On the SAC-BL Algorithm for Anomaly Detection
Authors: Xinsong Ma
Jie Wu
Weiwei Liu
Venue: NeurIPS 2025 poster ()
Keywords: SAC-BL algorithm
False positive rate
Primary area: general_machine_learning

Abstract: Visual anomaly detection is significant in safety-critical and reliability-sensitive scenarios. Prior studies mainly emphasize the design and training of scoring functions, while little effort has been devoted to constructing decision rules based on these score functions. A recent work  Ma et al. (2025b) highlights this issue and proposes the SAC-BL algorithm to address it. This method consists of a strong anomaly constraint (SAC) network and a betting-like (BL) algorithm serving as the decision rule. The SAC-BL algorithm can control the false discovery rate (FDR). However the performance of SAC-BL algorithm on anomalous examples, or its false positive rate (FPR), has not been thoroughly investigated. This paper provides a deeper analysis of this problem and explores how to theoretically reduce its FPR. First, we show that as the number of testing examples tends to infinity, the SAC-BL algorithm performs well on abnormal data if the scores follow the generalized Gaussian-like distribution family. But such conditions about the number of testing examples and the distribution of scores are overly restrictive for the real-world applications. So, we attempt to decrease the FPR of the SAC-BL algorithm under the condition of finite samples for practical anomaly detection. To this end, we redesign the BL algorithm by incorporating a randomization strategy and propose a novel stochastic BL (SBL) algorithm. The combination of the SAC network and the SBL algorithm yields our method, SAC-SBL. Theoretical results show that the SAC-SBL algorithm can achieve smaller FPR than SAC-BL algorithm while controlling its FDR. Finally, extensive experimental results demonstrate the superiority of our method over SAC-BL algorithm on multiple visual anomaly detection benchmarks.

On the SAC-BL Algorithm for Anomaly Detection
Xinsong Ma School of Computer Science Wuhan University maxinsongl 01 10gmmmlll.. com
Jie Wu School of Computer Science
Wuhan University
ren com
Weiwei Liu
School of Computer Science
Wuhan University
1 tiuweiwei g com
Abstract
Visual anomaly detection is significant in safety-critical and reliabilit--sensiiive scenarios. Prior studies mainly emphasize the design and training of scoring functions, while little effort has been devoted to constructing decision rules based on these score functions. recent work Ma al. (2025b) highlights this issue and proposes the SAC-BL algorithm to address it. This method consists of a strong anomaly constraint (SAC) network and betting-like (BL) algorithm serving as the decision rule. The SAC-BL algorithm can control the false discovery rate (FDR). However the performance false positive rate (FPR), has not been thoroughly investigated. This paper provides deeper analysis of this problem and explores how to theoretically reduce its FPR. First, we show that as the number of testing examples tends to infinity, the SAC-BL. algorithm performs well on abnormal data if the scores follow the generalized Gaussian- like distribution family. But such conditions about the number of testing examples and the distribution applications. So, we attempt to decrease the FPR of the SAC-BL algorithm under the condition of finite samples for practical anomaly detection. To this end, we redesign the BL algorithm by incorporating randomization strategy and propose a novel stochastic BL (SBL) algorithm. The combination the SAC network and the SBL algorithm yields method, SAC-SBL. Theoretical results show that the SAC-SBL algorithm can achieve smaller FPR than SAC-BL algorithm while controlling its FDR. Finally, extensive experimental results demonstrate the superiority of our method over SAC-BL algorithm on multiple visual anomaly detection benchmarks.
SAC-BL algorithm on anomalous examples, or its
scores are overly restrictive for the real-world
Introduction
Visual anomaly detection (AD) (Haselmann et al. 2018) is critical task in computer vision, where the objective to identify and spatially localize regions in visual data that deviate from normal patterns (Gong et al.] 2019; Tien et al.] 2023). The AD has wide-ranging applications, including industrial quality control (Bergmann et al. (019a), medical image analysis (Huang et al. (0244). video surveillance (Lv et al. (223)) and autonomous driving (Huang et al. 2020). Unlike standard supervised learning problems. The AD task is characterized by a scarcity or complete absence of labeled anomalous data, due to the inherent rarity, unpredictability. and diversity of anomalies in real-world scenarios (Bergmann et al. 2019b). Consequently, most existing approaches adopt an
"Corresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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unsupervised learning paradigm, where models are trained solely on normal samples and then identify the anomalies during inference based on the learned distribution information of normality (Wang
et al. 2021 Bergmann et al. 2019c Deng. Li 2022).
A substantial body of methods have been proposed to tackle the challenge of the AD. These methods
can be grouped into four principal categories: knowledge distillation-based methods (Deng, Li 2022
Tien et al. 2023), synthesizing-bee based methods (Li al., 2021 Yan al. 2021), embedding- based
methods (Detard et al. 2020; Roth et al. 2022) and reconstruction-basee
method (Akcay et al. 2018
Bergmann et al. (019c). The existing approaches mainly focus on designing the powerful score functions to learn the discriminative information from normal data. However, these methods neglect the deep investigation for the decision rules based on the proposed score functions. A recent work Ma et al. (2025b) points out this issue and tackle it from statistical perspective. Concretely, |a et al.) (2025b) frames the AD task as hypothesis testing problem and proposes a novel SAC-BI. algorithm to address it. The SAC-BL algorithm consists of a strong anomaly constraint (SAC) network and a betting-like (BI.) algorithm serving as the decision rule. Theoretically, the SAC-BL algorithm can control false discovery rate (FDR) at prescribed level. However, the performance of the SAC-BL algorithm on anomalous examples, or its false positive rate (FPR), has not been thoroughly analyzed. Factually, only controlling the FDR might result in poor performance on anomalous data in some worst-case scenario. For example, we consider a trivial decision rule d(.) which directly accepts all null hypotheses, equivalent to classify all testing examples as normal data. Obviously, the FDR of decision rule o(.) is zero, but its FPR is one. It is therefore imperative to conduct an in-depth study on the FPR of SAC-BL algorithm. Besides, review (Pang et al. (02I) highlights that existing anomaly detection methods methoos-pparticlaat particularly unsupervised methods- often suffer from high FPR, and how to reduce FPR remains one of the most important yet challenging problems in the field. Then, one natural question arises:
How to theoretically reduce the FPR of SAC-BL algorithm while controlling the FDR at prescribed level?
This paper attempts to address the problems mentioned above.
Based on the analytical framework in Ingster, Suslina (2003); ; Donoho, Jin (204)) Din, Ke (016)) we find that as the number of testing examples tends to infinity, the SAC-BL al algorithm performs well on anomalous data, provided that the score distribution belongs to the generalized Gaussian-like family. However, such assumptions are often overly restrictive for real-world applications. On the one hand, real-world data distributions are complex and typically unknown, making difficult to satisfy specific distributional requirements. On the other hand, the number of available testing samples is often limited. For example, the widely used AD benchmark MVTTe (Bergmann et al. 2019h), the testing set for the class "Pill" contains only 167 images. Hence, it is necessary to enhance the performance of SAC-BL algorithm on the anomalous data under the condition of finite examples for the real-world AD task. To this end, we redesign the BL algorithm by incorporating randomization strategy and propose a novel stochastic BL (SBL) algorithm. The combination of the SAC network and the SBL algorithm yields our method, SAC-SBL. Theoretical results show that SAC-SBL algorithm can achieve smaller FPR than the SAC-BL algorithm while controlling its FDR at the prespecified level. Finally, we conduct extensive experiments verify the effectiveness of SAC-SBL algorithm. For example, compared with the SAC-BL algorithm, our method reduces the image-level FPR from 13.26% to 29.08% while achieving the same TPR for the class "Pill" in MVTec
We summarize our main contributions as follows.
1. We demonstrate that as the number of testing examples tends to infinity, the SAC-BL algorithm achieves a well performance on anomalous testing examples if the distribution of scores belongs to the generalized Gaussian-like distribution family.
2. To improve the performance of SAC-BL algorithm on anomalous examples in practical AD task, we propose novel SAC-SBL. algorithm which is based on a randomization strategy for the p-values. Theoretically, our proposed method can reduce the FPR of SAC-BI. algorithm while controlling its FDR at prescribed level.
3. Extensive experimental results demonstrate the superiority of our method over SAC-BI. algorithm on multiple visual anomaly detection benchmarks.

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2 Background
Different from previous literature, Ma et al. (2025b) studies the AD problem from the perspective of multiple hypothesis testing, and propose the SAC-BL algorithm to tackle it. We first introduce the hypothesis testing framework introduced by Ma al. (20256) for the AD task. To avoid confusion, we use the same mathematical notations as Ma et al. (2025)) in our paper. Denote by X' the feature space of the normal examples, and X follows the underlying distribution D. In most cases, D is
unknown. Given a testing set Tlest {Xiest : Xg X3 ,,,, Xxaat)
Ma et al. (2025b) frames
the AD task as the following multiple hypothesis testing problem:
H1:0 X D, Hig : X D
H2,0 Xtest D, H2 Xge D
(1)
Hn,0 : X D Hn;1 Xx ~
where Hi,0 and Ha are called null hypothesis and alternative/non-nue hypothesis. respectively. In the context of anomaly detection, if H₃,0 is rejected, we declare that X: is anomalous.
In statisties, the decision to accept or reject the null hypothesis is determined by the concept of p-value. Its general definition is presented as follows.
Definition 2.1. (P-value (Casella, Berger, 2002)) Given a sample X A statistic p(X) is called
p-value corresponding to the null hypothesis Ho. if p( (X) satisfies
PP(( S H|Ho St
for every 0 5
small p-value usually provides strong evidence against the null hypothesis. If the cumulative distribution function F()) of testing statistic T for null hypothesis is known, then the corresponding p-value can be defined
p p(T) PT 11--)(())
where I't is the observation of T. It is easy demonstrate that p(T) satisfies the condition in Eq. (2) is noteworthy that the p-value has clear statistical interpretation. For example, suppose the p-value of anomalous testing example Xtest is 0.01. This means that for any coming testing example X the probability that Xftst is more anomalous than Xtest is 0.01. In other words, it is extremely difficult to find a more anomalous example than Xtest, Hence, we are highly confident that X is abnormal.
is known that the probability of type-I error should be controlled the prescribed significant level in single hypothesis testing. Similarly, in multiple hypothesis testing, the false discovery rate (FDR), as the generalization of probability of type-I error, should controlled. The statistical advantages of FDR have been detailedly discussed in (Benjamini, Hochberg 1995 Benjamini, Yekutieli] (000))
Given the null hypotheses (H 1,0 H2 ... Hn,)), let R be the set of indices of the rejected null hypotheses. Similarly, denote by Ho and Hi the set of indices for the true null hypotheses and false null hypotheses for 10 H2..0.... true null hypotheses. In statistics, if one null hypothesis is rejected, it is said to make a discovery. FDR the expected proportion of false discoveries among the rejected hypotheses.
) respectively. Besides, let no -HHl be the number of
Definition 2.2 (Bnjjaini,,,ni, Hochberg 1995)). Dnote by V the number of true null hypotheses rejected for the hypotheses H1,00 H2.0.... An.o Additionally, let U be the number of rejected hypotheses. The false discovery proportion (FDP) is defined as:
(V/U, if > 0,
FDP
0, otherwise.
The expectation of FDP is defined the FDR, namely
|R n Hol max [1, [
FDR E(FDP) E
2 Xxtest image or pixel.
AA sample means sequence of examples.
3

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We given a new insight for controlling the FDR. Following Ma et al. (2025)). the normal data is set positive. The FDP closely related to the TPR and FPR. Using the notations of confusion matrixm, the FDP is expressed as
FN FDP
FN + +N
1 1 TN FN
1+
N FP P-IP
I-PPR 1+ N I-TPR
Note that P= [Ho and N |H For given testing set, P and N are fixed. Is well-known that there is a tradeoff between the detection performance of normal and abnormal examples for a trained score functions. Therefore, (TPR) when designing the AD algorithm. Factually, an ideal AD algorithm should achieve low FPR while maintaining high TPR, which leads small FDP based Eq. (3) Thus, controlling FDR tends to achieve well tradeoff between the detection performance of both normal and abnormal examples * 6.
cannot only consider the true positive rate (TPR) or false positive rate
To control the FDR for the AD task, Ma et al. (2025b) proposes a new SAC-BL algorithm, which consists of SAC network serving the
function and BI. algorithm the decision rule. The SAC network is composed two parts: a reconstrunction network (R-net) and a discriminative network (D-net). The core of SAC network is apply strong constraints for the training process of discriminator by hard pairs generated from reconstruction network, which can improve the discriminative capability for weak anomalies. The details about SAC network can be found in Section 4 of Ma et al. (2025)). The BL algorithm is defined as follows.
Definition 2.3 (BL algorithm (Ma et al. 2025b)). Given the p-values P2. ... Pn corresponding to the null hypotheses H1,0, H2 un.u. let P(a) be the i-th order statistics from the smallest to the largest. For pre-specified level (0, 1), define
BB == max i
[1:
n]
5
(())
1
daci
n
max i [1:
:
dad
(4)
where 7, are two positive real numbers and satisfy y + 8 S Then, the null hypothesis H()) is rejected if i
In statistics, a is usually set to 0.05. Eq. indicates that the BL algorithm rejects the null hypothesis H5o if p? S 'BL and [R] LBL In practice, for a testing example X,est, if the distribution of testing statistic (or scores) is known, the p-value pt corresponding to null hypothesis Hi.0 can be presented as p; = F(s(X:)) where F(.) is cumulative distribution function of the scores and s()) is the score function. If the distribution information is unknown, the computation of the p-value relies on calibrated consisting of normal examples. Specifically, given a calibrated set Teal [X Xgal ..... X
J. pt is presented
Pt P p(Xtot) 1 1(s(X 2 s(Xtest) +
3 Asymptotic FPR of SAC-BL algorithm
False positive rate (FPR) is significant evaluation criterion for AD task, which can be expressed as
FP
Re n Hil
FPR FP+TN
[741
Although Ma al. (2025b) has proved that the SAC-BL algorithm can control FDR. However, relatively little is known about the theoretical properties of FPR SAC-BL algorithm. In this section, aim analyze the asymptotic performance of its FPR. Our analytical framework follows Donoho Uin (2004); Jin, Ke (2016)
In the vast majority of theoretical literature multiple hypothesis testing (Benjamini, Hochberg 1995 (enjamini, Yekutieli 200T; Storey 2002 Blanchard, Roquain (2088) the p-values are assumed to be available, equivalently, the distribution of test statistic for each null hypothesis is known. In this

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situation, the p-value can be represented as P. 1 (T; where T; is the observation of test statistic T; corresponding to hypothesis H, and (-) is the survival function of statistic. For analytical T, are continuous random variables. We first
simplicity, we suppose that the observation Tú, T2, ... define function class follows.
F -
V(I:: lim 2-000
- log V (t)
17
Based on the function class F, we have the following definition.
Definition 3.1 (Generalized Gaussian-like Distribution Family). A random variable X is said to follow the generalized Gaussian-like distribution with the location μ and the degree т, denote X G( r), if the survival function (-) of X satisfies (t - μ)) E F.
easy to verify that Gaussian distribution belongs to this distribution family. Similar to Donoho Jin 2004); Ingster, Suslina (2003); y; Jin, Ke (2016), our analytical framework relies on the above generalized Gaussian-like distribution family. Specifically, we assume that the test statistic 7 corresponding to the hypothesis H, satisfies:
T, [G(O,T), if E Ho G(,,T),
(6)
if E H1
Following Donoho, Jin 22004), we μ (rr log n// Besides, we focus on the sparse region in which the number of true null hypothesis larger than that of the true alternative hypothesis (Donoho Jin 2006). In this case, n1 = 1741 = n n- where 8 < < 1 (note that "1 + 0). Based on the Iramework above, we derive the asymptotic FPR of SAC-BL algorithm.
Theorem 3.2. Suppose that the test statistic T; corresponding H; satisfies the condition in Eq. (6) for E [n]. Then, for the p-values pt = (TT)) (TTz), Prn (Ta), the FPR of SAC-BL algorithm converges to zero in probability, namely
FP
FPRBL FP+TN
in probability
as 72 + OC.
The proof of Theorem 3.2 presented Appendix B.1 following theoretical result.
Based on the Theorem 3.2
we have the
Theorem 3.3. Suppose that the conditions in Theorem
3.2 hold. Then, we have
FP+TN
FP
000 afdd only if
1000
lim E
TP+TN
FP
-
The proof of Theorem
3.3 is presented in Appendix B.2
Theorem 3.2 indicates that the number of testing samples tends infinity, the SAC-BL algorithm performs well on anomalous examples, provided that the distribution of the scores belongs to the generalized Gaussian-lik distribution family. However, such conditions are overly restrictive in real-world applications, the distribution of real-world data is often complex and typically unknown. Besides, for the widely used AD benchmark MV'Tech (Bergmann et al., (019b), the testing set for the class "Pill" only contains 167 images and the SAC-BL. attains the image-level FPR of merely 43% on (see TT. Hence, we need to explore how improve the performance SAC-BL. algorithm on the anomalous data under the condition of finite examples for the practical AD task.
4 Decreasing the FPR of SAC-BL algorithm
In this section, we explore how to reduce the FPR of SAC-BL algorithm while controlling its FDR at the prescribed level. Our main focus is redesign the BL algorithm.
Under the null hypothesis, all of test statistic have the same function. So, we use same survival function. 'Note that the observations of test statistic can be regarded as its sample, thus the test statistic and its observations have the same distribution.
5

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Note that FPR =
where R is the complement of R. For given testing set, FP TN is equal the number of true anomalous examples in testing Therefore, we reduce the FPR by reducing FP, equivalent to increasing the number of rejecting null hypotheses [R]. Recall the BL algorithm: BB = max {i {iE [] n] Pla
FP
RUU
FP4TN
S bai
The null hypothesis Ha, is
rejected and only if P/)) I 6ax Bt. Denote
dai
18a 28a
nda
EBL BL
&
C
n
71
Note that {BL only take the value in C. For a given p-value ps satisfying P. > éax {BL our idea is to develop strategy that changes the value of p; to pi such that P7 εBL without violating the core conditions of controlling FDR for the SAC-BL algorithm. Then we could increase the number of rejecting null hypotheses and further reduce the FPR of SAC-BL algorithm.
n
Denote by A the random variable uniformly distributed on (0, 1) independent of p-values and define
(x)x) = x² - (x + (AG, < Ax" S Ex)
Moreover, for each E [n] == (1, ... ,nn, we denote
6
Σ 1 (p) S pa), Ø=1
11
x(i)
arg max J fze: Pj
there are multiple indices that satisfy the argmax, we take the largest index. Then, we define the stochastic BL (SBL) algorithm follows.
Definition 4.1. (Stochastic BL Algorithm) Given the random variable A uniformly distributed on (0, 1), p-values P1,22." ... Pn corresponding to the null hypotheses H1,0, H2.0. ... , and the prespecified level a (0, 1). For each i [n). define
T Mx(1) (p:) p? (p] Ex(s) + En(i) 1 (AEn(O App En(s))
and
1
daá
BL. max i [1 [1:n]]
max [1[1:0 P())
dai
P())
n
where y, 5 are two positive real numbers and satisfy 1. Then, the null hypothesis Hę is rejected
8a
T (pt)
BLL -L
The SBL algorithm in Definition 4.1 is mainly used to derive theoretical results. Additionally, have another simple version of SBL algorithm.
algorithm: version 2 ) Given p-values Pi,P2.... Pr, corre- sponding to the null hypotheses Hi,0, H2. Hn let P() be the i-th order statistics from the smallest to the largest. Denote by A the random variable uniformly distributed on (0, 1). For pre-specified level a E (0, 1), define
Definition 4.2. (Stochastic
na
SRL == max E [l:]]]]:
P(s)
5ezi
where y, 8 are two positive real number and satisfies 7 8 1. Then, the null hypothesis H(s), is rejected if < ASLL
The Following theory reveals the relation between the SBI. algorithms in Definition 4.1 and the one in Definition 4.2
Theorem 4.3. The SBL algorithm in Definition 4.1 is equivalent to the one in Definition 4.2
The proof of Theorem presented Appendix B.3 Obviously, the SBL algorithm Definition
has less computation steps than the one in Definition and thus is more suitable for the practical

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Algorithm 1: Practical SAC-SBL Algorithm
Input: Training set T calibrated set Teai = {Xpml X2at, > >.... xx testing set
Tiest {X1est X2 ...") preseribed level a E (0, 1), generator of synthetic
anomalous example G(.).
Utilize G(-) to construct the new training set T based on the training T
Train the SAC network on Tmir, and then obtain the score function s(-) for images or pixels.
Calculate the empirical p-value corresponding to Xtestt
= p(X(est) Σラ- ((((X)))) 2 s(X
(8)
k +
Draw a sample A from the uniform distribution on (0, 1). 5 Determine the index 'n
1
2 nA
SS. == max i [n]
P(
dai
Output: Declare that X test is anomalous if S 1'.LL
applications. The version of SBL algorithm in Definition 4.1 is used to establish the connection between the SBL algorithm and BL algorithm for the theoretical analysis.
In the context of anomaly detection, the underlying distribution information normal data usually unknown. Therefore, we can use the method in Eq. (5) to compute the p-values in practice. Combining the SAC network proposed by Ma al.] (2025b) and the SBL algorithm in Definition 4.2 yields our method, SAC-SBL. Its detailed steps are presented in Algorithm !
Now present our core theoretical results demonstrate the superiority of SAC-SBL algorithm over the SAC-BL algorithm in terms of FPR while controlling the FDR at the prescribed level.
Theorem 4.4. Given the random variable A uniformly distributed on (0. 1), the p-values P1.P2." Pn corresponding to the null hypotheses Hi.0, H2 ... ".,,H.HH in Eq. (8) and the pre- specified level a (0, 1). The following conclusions hold:
1. the FDR of SAC-SBL algorithm satisfies
FDRSAC-SHI. or
2. For the index sets RsAC-BL BL and RsAC we have
P (RSAC-BL S RSAC-SBL)
3. If there exists a p-value pt which satisfies P (2) Scdje
0, then we have
n
P P(RSAC-BLL < (RSAC-
0.
The proof of Theorem 4.4 is presented in Appendix B.4 Theorem 4.4 suggests that the SAC-SBL algorithm does not reject fewer null hypotheses than the SAC-BL algorithm almost surely while controlling the FDR the prescribed level. In other words, SAC-BL algorithm tends to classify more testing examples as anomalous data. Based on Theorem 4.4 we can easily obtain the following theoretical results.
Theorem 4.5. Suppose the conditions in Theorem 4.4 hold, then the FPR of the SAC-BL algorithm and that of SAC-SBL algorithm satisfy
P (F PRSAC-BII BLC 2 FPRSAC-SPS =1
Moreover, if there exists a p-value ps which satisfies P (p) BoeieL
then we have
P P(FPRSAC-BL FPRSAC-SBL) 0.
7

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Table 1: Experimental results (%) M We compare the performance between SAC-BL
algorithm and SAC-SBL based on the same trained SAC network. f indicates larger values are better and vice versa.
Image-level FPR +
Pixel-level
Category
Method
TPR
Fl-score
TPR
FPR
Fl-score
SAC-BL
100.0
35.78
54.12
99.25
41.82
93.24
Capsule
SAC-SBL
100.0
26.61
61.33
99.25
34.77
99.43
SAC-BL
95.00
3.17
92.68
95.40
3.63
99..2
Bottle
SAC-SBL
95.00
0.00
97.44
95.40
3.62
97.53
SAC-BL
100.0
13.73
80.00
90.39
4.94
94.87
Carpet
SAC-SBL
92.14
4.49
83.64
90.37
4,93
94.90
SAC:BL
100.0
4.35
94.12
98.19
4.24
99.06
Leather
SAC-SBL
100.0
0.00
100.0
98.21
4.16
99.08
SAC-Bİ.
T00.0
43.26
46.02
95.28
4.78
97.42
Pill
SAC-SBL
100.0
29.08
55.91
95.27
4.77
97.49
SAC-BL
T00.0
30.00
90,91
96.63
42.62
90.18
Transistor
SAC-SBL
100.0
30.00
90.91
96.62
31,58
97.22
SAC:BL
T00.0
T.19
98.51
95.01
0.77
97.38
Tile
SAC-SBL
100.0
0.00
100.0
95.01
0.77
97.38
SAC-BI.
98.28
21.74
84.44
93.33
19.89
95.96
Cable
SAC-SBL
96.55
15.22
87.50
92.85
16.95
96.01
SAC-BL
100.0
0.84
98.46
97.32
3.83
98.60
Zipper
SAC-SBL
100.0
0.84
98.46
97.32
3.83
98.60
SAC-BL
100.0
3.33
96.00
94.04
1.76
96.90
Toothbrush
SAC-SBI
100.0
0.00
100.00
94.05
1.71
96.92
SAC-BL
T00.0
1.08
97.78
96.31
3.63
97.64
Metal_nin nut
SAC-SBL
100.0
0.00
100.0
96.07
3.28
97.78
SAC-BL
100.0
I.43
98.77
98.99
10.24
95.27
Hazelnut
SAC-SBL
100.0
0.00
100.0
98.92
8.63
99.35
SAC-BL
100.0
72.27
48.81
99.40
12.80
95.67
Screw
SAC-SBI.
100.0
68.91
50.00
99.40
8.77
99.69
SAC-BL
100.0
19.30
79.25
97.56
0.63
98.76
Grid
SAC-SBL
100.0
1.75
97.67
97.57
0.61
98.77
SAC-BL
100.0
8.33
88.37
92.69
7.09
94.92
Wood
SAC-SBL
100.0
0.00
100.0
92.70
6.99
96.07
SAC-BL
99.55
17.45
83.22
95.99
10.85
96.22
Average
SAC-SBL
98.91
11.79
88.19
95.93
9.03
97.75
The proof of theorem 4.5 is presented in Appendix B.5 Theorem 4.5
indicates that the FPR of SAC-BL algorithm is not smaller than that SAC-SBL algorithm almost surely. Besides, the 5asju
(p.
condition "there exists p-value p; which satisfies P in practice, since it only excludes the trivial situation where all hypotheses are rejected. Therefore, our proposed method can achieve smaller FPR over the SAC-BL algorithm.
0" nearly always satisfied
Experiment
In this section, perform extensive comparison experiments to verify the effectiveness of SAC-SBL method.
5.1 Experimental Settings
Baseline. We compare the decision-making performance of our proposed SAC-SBL method with the SAC-BL method (Ma et al. 2025b), using the default settings for SAC-BL.
Datasets.
Experiments are conducted on three widely used anomaly detection datasets. The MVTec dataset (Bergmann et al.) 2019b) consists of 5354 images across 15 object and texture categories.
The VisA dataset (Zou al. (022) is an industrial anomaly dataset comprising 10821 images from
12 objects across domains. The BTAD dataset (Mishra et al., 2021) includes 2830 images industrial products, showcasing body and surface defects.
Evaluation Metrics. We use the true positive rate (TPR), false positive rate (FPR) and f1-score (F1)
to evaluate the effectiveness of the proposed method.

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Table 2: Experimental results (%) on VisA. We compare the performance between SAC-BL algorithm and SAC-SBL based on the same trained SAC network. 1 indicates larger values are better and vice versa.
Image-level
Pixel-level FPR +
Category
Method
TPR
FPR
F1-score
TPR
Fl-score
SAC-BL
100.0
49,00
80.97
99.88
$5.26
95.87
Candle
SAC-SBL
100.0
46.00
83.30
99.88
52.26
99.90
SAC-BL
100.0
52.00
77.59
98.39
9.83
"99.16"
Capsules
SAC-SBL
98.33
46.00
71.52
98.39
6.72
99.18
SAC-BL
100.0
87.00
54.05
99.30
67.93
90.88
Cashew
SAC-SBL
100.0
85.00
54.05
99.25
64.51
99.24
SAC-BL
100.0
13.00
86.96
94.38
4.94
97.10
Chewinggum
SAC-SBL
100.0
14.00
87.72
94.85
4.92
97.33
SAC-BT.
T00.0
67.00
56.88
99.44
65.78
92.19
Fryum
SAC-SBL
100.0
57.00
63.69
99.42
65.39
98.94
SAC-BL
T00.0
81.00
7I.17
99,91
18.00
99.95
Macaronil
SAC-SBL
100.0
79,00
76.68
99,91
18.21
99,95
SAC-BL.
T00.0
55.00
74.37
99.92
30.89
93.46
Macaroni2
SAC-SBL
100.0
48.00
80.65
99.92
25.98
99.95
SAC-BL 98.00
69.00
70.41
95.98
13.36
97.92
Peb1
SAC-SBL
98.00
64.00
73.68
95.98
13.36
97.92
SAC-BL 97.00
45,00
80.17
92.97
30.77
96.29
Peb2
SAC-SBL
96.00
31.00
83.56
92.98
30.69
96.33
SAC-BL 99.01
56.00
76.82
99.41
39.24
99.60
Peb3
SAC-SBL
98.02
50.00
79.20
99.41
38.94
99.66
SAC-BL 98.02
13.00
90.96
98.08
24.94
98.87
Peb4
SAC-SBL
97.03
7.00
95.15
98.08
24.92
98.95
SAC-BL"
100.0
100.0
550.00"
99.34
73.15
"91.7
Fipe_fryum
SAC-SBL
100.0
96.00
50.25
99.34
60.10
99.21
SAC-BL 99.34
57.42
71.95
98.08
36.20
96.09
Average
SAC-SBL
98.95
51.92
74.95
98.12
33.83
98.88
Implementation Details. SAC-SBL, we repeat the stochastic perturbation 100 times and take meaning value. All experiments are conducted on workstation with eight NVIDIA GeForce GTX 3090 GPUs and two 2.2GHZ Intel CPUs.
5.2 Experimental Results
We evaluate our proposed method against the SAC-BL. algorithm on three datasets, demonstrating its superior performance and generalization. The image-level and pixel-level results on MV'Tec are presented in Table [ and the results VisA are shown in Table 2 because of the space limitation, results in BTAD are reported Appendix A From the Tables and achieves the comparable performance in terms of TPR. For example, in Table TT] our method have the same image-level TPR as the BL algorithm in 13 classe among 15 classes. On average, the image-level and pixel-level TPRs of two methods also comparable. 2) Our method yields lower FPR and higher Fl-scores across all classes, regardless of image-level or pixel-level evaluation. As shown in Table 1 compared the SAC-BL algorithm, method reduce the image-level FPR from 21.74% to 15. 15.22%, and improve the image-level F1-score from 84.44% to 87 37.5%, the direct improvements of 6.52% and 3.06%, respectively. 3) In those classes where the SAC-BL algorithm struggles to identify anomalous examples, our method can considerably reduce the FPR. For instance, For classes "Carpet" and "Pill", our method achieve the improvements of 11.24% and 14.14% in terms of image-level FPR. Overall, these experimental results demonstrate the superiority of our method over the SAC-BL algorithm.
2 we can see: 1) Our mehtod
Conclusion
In this paper, we focus on investigating the SAC-BL algorithm in terms of FPR. First, we demonstrate that the FPR of SAC-BL algorithm converges to zero probability based on the generalized Gaussian- like distribution family. Then, we propose novel SAC-SBL algorithm which can reduce the FPR of prescribed level. Finally, we conduct extensive
SAC-BL algorithm while controlling the FDR experiments to verify the effectiveness of our proposed method.

---

Acknowledgment
This work is supported by the National Natural Science Foundation of China under Grant 624B2106, the Key R&D Program of Hubei Province under Grant 2024BAB038, the National Key R&D Program of China under Grant 2023YFC360400,, the Fundamental Research Funds for the Central Universities under Grant 2042025kf0045.
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Reviews:
[{"review_id": "FPpFAXlwTJ", "reviewer": "Reviewer_gRFq", "summary": "This paper improved the (Ma et al., 2025) by theoretically analyzing the FPR and proposing a stochastic version of the BL method. This paper discussed the decision rule of AD methods following (Ma et al., 2025), which is an important problem when applying AD to the real world. The author presented a theoretical guarantee and empirically showed the advantages over the original SAC-BL methods.", "questions": "- How to generalize the decision rule to any AD algorithm and any AD task? Since the p value is only calculated from the anomaly score, it should not be limited to the vision AD task and the reconstruction based SAC method. \n- How to obtain the calibrated set? Is it a subset of trianing set or it requires the labeled anomaly?\n- If the calibrated set is a subset, how to select the subset?\n- how the hyper-parameters, such as alpha and the synthetic strategies affect the performance?", "limitations": "The author does not discuss the potential negative societal impact.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n- This work focuses on FPR, which is more practical.\n- The proposed stochastic method is simple and effective.\n- This paper discussed an important problem in UAD, but it is not ready for publication.\n\nWeakness\n- The content in total is an incremental work of (Ma et al., 2025) so far. Compared with (Ma et al., 2025), only an extended FPR analysis and a stochastic variant of BL are not enough for NIPS acceptance.\n- It is not intuitive to see how the SBL can improve the FPR.\n- It seems that the BL method relys on a calibrated set while how this set is constructed is unclear in the main text.\n- No investigation on the related works. The authors should also compare the results with recent SOTA.\n- see more comments in the question.", "was_revised": true, "final_justification": "As other reviewers recognized, the FPR is more important.\n\nThe problem, threshold selection, itself is an important problem in AD applications. The author shows a clear respones to my main concern. So, I increase my score.\n\nAlthough the expeirment is still not enough, the authors give a strong theoretical analysis.", "rebuttal": "__W1__: About our contributions.\n\n__Ans1__: We are sorry to confuse you about our  contributions. We would like to clarify our core contributions in more detail as follows:\n- A recent survey [1] highlights that existing anomaly detection (AD) methods—particularly unsupervised methods—often suffer from high false positive rates (FPR), and that reducing FPR remains one of the most important yet challenging problems in the field. However, to our best knowledge, there is no prior work that systematically investigates the FPR of AD algorithms from a theoretical perspective. Our work fills this gap in the literature.\n- Previous studies focus primarily on designing  scoring functions but neglect the decision algorithms. [2] is perhaps the first to explicitly identify this limitation and propose a new decision algorithm, the BL procedure, to address it. Since theoretical analysis of FPR requires a well-defined decision rule, our work builds upon the BL algorithm in [2].  Then, we introduce a statistical framework to analyze the performance of AD algorithm about FPR and establish rigorous  theories of BL procedure. Theoretical results show that BL procedure performs well as the number of testing samples tends to infinity.\n-  To further reduce the FPR under finite-sample conditions, we introduce a new stochastic strategy and propose the SBL algorithm. We also provide a theoretically justified simplified version of SBL to facilitate practical implementation. We prove that SBL achieves a lower FPR than BL while maintaining FDR control at a prescribed level. Finally, we validate the effectiveness of our method through extensive experiments across diverse anomaly detection benchmarks.\n  \nOverall, [2] focuses mainly on a practical AD algorithm to improving AUROC, but our study aims to provide deeper theoretical insights into the FPR behavior of decision algorithms in AD. So, our research direction is fundamentally different from that of [2] and the contributions go well beyond an incremental extension of [2].\n\n[1] Pang, Guansong, et al. \"Deep learning for anomaly detection: A review.\"\n\n[2] Ma X, Wu J, Liu W. SAC-BL: A hypothesis testing framework for unsupervised visual anomaly detection and location[J].\n\n\n\n\n\n__W2__: About our intuition.\n\n__Ans2__: Our core intuition is outlined in lines 194–202 of the submission, and we elaborate on it here for clarity. We emphasize that our method is theory-driven. Since there is a tradeoff \nbetween the detection performance of both normal and abnormal examples, we cannot only consider the TPR or FPR when designing the AD decision algorithm. Note that $FPR = \\frac{FP}{FP+TN} = \\frac{ |\\mathcal{R}^c \\cup \\mathcal{H}_1|}{|\\mathcal{H}_1|}$. where $\\mathcal{R}$ is the set of indices of the rejected null hypothese, and  $\\mathcal{R}^c$ is the complement of $\\mathcal{R}$.So we can reduce the FPR by reducing FP, equivalent to increasing the number of rejecting null hypotheses $|\\mathcal{R}|$. \nHowever, simply increasing rejections can inflate false discoveries, so we must control the FDR at a prescribed level. As shown in lines 121–130 of our submission, controlling FDR leads to a better trade-off between detection performance on normal and abnormal examples. \nSo, our core intuition is to design a new stochastic strategy about p-values such that the decision algorithm can reject more null hypothesee than BL algorithm while satisfying the conditions for FDR control in [2]. The details about stochastic strategy can be found in line 203-216 of the submission.\n\n__W3, Q2, Q3__: About calibrated set.\n\n__Ans3__: We are sorry for the confusion regarding the  the calibrated set. In practice, we simply sample a small subset of normal examples from the training data to construct the calibrated set—no special procedure is required. The calibrated set is thus a subset of the training set and does not contain any anomaly data.  Since our work focuses on the unsupervised setting, no label information is needed for the calibrated set. In our experiments, we use 40% of the training data as the calibrated set.\n\n__W4__: About related work and and more comparison experiments with recent SOTA.\n\n__Ans4__: We would like to clarify that all significant relevant work has been cited and discussed in the Introduction and Background sections. To avoid any misunderstanding, we will include a dedicated Related Work section in new version.\nFollowing your suggestions, we have adopted several recent SOTA methods as baselines, including EfficientAD [3], RealNet [4], MambaAD [5], and DiAD [6]. The results, shown in Tables 1–3, consistently demonstrate the superior decision performance of our SAC-SBL method compared to these baselines.\n\nTable 1: the results on MVTech.\n|             |       | Image_level |          |       | Pixel_level |          |\n| ----------- | ----- | ----------- | -------- | ----- | ----------- | -------- |\n| Metric      | TPR   | FPR         | F1-Score | TPR   | FPR         | F1-Score |\n| EfficientAD | 97.52 | 18.51       | 79.86    | 93.85 | 18.33       | 86.51    |\n| RealNet     | 97.49 | 32.73       | 81.10    | 90.50 | 16.44       | 87.84    |\n| MambaAD     | 96.93 | 22.60       | 83.71    | 84.08 | 21.99       | 86.92    |\n| DiAD        | 95.90 | 41.32       | 80.76    | 92.38 | 18.37       | 88.74    |\n| SAC-SBL(ours)    | 98.91 | 11.79       | 88.19    | 95.93 | 9.03        | 97.75    |\n\n\nTable 2: the results on VisA.\n|             |       | Image_level |          |       | Pixel_level |          |\n| ----------- | :---: | ----------- | -------- | ----- | ----------- | :------: |\n| Metric      |  TPR  | FPR         | F1-Score | TPR   | FPR         | F1-Score |\n| EfficientAD | 95.97 | 64.17       | 73.48    | 94.13 | 44.19       |  85.04   |\n| RealNet     | 94.92 | 59.21       | 83.63    | 92.66 | 56.01       |  82.64   |\n| MambaAD     | 93.00 | 61.42       | 84.37    | 90.50 | 62.68       |  83.57   |\n| DiAD        | 95.75 | 56.44       | 89.01    | 92.05 | 45.02       |  89.63   |\n| SAC-SBL(ours)       | 98.95 | 51.92       | 94.95    | 95.12 | 33.83       |  98.88   |\n\nTable 3: the results on BTAD.\n|             |       | Image_level |          |       | Pixel_level |          |\n| ----------- | ----- | ----------- | -------- | ----- | ----------- | -------- |\n| Metric      | TPR   | FPR         | F1-Score | TPR   | FPR         | F1-Score |\n| EfficientAD | 94.38 | 41.04       | 62.71    | 96.39 | 83.41       | 88.61    |\n| RealNet     | 96.62 | 67.00       | 64.46    | 92.41 | 76.43       | 88.64    |\n| MambaAD     | 95.92 | 34.62       | 61.45    | 94.02 | 74.09       | 89.48    |\n| DiAD        | 96.71 | 40.94       | 67.20    | 89.11 | 80.30       | 81.36    |\n| SAC-SBL (ours)      | 99.67 | 30.65       | 73.33    | 97.54 | 66.98       | 97.61    |\n\n\n[3] Efficientad: Accurate visual anomaly detection at millisecond-level latencies. \n\n[4] Realnet: A feature selection network with realistic synthetic anomaly for anomaly detection.\n\n[5] Mambaad: Exploring state space models for multi-class unsupervised anomaly detection. \n\n[6] A diffusion-based framework for multi-class anomaly detection.\n\n__Q1__: About the generalization of SBL algorithm.\n\n__Ans5__: Note that our SBL algorithm imposes no restrictions on the architecture of  score function. Any score function can be directly integrated into SBL via empirical p-values, making it broadly applicable to anomaly detection (AD) tasks. Therefore, SBL is not limited to visual AD settings or reconstruction-based SAC methods. Additionally, the empirical p-values are computed based on scores of normal examples, rather than  anomaly scores.\n\n__Q4__: About the influence of $α$.\n\n__Ans6__: In statistics, $α$ is typically set to 0.05 and does not require adjustment in most cases. The experimental results on MVTech in Table 4 show that our SBL algorithm is not sensitive to $\\alpha$.\n\nTable 4: the experimental results of sensitivity on MVTech.\n|       |       | image_level |          |       | pixel_level |          |\n| ----- | ----- | ----------- | :------: | ----- | ----------- | :------: |\n| alpha | TPR   | FPR         | F1-score | TPR   | FPR         | F1-score |\n| 0.01  | 98.92 | 11.77       |  88.13   | 95.94 | 9.03        |  97.74   |\n| 0.05  | 98.91 | 11.79       |  88.15   | 95.93 | 9.03        |  97.75   |\n| 0.1   | 98.91 | 11.79       |  88.16   | 95.93 | 9.03        |  97.75   |\n| 0.2   | 98.91 | 11.8        |  88.16   | 95.93 | 9.03        |  97.75   |\n\nBesides, The synthetic strategies are used solely for training the score functions. However, our paper aims to investigate the FPR of the decision algorithm. Thus,  the synthetic strategies are not the focus of our research. For details on these strategies, please refer to Section 4 of [2]."}, {"review_id": "hpcOODsd13", "reviewer": "Reviewer_tkwZ", "summary": "This paper presents a theoretical analysis for the SAC-BL algorithm, focusing on its false positive rate (FPR), which has not been well-analyzed before. The authors propose SAC-SBL, a stochastic variant of SAC-BL, and demonstrate that it achieves a lower FPR while maintaining its false discovery rate (FDR) at under prescribed level.  Theoretical results are backed by experiments on various anomaly detection benchmarks.", "questions": "the calibrated set plays a key role in estimating empirical p-values. How does it influence the experimental results?", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths \n1, this paper provides a rigorous theoretical analysis for the asymptotic FPR of SAC-BL algorithm.\n2, The proposed method is a well-motivated extension of SAC-BL, the motivation and contributions are clear.\n3, The presentation is clear and the proofs in Appendix are sound. \n4, Experimental results across three benchmarks show the improvements in terms of FPR and F1-score, adding practical credibility to the theory.\n\nWeaknesses\n1, The title lacks clarity about the application domain. Readers unfamiliar with SAC-BL may not recognize this as a paper on visual anomaly detection.\n2, Apart from the Gaussian distribution, are there any other distributions that belong to the generalized Gaussian-like distribution family?\n3, There exist some typos in line 198 of submission.  \n4, Do the results in Theorem 4.5 depend on the distributional assumptions in Definition 3.1?", "was_revised": false, "final_justification": "", "rebuttal": "__W1__: The title lacks clarity about the application domain. Readers unfamiliar with SAC-BL may not recognize this as a paper on visual anomaly detection.\n\n__Ans1__: Thanks for your suggestions. We will rename the title of our submission as \"On the SAC-BL Algorithm for Anomaly Detection\".\n\n__W2__: Apart from the Gaussian distribution, are there any other distributions that belong to the generalized Gaussian-like distribution family? \n\n__Ans2__: Yes, the generalized Gaussian-like distribution family is quite\ngeneral. We can easily verify that many popular and widely used distributions in theoretical analysis, such as exponential location distribution E(µ, 1), Laplace distribution L(µ, 1), Gaussian distribution N(µ, 1) and Gamma distribution G(2, 1), belong to the generalized Gaussian-like distribution family. We set the scale parameter to 1 in these distributions just\nfor the analytical convenience.\n\n__W3__: There exist some typos in line 198 of submission.\n\n__Ans3__: Thanks for your careful review. We will correct the notation of $\\mathcal{C}$  as \"$\\mathcal{C} = [ \\frac{n} {1\\alpha }, \\frac{n} {2\\alpha}, \\cdots, \\frac{n} { n  \\alpha}  ]$\".\n\n__W4__: Do the results in Theorem 4.5 depend on the distributional assumptions in Definition 3.1?\n\n__Ans4__: We clarify that Theorem 4.5 does not depend on the assumptions in Definition 3.1. All required conditions for the result are explicitly presented in Theorem 4.5.\n\n__Q1__: the calibrated set plays a key role in estimating empirical p-values. How does it influence the experimental results?\n\n__Ans5__: In practice, we observe that a larger calibrated set tends to yield better detection performance."}, {"review_id": "0FeeIyOIUD", "reviewer": "Reviewer_mvCm", "summary": "This paper investigates the theoretical properties of the SAC-BL algorithm for visual anomaly detection, with a particular focus on its underexplored false positive rate (FPR). Specifically, the authors first demonstrate that the FPR of SAC-BL can be low under certain distributional assumptions. However, these assumptions may be too restrictive for real-world applications. To address this, the authors propose a new algorithm, SAC-SBL, which introduces stochasticity into the decision rule to reduce the FPR while maintaining the false discovery rate (FDR) control. Theoretical analysis shows that SAC-SBL achieves a lower FPR than SAC-BL almost surely. Experiments on MVTec, VisA, and BTAD datasets confirm consistent improvements in FPR and F1-score without compromising TPR. Overall, this paper is well-motivated, theoretically rigorous, and empirically validated, offering a novel perspective on the design of decision rules in anomaly detection.", "questions": "See weaknesses", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "## Strengths \n- Despite being theory-heavy, this paper is well-structured and readable. Motivations, algorithms, and results are presented clearly.\n- This paper proposes a novel SAC-SBL algorithm, and provides solid theoretical results showing that the proposed method can decrease the FPR control while maintaining FDR, compared the SAC-BL algorithm.\n- The proposed method is simple to implement and integrates naturally into existing SAC-BL frameworks. \n- This paper validates the proposed method across multiple challenging datasets and at both the image and pixel levels, showing consistent performance gains in terms of reduced FPR and improved F1-score, with comparable or improved TPR.\n\n## Weaknesses\nI have not identify the major weaknesses, but I have some recommendations:\n- It would be better if the title of the paper included the background of the research problem. For example, it could be revised to “On the SAC-BL Algorithm for Visual Anomaly Detection.”\n- The authors should provide more detailed experimental setup.\n- I am confused of the results of line 725 in Appendix, please give a more clear interpretations.", "was_revised": true, "final_justification": "All of my concerns have been well addressed. This paper offers a novel perspective on FPR and presents a method with strong theoretical guarantees and solid empirical performance. I will maintain my score for acceptance.", "rebuttal": "__W1__: It would be better if the title of the paper included the background of the research problem. For example, it could be revised to “On the SAC-BL Algorithm for Visual Anomaly Detection.”\n\n__Ans1__: Thanks for your suggestions. We will rename the title of our submission as \"On the SAC-BL Algorithm for Anomaly Detection\".\n\n__W2__: The authors should provide more detailed experimental setup.\n\n__Ans2__: Our experimental settings follows [1]. \nAll images are resized to 256$\\times$256 and normalized to the value in range of [0, 1]. The train batch size is 8. We use Pytorch 2.2 to implement our method and adopt AdamW to optimize it. Default training schedule is 700 epochs and initial learning rate is set to 0.0001. We conduct experiments on 8 NVIDIA GeForce GTX 4090 and 2 Intel(R) Xeon(R) Silver 4210 CPU @ 2.20GHz. Memory is 384GB. More details can be found in [1].\n\n\n[1] [2] Ma X, Wu J, Liu W. SAC-BL: A hypothesis testing framework for unsupervised visual anomaly detection and location[J].\n\n__W3__: I am confused of the results of line 725 in Appendix, please give a more clear interpretations.\n\n__Ans3__: Since $\\Psi(\\cdot ) \\in \\mathcal{F}$, we have  $\\Psi(\\rho_n) = \\frac{1}{n^{\\nu + o(1)}}$. \n\nNote that $\\rho_n - \\mu \\to -\\infty$, so $\\Psi(\\rho_n - \\mu) \\to 1$. It follows that $\t(1-\\eta)\\Psi(\\rho_n) + \\eta  \\Psi(\\rho_n -\\mu) \\sim \\frac{1}{n^{\\beta}} $ where $\\eta = \\frac{1}{n^{\\beta}}$. Similarly, the simple analysis can derive that \n$ \tO_{p}\\left(\\eta \\upsilon_{n_1}\\sqrt{\\Psi(\\rho_n-\\mu)(1-\\Psi(\\rho_n-\\mu))} \\right) = o_{p}(\\frac{1}{n^{\\beta}})$ and  $O_{p}\\left(  (1-\\eta )\\upsilon_{n_0}\\sqrt{\\Psi(\\rho_n )(1-\\Psi(\\rho_n ))}\\right)  = o_{p}(\\frac{1}{n^{\\beta}})$\nsince $\\kappa_{n_0} = o(\\frac{1}{n^{\\beta/2}}) $."}, {"review_id": "LWeLjecfdj", "reviewer": "Reviewer_aLKU", "summary": "In this paper, the authors provide a deeper analysis of this problem and explores how to theoretically reduce its False Positive rate. First, the authors show that as the number of testing examples tends to infinity, the SAC-BL algorithm performs well on abnormal data if the scores follow the generalized Gaussian-like distribution family.\n\nFirst, the authors show that as the number of testing examples tends to infinity, the SAC-BL algorithm performs well on abnormal data if the scores follow the generalized Gaussian-like distribution family. But such conditions about the number of testing examples and the distribution of scores are overly restrictive for the real-world applications. So, the authors attempt to decrease the FPR of the SAC-BL algorithm under the condition of finite samples for practical anomaly detection. To this end, the authors redesign the BL algorithm by incorporating a randomization strategy and propose a novel stochastic BL (SBL) algorithm.", "questions": "Algorithmic content could be clearer, see:\n\nhttps://shantoroy.com/latex/how-to-write-algorithm-in-latex/\n\nFor examples of algos, is that possible?\n\nIn the background, literature cited could focus more on limitations, is that possible?\n\nEquations not of authors should be cited when needed?\n\nI think the mathematics could be explained more in the text.\n\nExperimental analysis is short, such as Table 1 being discussed, \n\nMore graphical results should be presented.\n\nFuture work could be discussed in conclusion or before", "limitations": "More limitations could be expanded upon", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nPaper is well formatted, using LaTex formatting\n\nReferences are Up to date, perhaps more could be added\n\nBackground materials are up to date and well presented\n\n\n\nWeaknesses:\n\nAlgorithmic content could be clearer, see:\n\nhttps://shantoroy.com/latex/how-to-write-algorithm-in-latex/\n\nFor examples of algos\n\nIn the background, literature cited could focus more on limitations\n\nEquations not of authors should be cited\n\nI think the mathematics could be explained more in the text.\n\nExperimental analysis is short, such as Table 1 being discussed, \n\nMore graphical results should be presented.\n\nFuture work could be discussed in conclusion or before", "was_revised": true, "final_justification": "My ratings and decisions stand from the first round, while the author have addressed my comments I have still given the score that reflects the quality of the paper for NeurIPS.", "rebuttal": "__Q1__: About the algorithm presentation.\n\n__Ans1__: Thank you for your suggestions regarding Algorithm 1. We will revise it based on the examples provided in the link you shared.\n\n__Q2__: In the background, literature cited could focus more on limitations, is that possible?\n\n__Ans2__: Thank you for your valuable feedback. We discuss the limitations of existing literature in the Introduction section. Prior work has primarily focused on designing or training score functions, while little attention has been paid to systematically investigating the false positive rate (FPR) of anomaly detection (AD) algorithms from a theoretical perspective. Notably, reducing FPR remains one of the most important yet challenging problems in the field (see [1] for details). Our work aims to fill this gap in the literature. In the new version, we will also include a discussion of these limitations in the Background section.\n\n[1] Pang, Guansong, et al. \"Deep learning for anomaly detection: A review.\"\n\n__Q3,Q4__: About equations and mathematics in our submission.\n\n__Ans3__: Thanks for your suggestions about the equations and mathematics in our paper. We will  make revisions following your suggestions.\n\n__Q5, Q6__: About the experimental analysis and graphical results.\n\n__Ans4__: We appreciate your helpful suggestions regarding our experiments. In the new version, we will add detailed discussions and analyses of the results presented in Tables 1 and 2. Additionally, following your suggestions, we will provide graphical illustrations of the decision performance.\n\n__Q7__: Future work could be discussed in conclusion or before.\n\n__Ans5__:We sincerely appreciate your constructive feedback. Minimizing the FPR while controlling the FDR at a prescribed level is a valuable direction for future research. We will include relevant discussions in the new version."}]

Author rebuttal:

Meta-review: This paper provides a theoretical analysis of the strong anomaly constraint (SAC) network and a betting-like (BL) algorithm, focusing on the false positive rate (FPR) in anomaly detection. The reviewers agree that the paper makes a meaningful theoretical contribution. Building on this analysis, the authors propose a simple yet effective extension of SAC-BL by introducing stochasticity. The theoretical analysis shows the advantage of the proposed method. Experimental results further demonstrate its effectiveness. The author response successfully addressed the reviewers’ concerns, and the additional experiments included during the rebuttal phase further strengthen the paper.
Decision: Accept (poster)