2.4810.16
Fast- Dast-Sampling
1244622)
10
1.0853E+12+1.122-10
0.9570=0.0128
0.9207=0.0199
2.4520.09
Fast- Filtering
1027734+1221.699-009
0.9763=000026
0.975710.0027
108:1441
Past-Estimation
1.0288E+12:3.199-00
0.9709=0.0074
0.9679=t00100
4.07
meamst+
1.3425E+11.1..19--10
0.0176
0.6685 10,0442
0.13
Ergun
3.11675+111228754008
0.901
0.970010.0060
2.69
Det
3.070SE+11166.002
0,9882
0.0000
0.9866 0.9000
1.310
Fast- Sampling
3.0028E+110.935)
20
3.26632+1111.221644
0.9455
0.9118-10.0219
2.29:10.11
Fast-Filtering
0,9789=0.0033
0.974020.0053
0.50=0004
Fast- Listimation
3.864115-1112.5585-08
0.9787=000022
00.0312
5.98100227
k- k-means++
2:41344+11+44444.-
0.8562-0.0085
0.65614-0.024
0.15+-0.06
Frrgun Det
1.77100=+1111,66-000
0.981410.0020
10.0031
2.53..0.09
1.7480E=11::305::555
0.9000
0.9825110.0000
2.19100.5
Fast- Dast-Sampling
1.696E-11(88.06)
1.9463E+1133.11-000
0.9393=0.0084
0.8845+0.0224
2.20=
Fast- Pist-Filtering
117.
0,9563=0.0012
0.921420.0024
0.50:0002
Fast-Estimann Estat-stttttion
1.7302E++11+1.088-66
0.9783=0.0022
0.9009+0.0033
7.27=0.09
meamst+
1.770SE+1114.01E.
000070
0.6561 10.0249
0.16
Ergun
1.9471E=+1111.215000
0.9812=0.0024
0.971420.0048
2.73:00.15
Det
1.31177441111..335
0,9816 0.0000
0.969410.000
2.1910.06
Fast- Sampling
1.2380E-114).269) 40
11-66224+11144444---0
0.934620.0077
0.369310.0221
2.19:0.05
Fast-Filtering
112451E--11+4.92---00
0,9763=0.0014
0.9601
0.52=00.4
Pust-Estimation
1.30277=+1119.432-00
0.9798=0.0035
0.968420.0079
8.63:0.115
k- k-meass++
0.8544-0.00 0.0065
0.6206++0.022
0.19:-0.05
Eirgun
1. 1.189EE+111..88:4
0.9821 0.0017
10.0031
2.62..0.08
Dett
1. 1.7700EE-11100000--00
0.9785=000000
0.965320.00000
2.1020.04
Fast Sampling
9.8416E=10(50.111)
50
1.1.86E=+1
0.9321 000000
0.0000
2.12..0.44
Fast-Filtering
1.0107E-11:66338--0
0,9705=0.00
0.94562000006
0.50:0.02
Fast-Estimation
0.9750=0.0032
0.9572+0.0067
9.64+0.37
30

---

Published as a conference paper at ICLR 2025
Table 12: Comparisons dataset SUSY for varying h and fixed a 0.2
Dataset SUSY (5,000,00 18)
Method
Ref
k
Cost
NMI
ARI
Time(s)
k means++
0.221210.0495
0,2851
Ergun
3,1954F407+
0.8037+000014
0.7953++0.0017
129.61+-2.33
Del
3.1759E:+7710
0.8628..8
0,8696
159.18.12.90
Fast- Satt-Sanlling
2.9914E407(194066)
10
3.2683E4407
0.59990000199
0.8760=0.00221
12191111.6
Fast. Filtering
3.1814E+07
0.8662 10.0044
0.8747.1.0000
37,52.1
Fast- Estimation
3.1779E40717.75H6B
0.847910.0031
0.852120.0032
48.18:11.57
k k-means++
4.3848F+07-337FH405
0.4333-000107
0 10.71-00.009
9.27-0.24
Ergun
0.8507.L 1.
0.201110.000
135.43 2.5.4312..66
Det
2.6856F=07-D
0.8913-0
0.5805+0
155.56++1.55 1.50
Fast Sampling
24533E+07(307005))
20
2.6889E+07..3.EE
0.89910.0000
0.8778 1000013
125.01 12.09
Fast-billering
26121E=07=331--005
0.2477000009
0.789810.0144
39.78:1.07
Fast- Sest-Estimatio
2.6900440==6:9400
13807=0.0006
0.8664=0.0000
54.35:10.77
k meuns++
2.991.EE007
0.514010.0048
0.2509L1.0011
12.30.10.15
Frgun
2.4828F
0.8562-0
0.8133+000000
134.08--088
Del
2.4707E=071D
0.8915_00
0,875910
152.0911.88
Fast- Sast-aanli
2.1903EH07(20044g)
30
2.4700440748.5440
0,870000.0007
0.8753=0.0007
117.35
Fast. Filtering
18.6410.1161
0.827810.0288
06.72.1..79
Fast- Estimation
2.4646E407111774400
0.8705
0.843520.0345
$4.3720.95
k --mann++
2.7709E--7
0.5285+0.0152
0,2408- 0.0015
15.56-+0.88
Engun
0.893L0.0001
0.0002
444.4412.17
Det
2.3197F=07-0
0.8813-0
0.8525+0
149, 149.2440.977
Fast Sumpling
2,0262E477913.663)
40
0.872410.011
0.338510.0151
110.81 12.21
Fast-Filtering
2.20
0.846620.00 0065
0.7992=0.0120
15.08::.772
Fast- Estimation
0.8788
0.838810.088
55.65
k-means++
2:5509E-07+-2444455
0.5477 0082
0.0237=0000177
17.48-00.32
Ergun
08586--0.00
147.95 1,91
Del
2.1894E=07.D
0.8805
0,8371100
144.9911995
Fast- Samplinn
19136E400((3454223
50
2. 779999977 L.F
0.396--0.0289
0.7749 0.0068
101 3.4
Fast-Fillers Filtering
2.0645EH077...EE
0.871610.0065
0..1111000112
29.77.1.34
Fast- Estimation
2. 2.1435E40=441E440
0.833920.0026
0.7645=0.0046
54.1911.51
Table 13: Comparisons on dataset HIGGS for varying h and fixed a = 0.2
Dataset IHGGS (11,000,000 x 27)
Method
Cost
NMI
ARI
Time(s)
k-meanst+
2.3351E408 4.0--000-4.8SEE006 85E+06
0.3063
0.1806110,0030
14.92:10.12
Ergun
1 1,5833+00 44. 95
199263-0.0044
427.5011..17
Det
1. 1.5626E-48-1000 10.0000000000000
0.90300000000
0.9127-10.0000
481.121:.888
Fast-Sampling
L.87718008343.725)
10
1.5787E++08+3 68E+03
0.9465000006
0.9595=10.0077
443.06:26.99
Fast- Fillering
1.55590-408111.99E 1.99
0.9028=000067
0.907910.0006
129.32::5.94
Fast- Dast-Estimation
1. 156637E408:7..6EB
0.9176=0.0014
0.9260=t0.0010
164.52:22.6
memans++
2.0632E:408.12 2.4991-00
0.3454.0.000 0.0042
0.1658 10.009
20.5610..33
Ergun
1 44375E008111.22 1.261163
0.9066:10.0012 0.001
445.15::2.91
Det
42277E+088 1.000E+000
0,9148 0.0000
0.9135 1.33510.0000 0.9000
4...35..0.50
Fast- Sampling
1.9008+088706694
20
1. 111.777--00811..33---
0.000
0.8768110.0017
439.74
799.74:222222
Dast-Filterin
0,9017=0.0322
0.893720.0413
142.67=1882
Fast- Estimation
1. 42377E-408.2..11-1--0
0.9020=0000044
0.8971
173.38:11.6 1.69
k-meanst+
8 111
0.4060-0.0092
1772++0.0061
28.81++0.08
Ergun
1. 3..633H08 8,8998+03
0.908310.0000
0.8985 10.0000
450.15.19.77
Det
1.35 BA1.EEE----1.00000...... 00E+00
0.900
1.900000.0000
478.83226.9 633-66.944
Fast-Sampling
1.2411E00(1433.667))
1. +.355-44+08++66 70E+04
0.005
0.8522+000005
425 87:
Milt-Filtering
1..4255--08.8..00 988444
0.8739=0.0087
0.848220.0128
147.26:17.24
East-Estimation
1. 1.3525E+08++3.0 33
0.896=00.0002
0.8855+0.0003
175.96:
k-manss++
1. 1.7659E+488LL 111.EE-HH8.11.75E-- 11755
0.4153_.000122
0.1610.10,0086
35.46 0.559
Ergun
1.3153E+088422 7277
0.9046=0.0007
0.8921 10.0000
445.7919.25
Det
1.302 TE:488. 10.00E+00
0.9146 0.0000
10.0000
470.93
Fast- Saat--ampling
12204E-08(2001.98)
40
1. 1.3060E:+08-144 99E+0
0.0433
0.827610.0367
398. 501:9.999 65
Bast-Filtering
1. 1.00
0,8936=0 0068
0.8727+0.0101
139,57=0059
Fast- Esttmattin
1. 1.3002EH08:14
500+40
0.8958=0.0002
0.87752000013
172.35
k-means++
I 5F-05 74744-05
0,4342 00002
0.1611 +0.055
38.57 0.00
Ergun
1 27730E+08 1,455+04
0.8785 13785_0.000 0.0229
1..0333
436.17.10.26
Det
1 40.288--88--00000 00.0000000000
0.0000
0.893210.0000
443,46:20.36
Fast- Sampling
1.2336E-083354413bg
1 2645E=08 1.07E+05
0.8666 0.0028
0.8303 10.0324
364.05
Fast- Filterinn
0.8542=0.0015
0,8065
133.17:0.40
Ests--stttatio
22587E+08+1.53B--4
0.8958=0.0017
0.8710+0.0019
160.75:20999
31

---

Published as a conference paper at ICLR 2025
Table 14: Comparisons on dataset SIFT for varying i and fixed 0.2
Dataset SIFT (100,000.000
128)
Method
Ref
NMI
ARI
Timeo(s)
k-means++
1.8977=+344
141F+11
0.2994-000180
1.2026--000087
801.75 114.99
Ergun
0.906510.0000
0,9085 10.0000
23298.85 1009 69
Det
1.083:EE413:00000--44
0.911520.0000
0,9146=20.0000
94759-4924011.9 91
Fast: Sampling
1.05-2E+13844.18.
10
1.20831:-133.221155-11
0.5353.0.0306
0.431910.0255
14370.31 1.502.5
Fast-Filtering
1.0763EE-13:55.33::000
0.857920.0006
0.8959=000000
1013.03::7.22
Fast- Estimation
1.0856E=-1::3.276-666
0.9160=0.0004
0.9206=000008
14612.00++312.5
memmmt+
1.6112E:13.11.2662.
0.326910.0119
0.1353.10.0315
1329.66 173.552
Ergun
1.2000E=133+6--41:000
0.9012+000000
0.8012-40.000
20445 05 2-333 90
Det
9:9855SE=12210.000E:-00
0.002410.0000
0.891410.00000
111551991576.65
Fast- Dast-Sappling
9.705SE412(011.224)
1.1236E=13:225555
0.5600:0.03310
0.4002-0000483
13877.33:=778.0
Fast Fillering
9.8954EE-11211.4441::000
0.876810.0055
0.8637.10.0062
1059.94199,
Datt-Estimation
9,9979E-412:4.008---0
0.9077:5000022
0.9002:2000003
13987.
1017117
means++
1.52:8E=+13+2.51B..0
0.3730:-0.0112
0, 0.0094
2238.99 99 :46.66
Ergun
9.7328E=12.133 43.
0.001311..0000
0.8737100.0000
19099.7411 71.9944-11338 366
Det
9.5218E==12::00000F:
0.9027 -0027--0.00 0.0000
1.8745:1000010
14119.11 17 1024.58
Fast- Sampling
9.2478E-12(3300999)
30
1.1072E-413:4.911:::
0.542910.0058
0.3297:10.0074
13738. 711.63.40
Fast-Filtering
9:42822E-1::33388E-09
D 8863:200001
0.8590=0000082
1100.15:1044.2
Fast Estimation
9.53522=+12.16.386:-0
0.9072:10.0001
0.8832-10.0002
13633.175 18633.17.1144..77
k-means++
13:00
18 1.8P..
0. 0.3936+000072
0.1610=00.002
2976.74 7076.74-255.20
Ergun
9.4558E=12.11..2225
0.901910.0001
0.8699-10.000 10.0001
18493.35 995
Det
9.2487E-412:10000B22
0.9017110.0000
0.006000000000
13813, 1659.99..
Fast Sampling
8.97999-12213422739)))
1.0516E:13..1 1.6
2.582610000084
0.373610.0224
13533
460
Dill-rilieinng
9.1579E--12:333311:000
D. 888610000065
0.854910000076
1153.74..3.15
Fast-Estimation
9.25855E+12::446664
0.9070:00.0002
0.8780=20.0002
13920.46:2847,
means++
1.4553E+13.11 1.43E=+10
D.414610.0013
0,1681 10.0003
3410.89 157.58
Ergun
9.2450E=11::111 1.5b+407
0.0.0000000000
0.8656200.001
16390.58 1034,034,,4 .444
Det
9.0398E=+12.000 12000000::0000
1.8995.0.0000
0.8590
13948.
794.19
Fist- Sampling
8.7576E-12(1412.61)
50
1.040SE=43:11
0.5582:0.00135
0.3143:10,0204
13484.66 01484.681120 68:1622111
Fast-Filtering
8.9381E-12:1129888
+09
0.8 8890020.0003
0,8405=20.000
1157,97 133.86
Fast- Estimation
9.047115-122441111:0
0.9059:00.0003
0.8033
13367 68 1455 83
32

---

Published as a conference paper at ICLR 2025
A.6.2 VISUALIZATION FOR THE DISTRIBUTION OF THE PREDICTED CLUSTERS
In this section, we provide examples of visualizations for the coordinates distribution on some the datasets used in our experiments.
0.04
0.82
0.00
000
-0.02
2.02
0.
(a) coordinates cluster
(b) Plot coordinates cluster
Figure Examples of visualization of the coordinates distribution on dataset CIFARIO, where red points represent correctly predicted data points, and other colors indicate data points that are predicted incorrectly.
0.00
0.44
0.00
-0.00
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(a) Plot of coordinates cluster
(b) Plot coordinates cluster
Figure 4: Examples of visualization of the coordinates distribution dataset PHY, where red points represent correctly predicted data points, and other colors indicate data points that are predicted incorrectly.
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Reviews:
[{"review_id": "cUOaVe2m9o", "reviewer": "Reviewer_BxBK", "summary": "The authors provide asymptotically faster algorithms for the Learning-Augmented k-means problem by going beyond the initial sorting strategies and instead using sampling-based strategies. Their run time is $\\mathcal{O}(m \\cdot d) + \\mathcal{O}_{\\epsilon}(k\\cdot d)$ from \n$\\mathcal{O}(m \\cdot d \\log m)$. They further propose a heuristic algorithm using their sampling-based ideas that perform faster in real-world datasets compared to the existing learning-augmented methods.", "questions": "1) What is the quality of the clustering output for the K-Means algorithms proposed in the paper and those used for reference with respect to the ground truth labeling of the datasets (such as NMI and/or ARI values)?  \n\n2) When compared to K-Means++ (without any prediction), what is the improvement in performance on the Learning augmented K-Means algorithms on the datasets the authors have considered? (both in terms of K-Means cost as defined in the paper and ARI, NMI values w.r.t underlying ground truth labels)", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The learning augmented K-Means problem seems like an interesting setting (even though considerably less studied/without clear applications), and in practical scenarios, a speed-up of  $\\log m$ is a noticeable improvement. \n\nThe authors clearly explain their ideas and results and provide several experimental results.", "weaknesses": "1. The run-time of the theoretically guaranteed algorithms provided by the authors *does not improve* for the experimental datasets used compared to the existing algorithms.\n\n2. The authors provide a heuristic algorithm to improve their algorithms with guarantees that it has a faster runtime in practice; I think comparing its runtime with algorithms with theoretical guarantees seems unfair. There is a large (formal as well as informal) literature on many different variants of K-Means. One of the primary strengths of the papers by Ergun et al. and Nguyen et al. is that they have theoretical guarantees.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hwdmFTbwRU", "reviewer": "Reviewer_egdN", "summary": "In the k-means problem, the input is a set P of m points in d-dimensional Euclidean space and the goal is to find a set C of k centers that minimizes the sum of the square of the distances from P to C. K-means is important problem in machine learning, and has been proven to NP-hard to find a solution with approximation ratio smaller than 1.07. Therefore, the popular practical algorithms for k-means have a large approximation ratio, while the algorithms that promise a small approximation ratio have exponential run-time. To overcome the barrier of inapproximability and still have a fast running speed, the learning-augmented k-means problem becomes popular these year. In the learning-augmented k-means problem, we are given an oracle that predict the label of points with an error rate alpha. The goal of the problem is to solve the k-means problem with the help of such oracle in short time and low approximation ratio.\nIn this paper, the authors propose three algorithms to improve the run-time compared to the previous research. The best result of previous research provides an algorithm that generates an (1+O(alpha))-approximation in O(md log m) time, while the algorithm in this paper achieves O(md) + ~O(kd/alpha) run-time. The novelty of their algorithms is that the author use a random sampling strategy to generate the solution, while the previous researches all use a sorting-based techniques, which inevitably induce a log m dependency for run-time.\nIn this paper, the authors first propose the Fast-Sampling algorithm that generates an (1+O(alpha))-approximation in O(md log (kd)) time. They decompose the problem in each dimension, find a good approximation of the coordinates of the centers and then compose the coordinates back to R^d. In Fast-Sampling algorithm, candidate coordinates closes to the coordinates of optimal solution is identified first, and then they use these candidate coordinates to define intervals that capture the optimal solution more precisely.\nTo further remove the log (kd) dependency in run-time, they propose their second algorithm Fast-Estimation, which generates an (1+O(alpha))-approximation in O(md) + ~O(kd/alpha) run-time. They use a sampling-based strategy to construct an estimator, which can give accurate estimation of cluster cost. With the help of such estimator, they succeed to achieve the O(md) + ~O(kd/alpha) run-time.\nAlthough their algorithms have shorter run-time, the approximation ratio is worse than the algorithm in previous research. Hence, they give their third algorithm, Fast-Filtering, a heuristic that achieve low approximation ratio while still run fast. They also run experiments to test the performance of their algorithms.", "questions": "The experiments are only run for Fast-Sampling and Fast-Filtering. What is the performance of Fast-Estimation?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "They remove the log m dependency in the run-time compared to the previous algorithms, which is a huge improvement. Furthermore, they apply a new sampling-based technique to achieve such improvement, which is novel because the sorting-based technique is widely used for previous research. They provide both solid theoretical and practical support to verify their algorithms, which makes their algorithms very trustworthy.", "weaknesses": "Their Fast-Sampling and Fast-Estimation have worse approximation ratio compared to previous results. Although their Fast-Filtering performs well in approximation ratio, it is a heuristic and there lacks theoretical support for Fast-Filtering.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WO5JAdy1Do", "reviewer": "Reviewer_yxDp", "summary": "This paper considers the learning-augmented $k$-means problem. In this problem, the input consists of a data set and a predicted partition $\\{P_1,...,P_k\\}$. It is assumed that $P_i$ has a symmetric difference at most $\\alpha$ fraction to the $k$-th optimal cluster. The goal is still to find a set of $k$ centers to minimize the $k$-means objective (sum of the squared distance to the closest center). \n\nExisting results on this problem achieve a $1+O(\\alpha)$ approximation with running time $O(md\\log m)$ where $m$ is the size of the data set and $d$ is the dimension. The main contribution of this paper is to remove the $O(\\log m)$ factor by introducing sampling-based algorithms. The authors show that one can use sampling to find a good candidate for every dimension of each optimal center by employing some nice properties of $k$-means objectives and the condition that $P_i$ overlaps a lot with the true cluster. Two results are presented, one runs $O(\\epsilon^{-1/2} md\\log (kd))$ and the other runs in $O(md+\\alpha^{-1}\\epsilon^{-5}kd)$. Experimental results also show about 6 times speedup upon existing algorithms.\n\nI think this paper is a good addition to the study of learning-augmented $k$-means.", "questions": "1. In Line 153, the range of $\\alpha$ should be $(0,\\frac{1}{3}-\\frac{\\epsilon}{3})$? Why $\\alpha$ in the Sorting and Fast-Sampling algorithms can be 0 while in the Fast-Estimation case, it must be positive? Does that mean the formers can recover the ground truth (the optimal clusters) while the latter can not?\n\n2. The techniques seem to be specific to $k$-means clustering. Can one obtain similar results for $k$-median?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The running time is linear in the input size which improves previous results by avoiding using sorting.\n2. Interesting techniques are used and the theoretical results look solid.\n3. The algorithm is based on sampling and is easy to use in practice. The experiment also shows the improvement.", "weaknesses": "1. I am unsure if an $O(\\log m)$ factor improvement on this problem is super attractive to the community since $O(md\\log m)$ already sounds very fast in many cases. Can you explain why avoiding using sorting is so important in this problem?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CMRgDiJPmY", "reviewer": "Reviewer_Z7CR", "summary": "The paper studies the learning-augmented k-means problem. It proposes three sampling-based algorithms that achieve faster running times compared to previous works. The paper provides theoretical proofs for the algorithms and experimentally tests their cost and time performance.", "questions": "- Is this paper the first to use sampling techniques to improve the running time of the learning-augmented k-means problem? The paper does not mention other similar algorithms for comparison.\n\n- In Figure 1, Nguyen clearly outperforms Fast-Estimation; however, in the results presented in the other tables, the two methods show similar performance. Could the authors provide an explanation for this discrepancy?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper employs sampling-based techniques, which indeed improve the running time of existing algorithms.\n\n- The paper includes sufficient theoretical analysis, and the experimental evaluation is also very comprehensive.", "weaknesses": "- The paper's explanation and intuitive introduction to the learning-augmented k-means problem are not sufficient. I think (Nguyen et al., 2022) has a nice example on this front.\n\n- I'm not an expert in this area, but the technical innovations presented in the paper seem somewhat limited. Intuitively, the algorithm simply performs some sampling operations first and then moves on to subsequent steps, which appears to be a straightforward idea for improving the algorithm's running time.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)