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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf,len=1378
+page_content='PECAN: A Deterministic Certified Defense Against Backdoor Attacks  Yuhao Zhang 1 Aws Albarghouthi 1 Loris D’Antoni 1  Abstract  Neural networks are vulnerable to backdoor poi-  soning attacks, where the attackers maliciously  poison the training set and insert triggers into the  test input to change the prediction of the victim  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Existing defenses for backdoor attacks  either provide no formal guarantees or come with  expensive-to-compute and ineffective probabilis-  tic guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We present PECAN, an efficient  and certified approach for defending against back-  door attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The key insight powering PECAN  is to apply off-the-shelf test-time evasion certi-  fication techniques on a set of neural networks  trained on disjoint partitions of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We  evaluate PECAN on image classification and mal-  ware detection datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Our results demonstrate  that PECAN can (1) significantly outperform the  state-of-the-art certified backdoor defense, both  in defense strength and efficiency, and (2) on real  backdoor attacks, PECAN can reduce attack suc-  cess rate by order of magnitude when compared  to a range of baselines from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Introduction  Deep learning models are vulnerable to backdoor poisoning  attacks (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2019), which assume  that the attackers can maliciously poison a small fragment  of the training set before model training and add triggers to  inputs at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' As a result, the prediction of the victim  model that was trained on the poisoned training set will  diverge in the presence of a trigger in the test input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Effective backdoor attacks have been proposed for various  domains, such as image recognition (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2017), senti-  ment analysis (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021), and malware detection (Sev-  eri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For example, Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2021) can break  malware detection models as follows: The attacker poisons  a small portion of benign software in the training set by  modifying the values of the most important features so that  1Department of Computer Science, University of Wisconsin-  Madison, Madison, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Correspondence to: Yuhao Zhang  <yuhaoz@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='edu>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  the victim model recognizes these values as evidence of the  benign prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' At test time, the attacker inserts a trig-  ger by changing the corresponding features of malware to  camouflage it as benign software and thus making it bypass  the examination of the victim model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Thus, backdoor at-  tacks are of great concern to the safety and security of deep  learning models and systems, particularly as training data is  gathered from different sources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', via web scraping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Several works have studied defenses against various types  of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We identify two limitations with these defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  First, many existing approaches only provide empirical  defenses that are specific to certain attacks and do not  generalize to all backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Second, existing cer-  tified defenses—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', approaches that produce robustness  certificates—are either unable to handle backdoor attacks,  or are probabilistic (instead of deterministic), and therefore  expensive and ineffective in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Why certification?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  A defense against backdoor attacks  should construct effective certificates (proofs) that the  learned model can indeed defend against backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Empirical defenses (Geiping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2018)  do not provide certificates, can only defend against specific  attacks, and can be bypassed by new unaccounted-for at-  tacks (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Certification  has been successful at building models that are provably  robust to trigger-less poisoning attacks and evasion attacks,  but models trained to withstand such attacks are still weak  against backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The trigger-less attack (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Shafahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Aghakhani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Geiping  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021b) assumes the attacker can poison the training set  but cannot modify the test inputs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', adding triggers, while  the evasion attack (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2018) assumes the attacker  modifies the test inputs but cannot poison the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Existing certified defenses against trigger-less and evasion  attacks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', DPA (Levine & Feizi, 2021) and CROWN-  IBP (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020), cannot defend against backdoor  attacks as they can either defend against the poison in the  training data or the triggers at test time, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' As  we show in the experiments, we can break these certified  defenses using a backdoor attack (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Why determinism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  It is desirable for a certified defense  to be deterministic because probabilistic defenses (Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020) typically require one to  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='11824v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='CR]  27 Jan 2023  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  D  D1  D2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Dn  Step 1: Dataset Partitioning  AD1  AD2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  ADn  x  7  cert  1  abstain  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  7  cert  Step 2: Evasion Certification  Step 3: Aggregation  y∗: the top label, y′: runner-up label,  N1: # of certified y∗, N2: # of certified y′,  N3: # of abstain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  y∗: prediction of PECAN,  ⌊  N1−N2−N3−1y∗>y′  2  ⌋: certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Training  Testing  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' An overview of our approach PECAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  retrain thousands of models when performing predictions  for a single test input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Retraining can be mitigated by Bon-  ferroni correction, which allows reusing the trained models  for a fixed number of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' However, retraining is  still necessary after a short period, making it hard to deploy  these defenses in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' On the other hand, determin-  istic defenses (Levine & Feizi, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b)  can reuse the trained models an arbitrary number of times  when producing certificates for different test inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Fur-  thermore, probabilistic defenses for backdoor attacks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=',  BagFlip (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b), need to add noise to the train-  ing data, resulting in low accuracy for datasets that cannot  tolerate too much noise when training (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  PECAN  In this paper, we propose PECAN (Partitioning  data and Ensembling of Certified neurAl Networks), a de-  terministic certified defense against backdoor attacks for  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The key insight underlying PECAN is that  we can take any off-the-shelf technique for evasion certifi-  cation and use it to construct a certified backdoor defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  This insight results in a simple implementation and allows  us to seamlessly leverage future advances in evasion cer-  tification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Specifically, PECAN trains a set of  neural networks on disjoint partitions of the dataset, and  then applies evasion certification to the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' By  partitioning the dataset, we analytically bound the number  of poisoned data seen per neural network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' by employing eva-  sion certification, we bound the number of neural networks  that are robust in the face of triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Using this information,  we efficiently derive a backdoor-robustness guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Figure 1 illustrates the workflow of PECAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In Step 1, in-  spired by deep partition aggregation (Levine & Feizi, 2021),  PECAN deterministically partitions a dataset into multiple  disjoint subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' This step ensures that a poisoned data item  only affects a single partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In Step 2, PECAN trains an  ensemble of neural networks, one on each partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' At test  time, PECAN performs evasion certification to check which  neural networks are immune to triggers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' those that are not  immune (or that cannot be proven immune) abstain from  performing a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Finally, in Step 3, PECAN aggre-  gates the results of the ensemble and produces a prediction  together with a robustness certificate: the percentage of the  poisoned data in the training set that the training process  can tolerate, the certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We evaluate PECAN on two three datasets, MNIST, CI-  FAR10, and EMBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' First, we show that PECAN outper-  forms or competes with BagFlip, the state-of-the-art prob-  abilistic certified defense against backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Fur-  thermore, BagFlip takes hours to compute the certificate,  while PECAN only takes a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Second, when  we evaluate PECAN against a concrete known backdoor  attack (Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021), PECAN reduces the attack suc-  cess rate to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='85%, while DPA and CROWN-IBP fail to  defend against the backdoor attack on 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='05% and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='24%  of the cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The results show that PECAN  can defend against a known backdoor attack while other  baselines, such as DPA and CROWN-IBP, cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Related Work  Deep learning models are vulnerable to backdoor at-  tacks (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Although  many empirical defenses (Geiping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=',  2018) have been proposed, recent works (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022) show that new attacks can break these em-  pirical defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Therefore, certified defense is crucial for  defending against backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Certified defenses against backdoor attacks  Existing  certification approaches provide probabilistic certificates by  extending randomized smoothing (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Dvi-  jotham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2019), originally proposed to  defend against adversarial evasion attacks, to defend against  backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' BagFlip (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b) is the  state-of-the-art model-agnostic probabilistic defense against  feature-flipping backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We-  ber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2020) proposed backdoor-attack defenses that  are also model-agnostic, but are less effective than BagFlip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  PECAN is deterministic and therefore less expensive and  more effective than these defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Probabilistic defenses  are model-agnostic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' while PECAN is evaluated on neural  networks, it can work for any machine learning model as  long as a deterministic evasion certification approach of the  model is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2020) proposed a determin-  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  istic de-randomized smoothing approach for kNN classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Their approach computes the certificates using an expen-  sive dynamic programming algorithm, whereas PECAN’s  certification algorithm has constant time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Certified defenses against trigger-less attacks  Many  approaches provide certificates for trigger-less attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Jia  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2021) use bootstrap aggregating (Bagging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Chen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2020) extended Bagging with new selection strate-  gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Rosenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2020) defend against label-flipping  attacks on linear classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Differential privacy (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=',  2019) can also provide probabilistic certificates for trigger-  less attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' DPA (Levine & Feizi, 2021) is a deterministic  defense that partitions the training set and ensembles the  trained classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2022b) proposed FA, an  extension of DPA, by introducing a spread stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' A conjec-  ture proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2022a) implies that DPA and  FA are asymptotically optimal defenses against trigger-less  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2022) proposed to compute collective  certificates, while PECAN computes sample-wise certifi-  cates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Drews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  (2020) provide certificates for nearest neighborhood classi-  fiers and decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The approaches listed above only  defend against trigger-less attacks, while PECAN is a deter-  ministic approach for backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Certified defenses against evasion attacks  There are  two lines of certified defense against evasion attacks: com-  plete certification (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2019) and incomplete certification (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The com-  plete certified defenses either find an adversarial example  or generate proof that all inputs in the given perturbation  space will be correctly classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Compared to the complete  certified defenses, the incomplete ones will abstain from pre-  dicting if they cannot prove the correctness of the prediction  because their techniques will introduce over-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  The complete approaches do not have over-approximation  issues but require expensive verification algorithms such as  branch and bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Our implementation of PECAN uses an  incomplete certified approach CROWN-IBP (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=',  2020) because it is the best incomplete approach, trading off  between efficiency and the degree of over-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Problem Definition  Given a dataset D = {(x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' , (xn, yn)}, a (test) input  x, and a machine learning algorithm A, we write AD to  denote the machine learning model learned on dataset D  by the algorithm A, and AD(x) to denote the output label  predicted by the model AD on input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We assume the  algorithm will behave the same if trained on the same dataset  across multiple runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' This assumption can be guaranteed by  fixing the random seeds during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We are interested in certifying that if an attacker has poi-  soned the dataset, the model we have trained on the dataset  will still behave “well” on the test input with maliciously  added triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Before describing what “well” means, we  need to define the perturbation spaces of the dataset and  the test input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', what possible changes the attacker could  make to the dataset and the test input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Perturbation space of the dataset  Following Levine &  Feizi (2021), we define a general perturbation space over  the dataset, allowing attackers to delete, insert, or modify  training examples in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Given a dataset D and a  radius r ≥ 0, we define the perturbation space as the set of  datasets that can be obtained by deleting or inserting up to r  examples in D:  Sr(D) =  �  �D | |D ⊖ �D| ≤ r  �  ,  where A ⊖ B is the symmetric difference of sets A and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Intuitively, r quantifies how many examples need to be  deleted or inserted to transform from D to �D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' If the attacker modifies one training example  x ∈ D to another training example �x to form a poisoned  dataset �D = (D \\ {x}) ∪ {�x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Then �D ∈ S2(D) but  �D /∈ S1(D) because Sr(D) considers one modification as  one deletion and one insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Note that we assume a more general perturbation space  of the training set than the one considered by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  (2022b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' our work  allows inserting and deleting examples instead of just modi-  fying existing training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Perturbation space of the test input  We write π(x) to  denote the set of perturbed examples that an attacker can  transform the example x into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Formally, the perturbation  space π(x) can be defined as the lp norm ball with radius s  around the test input x,  π(x) = {�x | ∥x − �x∥p ≤ s}  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' BagFlip (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b) considers the  l0 feature-flip perturbation Fs(x), which allows the attacker  to modify up to s features in an input x,  Fs(x) = {�x | ∥x − �x∥0 ≤ s}  Threat models  Next, we define what type of guarantees  we are interested in our learning algorithm and model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We  consider backdoor attacks, where the attacker can perturb  both the training set and the test input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For the training set,  we assume we are given a perturbation space Sr(D) of the  training set D with a radius r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For the test input, we  assume a perturbation space π(x) of the test input x with a  given lp norm and the radius s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  We say that an algorithm A is robust to a backdoor attack  on a backdoored test input �x if the algorithm trained on any  perturbed dataset �D would predict the backdoored input �x  the same as AD(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Formally,  ∀ �D ∈ Sr(D), �x ∈ π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' A �  D(�x) = AD(x)  (1)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' When r = 0, Eq 1 degenerates to evasion  robustness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', ∀�x ∈ π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' AD(�x) = AD(x), because  S0(D) = {D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Given a large enough radius r, an attacker can always  change enough inputs and succeed at breaking robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Therefore, we will typically focus on computing the max-  imal radius r for which we can prove that Eq 1 for given  perturbation spaces Sr(D) and π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We refer to this quan-  tity as the certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Certified guarantees  This paper aims to design a cer-  tifiable algorithm A, which can defend against backdoor  attacks, and to compute the certified radius of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In our ex-  periments (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2), we suppose a given benign dataset  D and a benign test input x, and we certifiably quantify the  robustness of the algorithm A against backdoor attacks by  computing the certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3, we also experiment with how the certifiable  algorithm A defends the backdoor attacks if a poisoned  dataset �D and a test input �x with malicious triggers are  given, but the clean data is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We theoretically show  that we can still compute the certified radius if the clean  data D and x are unknown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The PECAN Certification Technique  Our approach, which we call PECAN (Partitioning data and  Ensembling of Certified neurAl Networks), is a determin-  istic certification technique that defends against backdoor  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Given a learning algorithm A, we show how to  automatically construct a new learning algorithm ¯A with  certified backdoor-robustness guarantees (Equation (1)) in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2, we prove the certified backdoor-  robustness guarantees (Equation (1)) provided by ¯A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We  further discuss how ¯A can defend against a backdoored  dataset and formally justify our discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Constructing Certifiable Algorithm ¯A  The key idea of PECAN is that we can take any off-the-shelf  technique for evasion certification and use it to construct  a certified backdoor defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Intuitively, PECAN uses the  evasion certification to defend against the possible triggers  at test time, and it encapsulates the evasion certification in  deep partition aggregation (DPA) (Levine & Feizi, 2021) to  defend against training set poisoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Given a dataset D, a test input x, and a machine learning  algorithm A, PECAN produce a new learning algorithm ¯A  as described in the following steps (shown in Figure 1),  Dataset Partitioning  We partition the dataset D into n  disjoint sub-datasets, denoted as D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' , Dn, using a hash  function that deterministically maps each training example  into a sub-dataset Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Train n classifiers AD1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' , ADn on  these sub-datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Evasion Certification  We certify whether the prediction  of each classifier ADi is robust under the perturbation space  π(x) by any evasion certification approach for the learn-  ing algorithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', CROWN-IBP for neural networks (Xu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Formally, the certification approach determines  whether the following equation holds,  ∀�x ∈ π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' ADi(x) = ADi(�x)  (2)  We denote the output of each certification as Aπ  Di(x), which  can either be Aπ  Di(x) = cert, meaning Eq 2 is certified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Otherwise, Aπ  Di(x) = abstain, meaning the certification  approach cannot certify Eq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Aggregation  We compute the top label y∗ by aggre-  gating all predictions from ADi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Concretely, y∗ ≜  argmax  y∈C  �n  i=1 1ADi(x)=y, where C = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='} is the set  of possible labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Note that if a tie happens when taking  the argmax, we break ties deterministically by setting the  smaller label index as y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We denote the runner-up label  as y′ as argmax  y∈C∧y̸=y∗  �n  i=1 1ADi(x)=y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We count the number  of certified predictions equal to y∗ as N1, the number of  certified predictions equal to y′ as N2, and the number of  abstentions as N3,  N1 =  n  �  i=1  1ADi(x)=y∗∧Aπ  Di(x)=cert,  N2 =  n  �  i=1  1ADi(x)=y′∧Aπ  Di(x)=cert,  N3 =  n  �  i=1  1Aπ  Di(x)=abstain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We set the prediction ¯AD(x) as y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We compute the cer-  tified radius r in the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' If N1 − N2 −  N3 − 1y∗>y′ < 0, we set r as ⋄, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', a value denoting no  certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In this case, PECAN cannot certify that ¯A is  robust to evasion attacks even if the dataset is not poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Otherwise, we compute r as ⌊  N1−N2−N3−1y∗>y′  2  ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' A spe-  cial case is r = 0, when PECAN can certify ¯A is robust  to evasion attacks, but cannot certify that it is robust if the  dataset is poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We note that the computation of the certified radius is equiv-  alent to DPA when no classifier abstains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', N3 = 0,  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  D  �  D  x  �x  D1  D2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Dn  �  D1  �  D2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  �  Dn  7  abstain  5  abstain  7  cert  7  cert  7  cert  1  cert  7 → y′  5 → y′  7 → y′  7 → y′  7  1  Dabs  Attacked by �x  Dbd  Attacked by �  D  Dsafe  Clean  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' An illustration of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' It shows  the worst case for PECAN, where the attacker can change all  predictions in Dabs and Dbd to the runner-up label y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Note  that we group Dabs, Dbd, and Dsafe together to ease illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Proving the Soundness of PECAN  In this section, we show that the prediction ¯AD(x) and the  certified radius r satisfy the certified backdoor-robustness  guarantees (Equation (1)) by proving the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1 (Soundness of PECAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Given a dataset D  and a test input x, PECAN computes the prediction ¯AD(x)  and the certified radius as r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Then, either r = ⋄ or  ∀ �D ∈ Sr(D), �x ∈ π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' ¯A �  D(�x) = ¯AD(x)  (3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For any poisoned dataset �D, we partition �D into  n sub-datasets { �D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' , �Dn} according to {D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' , Dn}  from the clean dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Note that we can determine such  a correspondence between Di and �Di because our hash  function is deterministic and only depends on each train-  ing example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We further divide {D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' , Dn} into three  disjoint parts Dabs, Dbd, and Dsafe in the following way,  Dabs = {Di | Aπ  Di(x) = abstain} are the sub-  datasets, on which A abstains from making the pre-  diction on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' From the definition of N3, we have  |Dabs| = N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Intuitively, Dabs contains the sub-  datasets that can possibly be attacked by the test input  �x with malicious triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Dbd are the sub-datasets on which A does not abstain  and are also poisoned, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', each of them has at least one  training example removed or inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Even though we  do not know the exact sub-datasets in Dbd, we know  |Dbd| ≤ r because �D ∈ Sr(D) constrains that there  are at most r such poisoned sub-datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Dsafe = {Di | Di = �Di ∧ Aπ  Di(x) = cert} contains  the clean sub-datasets, on which A does not abstain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We denote the numbers of the original top prediction y∗ and  the original runner-up prediction y′ on the backdoored data  �D and �x as �  Ny∗ and �  Ny′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Formally,  �  Ny∗ =  n  �  i=1  1A�  Di(�x)=y∗,  �  Ny′ =  n  �  i=1  1A�  Di(�x)=y′  Next, we prove Eq 3 for any backdoored data �D and �x by  showing that  �  Ny∗ ≥ �  Ny′ + 1y∗>y′  (4)  We prove Eq 4 by showing a lower bound of �  Ny∗ is N1 − r  and an upper bound of �  Ny′ is N2 + r + N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Together with  the definition of r, we can prove Eq 4 because we have,  �  Ny∗ − �  Ny′ − 1y∗>y′  ≥N1 − r − (N2 + r + N3) − 1y∗>y′  =N1 − N2 − 2r − N3 − 1y∗>y′  =N1 − N2 − 2⌊N1 − N2 − N3 − 1y∗>y′  2  ⌋ − N3 − 1y∗>y′  ≥N1 − N2 − (N1 − N2 − N3 − 1y∗>y′) − N3 − 1y∗>y′  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Note that the second last line holds iff N1 − N2 − N3 −  1y∗>y′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Otherwise, we have r = ⋄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  As shown in Figure 2, the lower bound of �  Ny∗ can be com-  puted by noticing that 1) the attacker can change any predic-  tion in Dbd from y∗ to another label because these datasets  are poisoned, 2) the attacker can change any prediction in  Dabs to another label because CROWN-IBP cannot certify  the prediction under the evasion attacks, and 3) the attacker  cannot change anything in Dsafe because of the guarantee  of CROWN-IBP and Dsafe is not poisoned,  ∀Di ∈ Dsafe, �x ∈ π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' ADi(x) = ADi(�x) = A �  Di(�x)  The upper bound of �  Ny′ can be computed by noticing that  1) the attacker can change any prediction in Dbd to y′, 2)  the attacker can change any prediction in Dabs to y′, and 3)  the attacker cannot change anything in Dsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We complete the proof by showing that the best attack strat-  egy of the attacker is to change the prediction of ¯A to the  runner-up label y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' If the attacker chooses to change the  prediction of ¯A to another label y′′, denoted the counts as  �  Ny′′, then the upper bound of �  Ny′′ will be always smaller  or equal to �  Ny′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' PECAN under the Backdoored Data  The above algorithm and proof of PECAN assume that a  clean dataset D and a clean test example x are already given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  However, we may be interested in another scenario where  the poisoned dataset �D ∈ Sr(D) and the input example  Pattern BackdoorPECAN: A Deterministic Certified Defense Against Backdoor Attacks  �x ∈ π(x) with malicious triggers are given, and the clean  data D and x are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In other words, we want to find  the maximal radius r such that ¯A �  D(�x) = ¯AD(x) for any D  and x that can be perturbed to �D and �x by the perturbation  Sr and π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Formally,  ∀D, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' �D ∈ Sr(D) ∧ �x ∈ π(x) =⇒  ¯A �  D(�x) = ¯AD(x)  (5)  Intuitively, Eq 5 is the symmetrical version of Eq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Ow-  ing to the symmetrical definition of Sr and π, if we apply  PECAN to the given poisoned data �D, �x, then the predic-  tion ¯A �  D(�x) and the certified radius r satisfy the certified  backdoor-robustness guarantee (Eq 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The following the-  orem formally states the soundness of PECAN under the  backdoored data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2 (Soundness of PECAN under the backdoored  data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Given a dataset �D and a test input �x, PECAN com-  putes the prediction ¯A �  D(�x) and the certified radius as r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Then, either r = ⋄ or Eq 5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Experiments  We implemented PECAN in Python and provided the im-  plementation in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In our evalua-  tion, we use CROWN-IBP, implemented in auto-LiRPA (Xu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2020), as the evasion defense approach for neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We also use CROWN-IBP to train the classifiers  in the dataset partitioning step since the classifiers trained  by CROWN-IBP can improve the certification rate in the  evasion certification step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2, we evaluate the effectiveness and efficiency  of PECAN by comparing it to BagFlip (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b),  the state-of-the-art probabilistic certified defense against  backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3, we evaluate the effective-  ness of PECAN under the backdoor attack (Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=',  2021) for malware detection and compare PECAN to other  baselines, DPA and CROWN-IBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Experimental Setup  Datasets  We conduct experiments on MNIST, CIFAR10,  and EMBER (Anderson & Roth, 2018) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' MNIST is  an image classification dataset containing 60,000 training  and 10,000 test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' CIFAR10 is an image classifica-  tion dataset containing 50,000 training and 10,000 test ex-  amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' EMBER is a malware detection dataset containing  600,000 training and 200,000 test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Each example  is a vector containing 2,351 features of the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Models  For image classification datasets MNIST and CI-  FAR10, we train fully-connected neural networks with four  layers for PECAN, while BagFlip uses CNN and ResNet for  MNIST and CIFAR10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We do not use CNN  and ResNet because CROWN-IBP used in PECAN has a  higher abstention rate for deeper and more complex neu-  ral network structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We use the same fully-connected  neural network for EMBER as in related works (Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We use the same data  augmentation for PECAN and other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Metrics  For each test input xi, yi, the algorithm ¯A will  predict a label and the certified radius ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In this section, we  assume that the attacker had modified R% examples in the  training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We denote R as the modification amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We  summarize all the metrics used as follows,  Certified Accuracy denotes the percentage of test examples  that are correctly classified and whose certified radii are  no less than R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 1  m  �m  i=1 1 ¯  AD(xi)=yi∧ ri  |D| ≥2R%, where  m and |D| are the sizes of test set and training set, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Notice that there is a factor of 2 on the modification  amount R because Sr(D) considers one modification as one  insertion and one deletion, as illustrated in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Normal Accuracy denotes the percentage of test examples  that are correctly classified by the algorithm without certifi-  cation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 1  m  �m  i=1 1 ¯  AD(xi)=yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Attack Success Rate (ASR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3, we are interested  in how many test examples are certified but wrongly clas-  sified by the classifier, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 1  m  �m  i=1 1 ¯  AD(xi)̸=yi∧ ri  |D| ≥2R%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We denote the above quantity as the attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We  note that a prediction can still be incorrect even if it is cer-  tified by PECAN because the classifier can have incorrect  predictions even when the data is clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Abstention Rate is computed as 1  m  �m  i=1 1 ri  |D| <2R%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Effectiveness and Efficiency of PECAN  We evaluate the effectiveness and efficiency of PECAN on  MNIST, CIFAR10, and EMBER under the backdoor attack  with the l0 feature-flip perturbation F1, which allows the  attacker to modify up to one feature in an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We  compare PECAN to BagFlip, the state-of-the-art probabilis-  tic certified defense against l0 feature-flip backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Moreover, we note that PECAN needs to construct harder  proofs than BagFlip because their definitions of perturbation  space are different, as discussed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  In Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2, we evaluate the effectiveness of PECAN  against the perturbation space with the l∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Summary of the results  PECAN achieves significantly  higher certified accuracy than BagFlip on CIFAR10 and  EMBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' PECAN achieves competitive results on MNIST  compared to BagFlip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' PECAN has similar normal accu-  racy as BagFlip for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' PECAN is more efficient  than BagFlip at computing the certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Setup  We use the same hyper-parameters for BagFlip as  reported in their paper for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For PECAN, we vary  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  0  2  4  10−2  0  20  40  60  80  100  Modification Amount R (%)  Certifiable Accuracy  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='5  2  10−2  BagFlip  PECAN-a  PECAN-b  (a) CIFAR10 F1  (b) EMBER F1  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparison to BagFlip on CIFAR10 and EMBER, show-  ing the normal accuracy (dotted lines) and the certified accuracy  (solid lines) at different modification amounts R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For CIFAR10:  a = 50 and b = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For EMBER: a = 200 and b = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='5  0  20  40  60  80  100  Modification Amount R (%)  Certifiable Accuracy  BagFlip  PECAN-600  PECAN-1200  PECAN-2000  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparison to BagFlip on MNIST, showing the normal  accuracy (dotted lines) and the certified accuracy (solid lines) at  different modification amounts R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  n, the number of partitions, to ensure a fair comparison be-  tween BagFlip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1 presents a detailed discussion  of hyper-parameter settings for BagFlip and PECAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We  denote PECAN with different settings of n as PECAN-n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  BagFlip achieves meaningful results only on MNIST, where  we also tune the parameter n of PECAN to 2000 to achieve  the same certified accuracy of BagFlip at R = 0 and com-  pare their results following the practice in related works (Jia  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Results  Figure 3 shows the comparison between PECAN  and BagFlip on CIFAR10 and EMBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' PECAN achieves  significantly higher certified accuracy than BagFlip  across all modification amounts R and the similar nor-  mal accuracy as BagFlip for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  BagFlip performs poorly on CIFAR10 and EMBER because  these two datasets cannot tolerate the high level of noise that  the BagFlip algorithm adds to the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Specifically,  BagFlip can add 20% noise to the training data of MNIST,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', a feature (pixel) in a training example will be flipped to  another value with 20% probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' However, for CIFAR10  and EMBER, this probability has to be decreased to 5% to  maintain normal accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Figure 4 shows the comparison between PECAN and  BagFlip on MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' PECAN achieves competitive results  compared to BagFlip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We find that two approaches have  similar normal accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Comparing PECAN-600 and  PECAN-1200 with BagFlip, we find that 1) PECAN-600  and PECAN-1200 achieves higher certified accuracy than  BagFlip when R ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='25] and R ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='17], respec-  tively, and 2) BagFlip has non-zero certified accuracy when  R ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='5], where the certified accuracy of PECAN-600  and PECAN-1200 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparing PECAN-2000 with  BagFlip, we find that BagFlip outperforms PECAN-2000  across all modification amounts R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We argue that the gap of certified accuracy between PECAN-  2000 and BagFlip mainly comes from the different def-  initions of the perturbation spaces as discussed in Ap-  pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Moreover, the root cause of this difference  is owing to the probabilistic nature of BagFlip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  PECAN is more efficient than BagFlip at computing the  certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' PECAN computes the certified radius in a  constant time complexity via the closed-form solution in the  aggregation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' However, in our experiment of the MNIST  dataset, BagFlip requires 8 hours to prepare a lookup table  because BagFlip does not have a closed-form solution for  computing the certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' PECAN under the Backdoored Data  We evaluate the effectiveness of PECAN under the back-  door attack (Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021) for malware detection on the  EMBER dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We do not compare PECAN to BagFlip  because BagFlip has poor certified accuracy on EMBER, as  shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We also evaluate other baselines, DPA  and CROWN-IBP, which do not aim to defend against back-  door attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' DPA is the certified defense against trigger-less  attacks, and CROWN-IBP is the certified defense against  evasion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We also present the results of the victim  classifiers without any defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='4 shows that  the empirical defense spectral signatures (Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2018)  cannot defend against the backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Summary of the results  PECAN reduces the ASR of  the victim model on the test set with malicious triggers  from 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='33% to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='85%, while the other baselines fail  to defend against the backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Being the most  conservative, PECAN has the highest abstention rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Setup  We use Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2021) to generate backdoored  data by modifying 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1% training examples and adding trig-  gers into the test inputs that should be labeled as malware to  fool the victim model to predict the malware with malicious  triggers as benign software (non-malware).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We generate  three poisoned datasets �D1, �D2, �D3 and their correspond-  ing test sets with triggers by perturbations F1, F2, and F3,  which allow the attacker to modify up to one, two, and three  features in an example, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We report the results of all approaches on the malware test  sets with triggers and the malware test sets without trig-  gers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', the original malware test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The results on the  non-malware test sets without triggers can be found in Ap-  pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' ASR on malware is a much more critical metric  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  0  20  40  60  80  100  ASR  Correct  Abstain  PECAN  DPA  C-IBP  NoDef  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Results of PECAN, DPA, CROWN-IBP (C-IBP), and  vanilla model without defense (NoDef) trained on three poisoned  EMBER datasets when evaluated on the malware test set with  malicious triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We note that NoDef does not have abstention  rates because it does not use any defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  0  20  40  60  80  100  ASR  Correct  Abstain  PECAN  DPA  C-IBP  NoDef  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Results of PECAN, DPA, C-IBP, and NoDef when evalu-  ated on the (original) malware test set without malicious triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  than the ASR on non-malware, because the former shows  how many pieces of malware can bypass the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  For PECAN and DPA, we show their results at modification  amount R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We show CROWN-IBP results against  the perturbations F1, F2, and F3 regardless of R because  CROWN-IBP does not consider R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Results  Figures 5 and 6 show the ASR, accuracy, and ab-  stention rate of all the approaches on the malware test set  with and without triggers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Table 1 in the ap-  pendix shows the detailed numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Note that PECAN is  the only certified approach for backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The  results of other baselines can be seen as empirical be-  cause DPA and CROWN-IBP certify a different goal,  and NoDef has no defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  PECAN can defend against the backdoor attack on the  EMBER dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Figures 5 and 6 show that PECAN has  the lowest ASR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='85% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='03% on both malware test  sets with and without triggers on average, compared to  DPA (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='05%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='98%), CROWN-IBP (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='24%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='82%),  and NoDef (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='33%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='12%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  0  5 · 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1  0  20  40  60  80  100  Modification Amount R (%)  Test Set Percentage  0  5 · 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1  Correct  ASR  Abstain  (a) PECAN  (b) DPA  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparison between PECAN and DPA trained on �D3  across all modification amount R when evaluated on the malware  test set with triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  DPA and CROWN-IBP fail to defend against the back-  door attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The average ASR of DPA and CROWN-IBP  on the malware test set with triggers are 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='05% and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='24%  in Figure 5, respectively, meaning that many malware with  triggers can bypass their defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The average ASR of  DPA on the malware test set without triggers, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='98%, is  much lower than its ASR on the one with triggers, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='05%,  which shows that DPA successfully defends against trigger-  less attacks when the test input does not have any trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  CROWN-IBP has high ASR on both the malware test sets  with and without triggers, as CROWN-IBP cannot defend  against the poison in the training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  PECAN has higher abstention rates than other ap-  proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  On average, PECAN abstains from 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='41%  predictions compared to DPA (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='73%) and CROWN-IBP  (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='44%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We further compare the accuracy, ASR, and ab-  stention rate of PECAN and DPA across all modification  amount R when trained on �D3 in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The results on  �D1 and �D2 are shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We can observe  that PECAN has a much lower ASR than DPA across all  modification amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Meanwhile, Figure 7 shows that  the certification of PECAN might be over-conservative be-  cause the ASR is low (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='17%) even when we regard �D3 as  non-poisoned (when R = 0), yet �D3 is actually poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Conclusion, Limitations, and Future Work  We presented PECAN, a deterministic certified approach  to effectively and efficiently defend against backdoor at-  tacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We foresee many future improvements to PECAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  First, we implemented PECAN as a certified defense special-  ized for neural networks because the evasion certification  step, CROWN-IBP, is limited to neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' However,  we can replace CROWN-IBP with an evasion certification  approach for another machine learning model to get a cor-  responding backdoor defense for that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Second, we  adopt the idea of deep partition aggregation (DPA) to de-  sign the partition and aggregation steps in PECAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We can  improve these steps by using finite aggregation (FA) (Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b), which extends DPA and gives higher certified  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Third, during the certification of evasion attacks,  we need to propagate the abstraction of the same test input  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  through thousands of neural networks that have different  weights but the same architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Sharing the propaga-  tion results among different neural networks (Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=',  2022) can greatly improve the efficiency of PECAN and  may enable using complete certification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  References  Aghakhani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', Meng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', Kruegel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', and Vigna,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Bullseye polytope: A scalable clean-label poison-  ing attack with improved transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='1109/Euro  SP51992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content=' EMBER: an open dataset  for training static PE malware machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  CoRR, abs/1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='04637, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content=' A framework  of randomized selection based certified defenses against  data poisoning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' CoRR, abs/2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='08739, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content=' On collec-  tive robustness of bagging against data poisoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content=' ), International Conference on  Machine Learning, ICML 2022, 17-23 July 2022, Balti-  more, Maryland, USA, volume 162 of Proceedings of  Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' URL http://pr  oceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='press/v97/cohen19c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='  Drews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', Albarghouthi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' ), Proceedings of the 36th International Confer-  ence on Machine Learning, ICML 2019, 9-15 June 2019,  Long Beach, California, USA, volume 97 of Proceedings  of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1 tells that either r = ⋄ or the following  equation holds,  ∀D′ ∈ Sr( �D), x′ ∈ π(�x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' ¯A �  D(�x) = ¯AD′(x′)  (6)  By the symmetrical definition of Sr and π, we have  ∀D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' �D ∈ Sr(D) =⇒ D ∈ Sr( �D)  (7)  ∀x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' �x ∈ π(x) =⇒ x ∈ π(�x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  (8)  Then, for all possible clean data D and x, we have  �D ∈ Sr(D) ∧ �x ∈ π(x)  =⇒ D ∈ Sr( �D) ∧ x ∈ π(�x)  (By Eq 7 and Eq 8)  =⇒ ¯A �  D(�x) = ¯AD(x)  (By Eq 6)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Experiment  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Detailed Setup of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2  Following the BagFlip paper (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b), we set k,  the number of training examples in a bag used in BagFlip,  as 100, 1000, and 3000 for the MNIST, CIFAR10, and  EMBER dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For PECAN, we vary n, the  number of partitions, according to the value of k in BagFlip  by setting n = |D|  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  BagFlip defines their perturbation space S′  r(D) that is dif-  ferent from PECAN,  S′  r(D) =  �  �D | max(|D \\ �D|, | �D \\ D|) ≤ r  �  ,  where A \\ B is the set difference, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', the elements in A  but not in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Notice that with the same radius r, the above  definition gives a larger S′  r(D) than Sr(D) as the following  example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' If the attacker modifies one training example  x ∈ D to another training example �x to form a poisoned  dataset �D = D \\ {x} ∪ {�x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Then �D ∈ S2(D) but �D /∈  S1(D) because Sr(D) considers one modification as one  deletion and one insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' However, we have �D ∈ S′  1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2022) show that S′  r(D) works when the ap-  proach uses non-deterministic sub-sampling (Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' However, the certification of determin-  istic approaches only works under the definition of Sr(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  We adjust the computation of certified accuracy for BagFlip  as 1  m  �m  i=1 1 ¯  AD(xi)=yi∧ ri  |D| ≥R% by removing the factor 2  on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Thus, we are also interested in the performance of  PECAN when n = |D|  2k to compensate the removed factor  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='4  0  20  40  60  80  100  Modification Amount R (%)  Certifiable Accuracy  0  2  4  10−2  PECAN-a  PECAN-b  (a) MNIST s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1  (b) CIFAR10 s = 2/255  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Results of PECAN on CIFAR10 and EMBER, showing  the normal accuracy (dotted lines) and the certified accuracy (solid  lines) at different modification amounts R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For MNIST: a = 600  and b = 1200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For CIFAR10: a = 50 and b = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Evaluation on the l∞ Perturbation Space  Setup  As the CROWN-IBP used in PECAN can handle  π with different lp norms, PECAN can handle different lp  norms as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We evaluate PECAN on the l∞ norm with  distance s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1 and s = 2/255 on MNIST and CIFAR10,  respectively, because the l∞ norm is widely applied to eval-  uate the robustness of image classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' In this experimental  setting, we use two CNN models for MNIST and CIFAR10  because CROWN-IBP works better for CNN on l∞ norm  than on l0 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For training on MNIST and CIFAR10, we  train on s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2 and s = 5/255 but test on s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1 and  s = 2/255 to overcome the overfitting issue when s is small,  following the practice in the original paper of CROWN-IBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  For the experiments on l0 (Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3), we set  the κstart = 0 and κend = 0 for CROWN-IBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' For the ex-  periments on l∞, we set the κstart = 1 and κend = 0 for  CROWN-IBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Results  Figure 8 shows the results of PECAN against l∞  perturbation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' The results show that PECAN achieves  certified accuracy similar to F1 as shown in Figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparison to DPA, CROWN-IBP, and NoDef on  the Non-Malware Test Set without Trigger  Figure 9 shows that NoDef has the lowest ASR of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='70% on  the non-malware set without trigger than all three defenses  because the backdoor attack does not aim to attack the  prediction of non-malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' However, PECAN still achieves  the lowest ASR of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='73% compared to DPA (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='85%) and  CROWN-IBP (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='68%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparison to Spectral Signatures  We followed the experiment in Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (2021) to filter  out poisoned examples in the training dataset �D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' After  removing the top 15% outliers in the non-malware training  set, we observe that only 14% (84 out of 600) of the poison  is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Then we train a new model using the filtered  training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We find the ASR of the new model on the  PECAN: A Deterministic Certified Defense Against Backdoor Attacks  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Results of PECAN, DPA, CROWN-IBP (C-IBP) and vanilla model without defense (NoDef) trained on three backdoored EMBER  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Malware with triggers is the backdoored test data that should be labeled as malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Malware w/o triggers is the original test  data that should be labeled as malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Non-malware w/o triggers is the original test data that should be labeled as non-malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  Test sets  Malware with triggers  Malware w/o triggers  Non-Malware w/o triggers  Approaches  PECAN  DPA  C-IBP  NoDef PECAN  DPA  C-IBP  NoDef PECAN  DPA  C-IBP  NoDef  �D1  ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (↓)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='38%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='68%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='72% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='15%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='27%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='00%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='10%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='92%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='48%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='81%  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='41%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='94%  Correct Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (↑) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='42% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='57% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='86% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='85% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='01% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='34% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='21% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='08% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='82% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='29% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='66% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='06%  Abstention Rate  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='20% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='75% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='42%  N/A 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='72% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='66% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='69%  N/A 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='70% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='90%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='93%  N/A  �D2  ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='64%  Correct Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (↑) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='33% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='95% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='03% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='36%  Abstention Rate  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='69% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='64%  N/A 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='27% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='70% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='95%  N/A 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='50% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='14%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='21%  N/A  �D3  ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (↓)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='42% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='67%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='71%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='97% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='32%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='91%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='51%  Correct Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' (↑) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='72% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='40% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='96% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
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+page_content='49%  Abstention Rate  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='33% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='54% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='99%  N/A 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='84% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='45% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='58%  N/A 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='44% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='14%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='59%  N/A  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  �  D1 �  D2 �  D3  0  20  40  60  80  100  ASR  Correct  Abstain  PECAN  DPA  C-IBP  NoDef  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Results of PECAN, DPA, CROWN-IBP (C-IBP), and  vanilla model without defense (NoDef) trained on three poisoned  EMBER datasets when evaluated on the (original) non-malware  test set without triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  malware set with triggers, the malware set without triggers,  and the non-malware set without triggers are 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='55%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='38%,  and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='91%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' These ASRs are all higher than  PECAN’s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='19%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='71%, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='16% on the three parts of  the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparison to DPA on �D1 and �D2  Figures 10 and 11 show the comparison between PECAN  and DPA on �D1 and �D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' We can observe that PECAN  has much higher ASRs than DPA across all modification  amounts on �D1 and �D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  0  5 · 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1  0  20  40  60  80  100  Modification Amount R (%)  Test Set Percentage  0  5 · 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1  Correct  ASR  Abstain  (a) PECAN  (b) DPA  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparison between PECAN and DPA trained on �D1  across all modification amount R when evaluated on the malware  test set with triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='  0  5 · 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1  0  20  40  60  80  100  Modification Amount R (%)  Test Set Percentage  0  5 · 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content='1  Correct  ASR  Abstain  (a) PECAN  (b) DPA  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}
+page_content=' Comparison between PECAN and DPA trained on �D2  across all modification amount R when evaluated on the malware  test set with triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FKT4oBgHgl3EQfeS7D/content/2301.11824v1.pdf'}