Theoretical Analysis for SAC-BL Algorithm
Xinsong Ma, Jie Wu, Weiwei Liu
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. (2025)  is the first to highlight 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.