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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf,len=1569
+page_content='REaaS: Enabling Adversarially Robust Downstream  Classifiers via Robust Encoder as a Service  Wenjie Qu1, Jinyuan Jia2, Neil Zhenqiang Gong3  1 Huazhong University of Science and Technology, wen jie qu@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='com  2 University of Illinois Urbana-Champaign, jinyuan@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='edu  3 Duke University, neil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='gong@duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='edu  Abstract—Encoder as a service is an emerging cloud service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Specifically, a service provider first pre-trains an encoder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', a  general-purpose feature extractor) via either supervised learning  or self-supervised learning and then deploys it as a cloud service  API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A client queries the cloud service API to obtain feature  vectors for its training/testing inputs when training/testing its  classifier (called downstream classifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A downstream classifier  is vulnerable to adversarial examples, which are testing inputs  with carefully crafted perturbation that the downstream classifier  misclassifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, in safety and security critical applications,  a client aims to build a robust downstream classifier and certify  its robustness guarantees against adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  What APIs should the cloud service provide, such that a  client can use any certification method to certify the robustness  of its downstream classifier against adversarial examples while  minimizing the number of queries to the APIs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' How can a service  provider pre-train an encoder such that clients can build more  certifiably robust downstream classifiers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We aim to answer the  two questions in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For the first question, we show that  the cloud service only needs to provide two APIs, which we  carefully design, to enable a client to certify the robustness of  its downstream classifier with a minimal number of queries to  the APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For the second question, we show that an encoder pre-  trained using a spectral-norm regularization term enables clients  to build more robust downstream classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  INTRODUCTION  In an encoder as a service, a service provider (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', OpenAI,  Google, and Amazon) pre-trains a general-purpose feature  extractor (called encoder) and deploys it as a cloud service;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  and a client queries the cloud service APIs for the feature  vectors of its training/testing inputs when training/testing a  downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For instance, the encoder could be pre-  trained using supervised learning on a large amount of labeled  data or self-supervised learning [1], [2] on a large amount of  unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A client could be a smartphone, IoT device,  self-driving car, or edge device in the era of edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Encoder as a service has been widely deployed by industry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=',  OpenAI’s GPT-3 API [3] and Clarifai’s General Embedding  API [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In the Standard Encoder as a Service (SEaaS), the  service provides a single API (called Feature-API) for clients  Wenjie Qu performed this research when he was an intern in Gong’s group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  and the encoder is pre-trained without taking the robustness  of downstream classifiers into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A client sends its  training/testing inputs to the Feature-API, which returns their  feature vectors to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  A downstream classifier is vulnerable to adversarial exam-  ples [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Suppose a testing input is correctly classified by the  downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' An attacker adds a small carefully crafted  perturbation to the testing input to induce misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Such testing input with carefully crafted perturbation is called  an adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, in security and safety  critical applications such as user authentication and traffic sign  recognition, a client desires to build a downstream classifier  robust against adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Many methods have  been developed for an attacker to craft adversarial examples  and the community keeps developing new, stronger ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Therefore, instead of defending against a specific class of  adversarial examples, a client aims to defend against all  bounded adversarial perturbations via building a certifiably  robust downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A classifier is certifiably robust if  its predicted label for a testing input is unaffected by arbitrary  perturbation added to the testing input once its size (measured  by ℓ2-norm in this work) is less than a threshold, which is  known as certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A larger certified radius indicates  better certified robustness against adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In general, there are two categories of complementary  methods to build a certifiably robust classifier and derive  its certified radius for a testing input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', base classifier  (BC) based certification [7], [8], [9], [10] and smoothed  classifier (SC) based certification (also known as randomized  smoothing) [11], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' BC based certification aims to  directly derive the certified radius of a given classifier (called  base classifier) for a testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' BC based certification  requires white-box access to the base classifier as it often  requires propagating the perturbation from the input layer to  the output layer of the base classifier layer by layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' SC based  certification first builds a smoothed classifier based on the  base classifier via adding random noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', Gaussian noise)  to a testing input and then derives the certified radius of the  smoothed classifier for the testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' To increase the testing  inputs’ certified radii, SC based certification often requires  training the base classifier using training inputs with random  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, to derive the predicted label and certified  radius for a testing input, SC based certification requires the  base classifier to predict the labels of multiple noisy versions  of the testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  SEaaS faces three challenges when a client aims to build a  certifiably robust downstream classifier and derive its certified  Network and Distributed System Security (NDSS) Symposium 2023  28 February - 4 March 2023, San Diego, CA, USA  ISBN 1-891562-83-5  https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='14722/ndss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='24444  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='ndss-symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='org  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='02905v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='CR]  7 Jan 2023  SEaaS  REaaS  Feature-API  Feature-API  F2IPerturb-API  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='1, ⋯, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='2]  Cloud Server  Client  Client  Cloud Server  Encoder  Encoder  �  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='1, ⋯, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='2]  Downstream  Classifier  Step 1  Step 2  Step 3  BC/SC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' 1: SEaaS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  radii for testing inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The first challenge is that a client  cannot use BC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, the composition  of the encoder and the client’s downstream classifier is the  base classifier that the client needs to certify in BC based  certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, the client does not have white-box  access to the encoder deployed on the cloud server, making  BC based certification not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The second challenge  is that, although a client can use SC based certification by  treating the composition of the encoder and its downstream  classifier as a base classifier, it incurs a large communication  cost for the client and a large computation cost for the cloud  server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, the client needs to query the Feature-API  once for each noisy training input in each training epoch of  the downstream classifier because SC based certification trains  the base classifier using noisy training inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, the  client requires e queries to the Feature-API per training input,  where e is the number of epochs used to train the downstream  classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, to derive the predicted label and certified  radius for a testing input, SC based certification requires the  base classifier to predict the labels of N noisy testing inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Therefore, the client requires N queries to the Feature-API  per testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that N is often a large number (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=',  10,000) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The large number of queries to the Feature-  API imply 1) large communication cost, which is intolerable  for resource-constrained clients such as smartphone and IoT  devices, and 2) large computation cost for the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The third challenge is that SC based certification achieves  suboptimal certified radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' This is because the base classifier  is the composition of the encoder and a client’s downstream  classifier, but a client cannot train/fine-tune the encoder as it  is deployed on the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Our work: We propose Robust Encoder as a Service (REaaS)  to address the three challenges of SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Figure 1 compares  SEaaS with REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our key idea is to provide another API  called F2IPerturb-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='1 A downstream classifier essentially  takes a feature vector as input and outputs a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our  F2IPerturb-API enables a client to treat its downstream classifier  alone as a base classifier and certify the robustness of its  base or smoothed downstream classifier in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Specifically, a client performs three steps to derive the certified  radius of a testing input in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' First, the client obtains the  feature vector of the testing input via querying the Feature-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Second, the client views its downstream classifier alone as a  base classifier and derives a feature-space certified radius RF  for the testing input using any BC/SC certification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  client’s base or smoothed downstream classifier predicts the  same label for the testing input if the ℓ2-norm of the adversarial  perturbation added to the testing input’s feature vector is less  1‘F’ stands for Feature and ‘I’ stands for Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  than RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Third, the client sends the testing input and its feature-  space certified radius RF to query the F2IPerturb-API, which  returns the corresponding input-space certified radius R to  the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our input-space certified radius R guarantees the  client’s base or smoothed downstream classifier predicts the  same label for the testing input if the ℓ2-norm of the adversarial  perturbation added to the testing input is less than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The key challenge of implementing our F2IPerturb-API is  how to find the largest input-space certified radius R for a  given testing input and its feature-space certified radius RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  To address the challenge, we formulate finding the largest R  as an optimization problem, where the objective function is  to find the maximum R and the constraint is that the feature-  space perturbation is less than RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, the optimization  problem is challenging to solve due to the highly non-linear  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' To address the challenge, we propose a binary search  based solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The key component of our solution is to check  whether the constraint is satisfied for a specific R in each  iteration of binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Towards this goal, we derive an  upper bound of the feature-space perturbation for a given R  and we treat the constraint satisfied if the upper bound is less  than RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our upper bound can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  F2IPerturb-API addresses the first two challenges of SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Specifically, BC based certification is applicable in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Moreover, SC based certification requires much less queries to  the APIs in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, for any certification method, a  client only requires one query to Feature-API per training input  and two queries (one to Feature-API and one to F2IPerturb-API)  per testing input in our REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  To address the third challenge of SEaaS, we propose a new  method to pre-train a robust encoder, so a client can derive  larger certified radii even though it cannot train/fine-tune the  encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our method can be combined with standard supervised  learning or self-supervised learning to enhance the robustness of  a pre-trained encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' An encoder is more robust if it produces  more similar feature vectors for an input and its adversarially  perturbed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our key idea is to derive an upper bound for  the Euclidean distance between the feature vectors of an input  and its adversarial version, where our upper bound is a product  of a spectral-norm term and the perturbation size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The spectral-  norm term depends on the parameters of the encoder, but it  does not depend on the input nor the adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  An encoder with a smaller spectral-norm term may produce  more similar feature vectors for an input and its adversarial  version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Thus, we use the spectral-norm term as a regularization  term to regularize the pre-training of an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We perform a systematic evaluation on multiple datasets  including CIFAR10, SVHN, STL10, and Tiny-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our  2  evaluation results show that REaaS addresses the three chal-  lenges of SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' First, REaaS makes BC based certification ap-  plicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Second, REaaS incurs orders of magnitude less queries  to the cloud service than SEaaS for SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For  instance, REaaS reduces the number of queries to the cloud  service APIs respectively by 25× and 5, 000× per training and  testing input when a client trains its downstream classifier for  e = 25 epochs and uses N = 10, 000 for certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Third,  in the framework of REaaS, our robust pre-training method  achieves larger average certified radius (ACR) for the testing  inputs than existing methods to pre-train encoders for both BC  and SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For instance, when the encoder is  pre-trained on Tiny-ImageNet and the downstream classifier is  trained on SVHN, the ACRs for MoCo (a standard non-robust  self-supervised learning method) [1], RoCL (an adversarial  training based state-of-the-art robust self-supervised learning  method) [14], and our method are respectively 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='011, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='014,  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='275 when a client uses SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In summary, we make the following contributions: We propose REaaS, which enables a client to build  a certifiably robust downstream classifier and derive  its certified radii using any certification method with a  minimal number of queries to the cloud service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We propose a method to implement F2IPerturb-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We propose a spectral-norm term to regularize the  pre-training of a robust encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We extensively evaluate REaaS and compare it with  SEaaS on multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  RELATED WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Adversarial Examples  We discuss adversarial examples [5], [15] in the context of  encoder as a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We denote by f a pre-trained encoder  and g a downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given a testing input x, the  encoder outputs a feature vector for it, while the downstream  classifier takes the feature vector as input and outputs a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  For simplicity, we denote by f(x) the feature vector and g◦f(x)  the predicted label for x, where ◦ represents the composition  of the encoder and downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In an adversarial  example, an attacker adds a carefully crafted small perturbation  δ to x such that its predicted label changes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', g ◦f(x+δ) ̸=  g ◦ f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The carefully perturbed input x + δ is called an  adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Many methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', [5], [6], [16]) have  been proposed to find an adversarial perturbation δ for a given  input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In our work, we focus on certified defenses, which  aim to defend against any bounded adversarial perturbations  no matter how they are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, we omit the details  on how an attacker can find an adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Certifying Robustness of a Classifier  Definition of certified radius: A classifier is certifiably robust  against adversarial examples if its predicted label for an input  is unaffected by any perturbation once its size is bounded [7],  [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Formally, a classifier h is certifiably robust if we  have the following guarantee for an input x:  h(x + δ) = h(x), ∀ ∥δ∥2 < R,  (1)  where R is known as certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that certified radius  R may be different for different inputs x, but we omit the  explicit dependency on x in the notation for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  A certification method against adversarial examples aims  to build a certifiably robust classifier and derive its certified  radius R for any input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' There are two general categories  of certification methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', base classifier (BC) based  certification [7], [8], [9], [10] and smoothed classifier (SC)  based certification [11], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Both categories of methods  may be adopted in different scenarios depending on certification  needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' On one hand, BC based certification often produces  deterministic guarantees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', the derived certified radius is  absolutely correct), while SC based certification often provides  probabilistic guarantees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', the derived certified radius may  be incorrect with a small error probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' On the other hand,  SC based certification often derives a larger certified radius  than BC based certification due to its probabilistic guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Base classifier (BC) based certification:  BC based certi-  fication aims to directly derive the certified radius R of a  given classifier (called base classifier) for an input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' These  methods often propagate perturbation from the input x to  the output of the base classifier layer by layer in order to  derive the certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, they require white-box  access to the base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Suppose F is a base classifier  that maps an input x to one of c classes {1, 2, · · · , c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We use  H(x) to denote the base classifier’s last-layer output vector  for x, where Hl(x) represents the lth entry of H(x) and  l = 1, 2, · · · , c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' F(x) denotes the predicted label for x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=',  F(x) = argmaxl=1,2,··· ,c Hl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Next, we overview how to  derive the certified radius R using CROWN [9], a state-of-the-  art BC based certification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' CROWN shows that each  entry Hl(x) can be bounded by two linear functions HL  l (x) and  HU  l (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Suppose the base classifier predicts label y for x when  there is no adversarial perturbation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', F(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' CROWN  finds the largest r such that the lower bound of the yth entry  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', min∥δ∥2<r HL  y (x + δ)) is larger than the upper bounds  of all other entries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', maxl̸=y max∥δ∥2<r HU  l (x + δ)) and  views it as the certified radius R for the input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The complete  details of CROWN can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In the context  of encoder as a service, the composition of the encoder and  downstream classifier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', g ◦ f) is a base classifier F, whose  certified radius a client aims to derive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, in SEaaS, a  client does not have white-box access to the encoder g since it  is deployed on the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' As a result, a client cannot use  BC based certification to derive the certified radius of g ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Smoothed classifier (SC) based certification:  SC based  certification first builds a smoothed classifier based on the base  classifier and then derives the certified radius R of the smoothed  classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In SEaaS, a client builds a smoothed classifier h based  on the base classifier g ◦ f via adding random Gaussian noise  N(0, σ2I) to an input x, where σ is the standard deviation of  the Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, given a testing input x, the  client constructs N noisy inputs x + n1, x + n2, · · · , x + nN,  where ni (i = 1, 2, · · · , N) is sampled from N(0, σ2I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  client uses the base classifier g ◦ f to predict the label of  each noisy input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, the client computes the label  frequency Nl of each label l among the noisy inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=',  Nl = ∑︁N  j=1 I(g ◦ f(x + nj) = l), where I is an indicator  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The smoothed classifier predicts the label with the  3  largest label frequency for the original testing input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover,  the client can derive the certified radius R of the smoothed  classifier for x based on the label frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Due to the  random sampling, the derived certified radius may be incorrect  with an error probability α, which can be set by the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In Appendix C, we take Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' [13] as an example to  discuss more technical details on SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  To improve certified radius, the base classifier g ◦f is often  trained using noisy training inputs [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In encoder as a service  (both SEaaS and REaaS), a client does not have white-box  access to the encoder g and thus can only train its downstream  classifier f using noisy training inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, for SEaaS,  in each epoch of training the downstream classifier, a client  adds random Gaussian noise from N(0, σ2I) to each training  input, queries the Feature-API to obtain the feature vector of  each noisy training input, and uses the feature vectors to update  the downstream classifier via stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  SC based certification faces two challenges in SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' First, a  client needs to query the cloud service many times, leading to a  large communication cost for the client and a large computation  cost for the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, for each testing input x,  a client needs to query the Feature-API N times to obtain the  feature vectors of the N noisy inputs in order to compute the  label frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, the client queries the Feature-API  e times per training input, where e is the number of epochs used  to train the downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We note that [17] proposed  to prepend a denoiser to a base classifier instead of training  it with noisy training inputs, which can reduce the number of  queries from e to 1 per training input when applied to SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  However, it is hard for a client with a small amount of data  to train such a denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Second, the derived certified radius  is suboptimal because a client cannot fine-tune the encoder,  which is not pre-trained to support certified robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Pre-training an Encoder  1) Pre-training Non-robust Encoders: We discuss both  standard supervised learning and self-supervised learning  methods to pre-train encoders, which do not take robustness  against adversarial examples into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Supervised learning: The idea of using supervised learning  to pre-train an encoder is to first train a deep neural network  classifier using labeled training data and then use the layers ex-  cluding the output layer as an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, supervised  learning defines a loss function l(i) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', cross-entropy loss) for  each labeled training example (xi, yi), where yi is the ground  truth label of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then, supervised learning iteratively trains a  deep neural network classifier by minimizing the sum of the  losses over the labeled training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Such paradigm of  training a deep neural network classifier via supervised learning  and using the layers excluding the output layer as an encoder  is also known as transfer learning [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Self-supervised learning:  Unlike supervised learning, self-  supervised learning [1], [2] aims to pre-train an encoder using  unlabeled data, which has attracted growing attention in the  past several years in the AI community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A basic component  of self-supervised learning is data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically,  given an image, the data augmentation component applies a  series of (random) data augmentation operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', random  cropping, color jitter, and flipping) sequentially to produce  an augmented image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Roughly speaking, the main idea of  self-supervised learning is to pre-train an encoder such that  it produces similar feature vectors for two augmented images  produced from the same image, but dissimilar feature vectors for  two augmented images produced from different images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Next,  we take MoCo [1], a state-of-the-art self-supervised learning  algorithm, as an example to elaborate more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  MoCo uses an auxiliary encoder (called momentum encoder)  that has the same architecture as the encoder and a queue  (denoted as Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, the queue is used to cache the  output of the momentum encoder for the augmented images  and is dynamically updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For simplicity, we respectively  use f and fe to denote the encoder and the momentum  encoder, and use θ and θe to denote their encoder parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Suppose we have a mini-batch of unlabeled images which are  denoted as xi, i = 1, 2, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We apply the data augmentation  component to each image in the mini-batch twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We use x1  i  and x2  i to denote the two augmented images produced for the  image xi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given x1  i , x2  i , and the queue Γ, MoCo  defines a loss function for xi as follows:  ℓ(i) = − log(  exp(Sim(f(x1  i ), fe(x2  i ))/τ)  ∑︁  z∈Γ∪{fe(x2  i )} exp(Sim(f(x1  i ), z)/τ)),  (2)  where τ is a temperature parameter and Sim(·, ·) measures the  similarity of two feature vectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', cosine similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' MoCo  uses gradient descent to minimize  1  m · ∑︁m  i=1 ℓ(i) to update the  parameters θ in the encoder f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The queue Γ is dynamically  updated in each step, where {fe(x2  1), fe(x2  2), · · · , fe(x2  m)} are  enqueued and the m “oldest” vectors are dequeued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  2) Pre-training Empirically Robust Encoders: Adversarial  training [15], [16] is a standard method to train empirically  robust classifiers in supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The key idea is to  generate adversarial examples based on training examples  during training, and use the adversarial examples to augment  the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The layers excluding the output layer of the  classifier are then used as an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Several studies [19],  [14], [20] generalized adversarial training to pre-train a robust  encoder in self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Roughly speaking, the idea  is to first generate adversarial examples that incur large loss  and then use them to pre-train an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For instance, Kim  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' [14] proposed Robust Contrastive Learning (RoCL) to  pre-train robust encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, given a training image,  RoCL uses Projected Gradient Descent (PGD) [16] to generate  an adversarial perturbation for an augmented version of the  training image such that the adversarially perturbed augmented  version incurs a large loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then, RoCL pre-trains an encoder  such that it outputs similar feature vectors for the adversarially  perturbed augmented version and other augmented versions of  the training image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The goal of these studies is to pre-train empirically rather  than certifiably robust encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' As a result, the pre-trained  encoders achieve suboptimal certified radii for a client as shown  by our experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In this work, we propose a new  method, which can be combined with either supervised learning  or self-supervised learning to pre-train a robust encoder that  can achieve larger certified radii against adversarial examples  for a client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  4  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  PROBLEM FORMULATION  Threat model: We consider an attacker can use adversarial  examples to induce misclassification for a client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically,  given a testing input, an attacker can add a carefully crafted  perturbation to it such that the client’s downstream classifier  predicts a different label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' To defend against adversarial ex-  amples, the client aims to build a certifiably robust classifier,  which provably predicts the same label for a testing input no  matter what perturbation is added to it once the ℓ2-norm of the  perturbation is less than a threshold (called certified radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Note that we do not constrain on what method an attacker can  use to find the perturbation since we aim to defend against all  bounded perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Problem definition: We aim to design an encoder as a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In particular, when designing an encoder as a service, we  essentially aim to answer two key questions: 1) what APIs  should the cloud service provide for a client?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' and 2) how to  pre-train the encoder?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Design goals:  We aim to design an encoder as a service  to achieve three goals, generality, efficiency, and robustness,  which we elaborate in the following: Generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' As we discussed in Section II-B, BC and  SC based certification methods are complementary  and may be adopted by different clients due to their  different needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We say an encoder as a service achieves  the generality goal if a client can use any certification  method to build a certifiably robust classifier and derive  its certified radius for any given testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We note  that SEaaS can only support SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We use the number of queries sent to  the cloud service to measure the communication cost  between a client and the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, we  use computation time to measure the computation cost  for a client and the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We aim to design an  encoder as a service to achieve a small communication  cost and computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A certified radius measures the certified  robustness of a classifier for a testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We use the  average certified radius of testing inputs to measure the  certified robustness of a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We aim to design  an encoder as a service that enables a client to build  a downstream classifier with a large average certified  radius in both BC and SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We note that SEaaS does not achieve any of the three  goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, SEaaS cannot support BC based certification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  SEaaS incurs a large communication cost between a client and  the cloud server as well as a large computation cost for the  cloud server in SC based certification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' and SEaaS achieves  suboptimal average certified radius for SC based certification  because certified robustness is not taken into consideration  when pre-training the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  OUR REAAS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Overview  To achieve the generality and efficiency goals, our key idea  is to enable a client to treat its own downstream classifier as a  base classifier and certify the robustness of its base downstream  classifier or smoothed downstream classifier in the feature  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Towards this goal, other than the Feature-API provided  in SEaaS, our REaaS provides another API (called F2IPerturb-  API).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, since a downstream classifier takes a feature  vector as input, we propose a client first derives a feature-  space certified radius RF of its base downstream classifier  or smoothed downstream classifier for a testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then,  the client transforms the feature-space certified radius RF to  the input-space certified radius R by querying the F2IPerturb-  API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' To achieve the robustness goal, we further propose a new  method to pre-train a robust encoder, which uses a spectral-  norm term to regularize the pre-training of an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our  pre-trained encoder aims to produce similar feature vectors for  an input and its adversarially perturbed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Feature-API and F2IPerturb-API  1) Feature-API: We first introduce the input and output of  Feature-API, and then its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Input and output for a client:  In Feature-API, the input  from a client is an image x and the output returned to the  client is the input’s feature vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Formally, Feature-API is  represented as v = Feature-API(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Implementation on the server: Given an input x, the cloud  server uses a pre-trained encoder f to compute its feature vector  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, we have v = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We note that SEaaS only has this Feature-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  2) F2IPerturb-API: Like Feature-API, we first introduce the  input and output of F2IPerturb-API, and then its implementation  on the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Input and output for a client: F2IPerturb-API transforms a  feature-space certified radius to an input-space certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The input from a client contains an input image x and a  feature-space certified radius RF for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, the client’s  downstream classifier predicts the same label for x once the  ℓ2-norm of the perturbation to x’s feature vector v is bounded  by RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The client can use any BC or SC based method to  derive RF for x by treating its downstream classifier alone as  a base classifier and x’s feature vector v as an “input” to the  base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The output of F2IPerturb-API is an input-space certified  radius R such that when the ℓ2-norm of the adversarial  perturbation added to the input image x is smaller than R,  the ℓ2-norm of the perturbation introduced to the feature vector  v is smaller than RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Formally, given an input image x and a  feature-space certified radius RF , F2IPerturb-API is represented  as follows: R = F2IPerturb-API(x, RF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Implementation on the server:  A larger R enables the  client to derive a larger certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The key challenge of  implementing the F2IPerturb-API is how to derive the largest  R for a given input x and RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' To address the challenge, we  formulate the input-space certified radius R as the solution to  the following optimization problem:  R = max  r  r  (3)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  max  ∥δ∥2<r ∥f(x + δ) − f(x)∥2 < RF ,  (4)  5  Algorithm 1: F2IPerturb-API  Input from client: image x and feature-space certified  radius RF  Output for client: image-space certified radius R  ρL, ρU ← 0, large value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', 10)  while ρU − ρL > β do  ρk = ρL+ρU  2  for i = 1, 2, · · · , d do  f L  i , f U  i ← CROWN(i, x, f)  Li = min∥δ∥2≤ρk f L  i (x + δ) − fi(x)  Ui = max∥δ∥2≤ρk f U  i (x + δ) − fi(x)  end for  R′  F =  √︂∑︁d  i=1 max(L2  i , U 2  i )  if R′  F < RF then  ρL = ρk  else  ρU = ρk  end if  end while  return ρL  where f is an encoder and δ is an adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  However, the optimization problem is challenging to solve  because the constraint is highly non-linear when the encoder is  a complex neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' To address the challenge, we propose  a binary search based method to solve R in the optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, we search in the range [ρL  k , ρU  k ] in the  kth round of binary search, where we set ρL  1 to be 0 and ρU  1  to be a large value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', 10 in our experiments) in the first  round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, we denote ρk = ρL  k +ρU  k  2  for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In the  kth round, we check whether r = ρk satisfies the constraint in  Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' If the constraint is satisfied, then we can search  the range [ρk, ρU  k ] in the (k + 1)th round, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', ρL  k+1 = ρk and  ρU  k+1 = ρU  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Otherwise, we search the range [ρL  k , ρk] in the  (k + 1)th round, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', ρL  k+1 = ρL  k and ρU  k+1 = ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We stop the  binary search when ρU  k − ρL  k ≤ β and treat ρL  k as R, where β  is a parameter characterizing the binary-search precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Our binary search based solution faces a key challenge,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', how to check whether r = ρk satisfies the constraint  in Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our key idea to address the challenge is to  derive an upper bound for the left hand side of the constraint  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', max∥δ∥2<ρk ∥f(x + δ) − f(x)∥2) and decide that the  constraint is satisfied if the upper bound is smaller than RF ,  where the upper bound can be efficiently computed for any ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Suppose the encoder f maps an input x to a d-dimensional  feature vector f(x), where fi(x) represents the ith entry of  f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' An encoder f is essentially a deep neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Therefore, according to CROWN [9], we have the following  lower bound and upper bound for fi(x + δ) when ∥δ∥2 < ρk:  min  ∥δ∥2<ρk f L  i (x + δ) ≤ fi(x + δ) ≤  max  ∥δ∥2<ρk f U  i (x + δ),  (5)  where f L  i  and f U  i  are two linear functions and i  =  1, 2, · · · , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In Appendix D, we show that Equation 5 is tight  when f consists of one linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' As min∥δ∥2≤ρk f L  i (x +  δ) ≤ min∥δ∥2<ρk f L  i (x + δ) and max∥δ∥2<ρk f U  i (x + δ) ≤  max∥δ∥2≤ρk f U  i (x+δ), we have the following when ∥δ∥2 < ρk:  min  ∥δ∥2≤ρk f L  i (x + δ) ≤ fi(x + δ) ≤  max  ∥δ∥2≤ρk f U  i (x + δ),  (6)  Therefore, we have the following inequalities for ∀ ∥δ∥2 < ρk:  fi(x + δ) − fi(x) ≥ Li,  (7)  fi(x + δ) − fi(x) ≤ Ui,  (8)  where Li  = min∥δ∥2≤ρk f L  i (x + δ) − fi(x) and Ui  =  max∥δ∥2≤ρk f U  i (x + δ) − fi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Based on the above two  inequalities, we have the following:  max  ∥δ∥2<ρk(fi(x + δ) − fi(x))2 ≤ max(L2  i , U 2  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (9)  Therefore,  we  can  derive  an  upper  bound  for  max∥δ∥2<ρk ∥f(x + δ) − f(x)∥2 as follows:  max  ∥δ∥2<ρk ∥f(x + δ) − f(x)∥2  (10)  ≤  ⌜  ⃓  ⃓  ⎷  d  ∑︂  i=1  max  ∥δ∥2<ρk(fi(x + δ) − fi(x))2  (11)  ≤  ⌜  ⃓  ⃓  ⎷  d  ∑︂  i=1  max(L2  i , U 2  i )  (12)  ≜R′  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (13)  If the upper bound R′  F is smaller than RF , then we have r = ρk  satisfies the constraint in Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We note that r = ρk  may also satisfy the constraint even if the upper bound R′  F is  no smaller than RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, such cases do not influence the  correctness of our binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that min∥δ∥2≤ρk f L  i (x +  δ) and max∥δ∥2≤ρk f U  i (x + δ) have closed-form solutions for  i = 1, 2, · · · , d [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, Li, Ui, and the upper bound R′  F  can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In other words, we can efficiently  check whether r = ρk satisfies the constraint in Equation (4)  for any ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Algorithm 1 shows our F2IPerturb-API, where the  function CROWN obtains the lower bound and upper bound  linear functions for each fi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Our binary search based solution correctly finds a lower  bound of the optimal R of the optimization problem in Equa-  tion (3) because the constraint in Equation (4) is guaranteed to  be satisfied in each round of our binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Image rescaling: We note that, in our above discussion on  the two APIs, a client’s input image size is the same as  the input size of the cloud server’s encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' When the size  of a client’s input image is different, the cloud server can  rescale it to be the input size of its encoder using the standard  bilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The bilinear interpolation can be viewed  as a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, suppose xb and xa  respectively represent the image before and after rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Then, we have xa = W · xb, where W is the matrix used  to represent the linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The cloud server can  implement this linear transformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', rescaling) by adding  a linear layer whose weight matrix is W before the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Moreover, the cloud server can view the linear layer + the  encoder as a “new” encoder to implement the two APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  6  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Pre-training Robust Encoder  Our REaaS is applicable to any encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, a more  robust encoder enables a client to derive a larger certified radius  for its testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, we further propose a method  to pre-train robust encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' An encoder f is more robust if  it produces more similar feature vectors for an input and its  adversarially perturbed version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', if f(x + δ) and f(x) are  more similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, based on our implementation of  the F2IPerturb-API, if ∥f(x + δ) − f(x)∥2 is smaller for any  adversarial perturbation δ, then F2IPerturb-API would return  a larger input-space certified radius to a client for a given  feature-space certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, our key idea is to  reduce ∥f(x + δ) − f(x)∥2 when pre-training an encoder f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Next, we derive an upper bound of ∥f(x + δ) − f(x)∥2, based  on which we design a regularization term to regularize the  pre-training of an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  A neural network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', an encoder) can often be de-  composed into the composition of a series of linear trans-  formations [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, we can do so if each layer  of the neural network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', linear layer, convolutional layer,  and batch normalization layer) can be expressed as a linear  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We denote an encoder as the composition of n  linear transformations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', f(·) = T n ◦ T n−1 ◦ · · · ◦ T 1(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' [5]  showed that the difference between the outputs of any neural  network f (f is an encoder in our case) for an input and its  adversarially perturbed version can be bounded as follows:  ∥f(x + δ) − f(x)∥2 ≤  n  ∏︂  j=1  ⃦⃦T j⃦⃦  s · ∥δ∥2 ,  (14)  where x is an input, δ is an adversarial perturbation, and ∥·∥s  represents spectral norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The product of the spectral norms of  the n linear transformations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', ∏︁n  j=1  ⃦⃦T j⃦⃦  s) is independent  with input x and adversarial perturbation δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, our idea  is to add ∏︁n  j=1  ⃦⃦T j⃦⃦  s as a regularization term (called spectral-  norm regularization) when pre-training an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Minimizing  such regularization term may enforce the encoder to produce  more similar feature vectors for an input and its adversarially  perturbed version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', ∥f(x + δ) − f(x)∥2 may be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In  particular, we minimize the following loss function for each  mini-batch of inputs when pre-training an encoder:  1  m ·  m  ∑︂  i=1  ℓ(i) + λ ·  n  ∏︂  j=1  ⃦⃦T j⃦⃦  s ,  (15)  where ℓ(i) is a loss for a training input in pre-training, m  is batch size, and λ is a hyper-parameter used to balance the  two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For instance, when using supervised learning to  train a classifier, whose layers excluding the output layer are  used as an encoder, the loss ℓ(i) is often the cross-entropy loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  when using self-supervised learning algorithm MoCo [1] to  pre-train an encoder, ℓ(i) is defined in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We adopt  the power method [21] to estimate the spectral norms of the  linear transformations when pre-training an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Certifying Robustness for a Client  In REaaS, a client can treat its own downstream classifier  as a base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We discuss how a client can use our  two APIs to train a base downstream classifier and derive  the certified radius of the base downstream classifier in BC  TABLE I: Comparing the communication and computation  cost per training/testing input in SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' e is  the number of epochs used to train a base downstream  classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' N is the number of noisy inputs per testing input  in SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Tf (or Tg) and Mf (or Mg) are respectively the  number of layers and the maximum number of neurons  in a layer in an encoder (or a downstream classifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Kf  (or Kg) is the number of parameters in an encoder (or a  downstream classifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='(a) Communication cost  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Service  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='#Queries  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Per training input Per testing input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='BC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='SC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='BC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='SC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='SEaaS N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='REaaS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='(b) Computation cost  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Service Entity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Computational complexity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Per training input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Per testing input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='BC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='SC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='BC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='SC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='SEaaS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Cloud  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='server  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='O(e · Kf )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='O(N · Kf )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Client  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='O(e · Kg)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='O(N · Kg)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='REaaS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Cloud  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='server  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='O(Kf )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='O(Kf )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='O(Kf + T 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='f · M 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='f ) O(Kf + T 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='f · M 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='f )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='Client O(e · Kg) O(e · Kg) O(Kg + T 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g · M 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='O(N · Kg)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='based certification or the smoothed downstream classifier in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='SC based certification for a testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  BC based certification: When training a base downstream  classifier, a client queries the Feature-API to obtain a feature  vector for each training input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then, given the feature vectors  and the corresponding training labels, the client can use any  training method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', standard supervised learning) to train a  base downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given a testing input, the client  queries the Feature-API to obtain its feature vector and uses the  base downstream classifier to predict its label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, the  client can use any BC based certification method to derive a  feature-space certified radius for the testing input by treating its  feature vector as an “input” to the base downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Then, the client queries the F2IPerturb-API to transform the  feature-space certified radius to an input-space certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  SC based certification: Similar to BC based certification, a  client queries the Feature-API to obtain a feature vector for  each training input when training a base downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  However, unlike BC based certification, the client adds noise  to the training feature vectors in SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In  particular, the client adds random noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', Gaussian noise) to  each feature vector in each mini-batch of training feature vectors  in each training epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that the client does not need to  query the Feature-API again for the noisy feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given  a testing input, the client queries the Feature-API to obtain its  feature vector and uses the smoothed downstream classifier to  predict its label and derive its feature-space certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In particular, the client constructs N noisy feature vectors  by adding random noise to the feature vector and uses it’s  base downstream classifier to predict their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Based on  the predicted labels, the client can derive the predicted label  and feature-space certified radius for the original feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Then, the client queries the F2IPerturb-API to transform the  feature-space certified radius to an input-space certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  7  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Theoretical Communication and Computation Cost Analysis  Communication cost:  The number of queries to the APIs  characterizes the communication cost for a client and the cloud  server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In both BC and SC based certification, a client only  queries the Feature-API once for each training input in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Therefore, the number of queries per training input is 1 in  REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In both BC and SC based certification, a client only  queries the Feature-API and F2IPerturb-API once to derive  the predicted label and certified radius for a testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Therefore, the number of queries per testing input is 2 in  REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that the client only sends an image x to the cloud  server when querying the Feature-API, while it also sends  the feature-space certified radius Rf to the cloud server when  querying the F2IPerturb-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, Rf is a real number  whose communication cost is negligible, compared to that of  the image x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Thus, we consider querying Feature-API and  querying F2IPerturb-API have the same communication cost in  our analysis for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Table Ia compares the number  of queries per training/testing input in SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Compared with SEaaS, REaaS makes BC based certification  applicable and incurs a much smaller communication cost in  SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Computation cost:  Table Ib compares the computational  complexity of REaaS and SEaaS for the cloud server and  a client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In both REaaS and SEaaS, the computation cost for  the cloud server to process a query to the Feature-API is linear  to the number of encoder parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', O(Kf), where Kf  is the number of parameters in the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In REaaS, we use  binary search to process a query to the F2IPerturb-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given  the initial search range [ρL  1 , ρU  1 ] and binary-search precision  β, the number of rounds of binary search is ⌈log2( ρU  1 −ρL  1  β  )⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In practice, we can set ρL  1 , ρU  1 , and β to be constants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=',  ρL  1 = 0, ρU  1 = 10, and β = 10−50, and thus ⌈log2( ρU  1 −ρL  1  β  )⌉  can be viewed as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' From [9], the computational  complexity is O(T 2  f · M 3  f ) in each round of binary search,  where Tf and Mf are respectively the number of layers and  the maximum number of neurons in a layer in an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Thus,  the computational complexity for the cloud server to process a  query to the F2IPerturb-API is O(T 2  f · M 3  f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  On the client side, the computational complexity of gradient  descent is O(Kg) for each training input per epoch when  training a base downstream classifier in both BC and SC  based certification, where Kg is the number of parameters  in the base downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, the computational  complexity of training a base downstream classifier is O(e·Kg)  per training input, where e is the number of training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  computational complexity for a client to derive the feature-space  certified radius of a testing input is O(T 2  g · M 3  g ) in BC based  certification [9], where Tg and Mg are respectively the number  of layers and the maximum number of neurons in a layer in  the base downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, the computational  complexity of using the base downstream classifier to predict  a label for a (noisy) feature vector is O(Kg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  As SEaaS does not support BC based certification, we focus  on comparing the computation cost of SEaaS and REaaS for  SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' First, we observe that the computation  cost per training/testing input is the same for a client in SEaaS  and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Second, REaaS incurs a smaller computation cost  per training input for the cloud server than SEaaS, because  REaaS incurs much fewer queries than SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Third, REaaS  often incurs a smaller computation cost per testing input for the  cloud server than SEaaS, because N is often large to achieve  a large certified radius as shown in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  EVALUATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Experimental Setup  Datasets: We use CIFAR10 [22], SVHN [23], STL10 [24],  and Tiny-ImageNet [25] in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' CIFAR10 has  50,000 training and 10,000 testing images from ten classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  SVHN contains 73,257 training and 26,032 testing images from  ten classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' STL10 contains 5,000 training and 8,000 testing  images from ten classes, as well as 100,000 unlabeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Tiny-ImageNet contains 100,000 training and 10,000 testing  images from 200 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We rescale each image in all datasets to 32 × 32 by the  standard bi-linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, the input image size  in a downstream dataset is the same as the input size of the pre-  trained encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, we will also explicitly explore the  scenarios in which the input image size of a downstream dataset  is different from the input size of the pre-trained encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Pre-training encoders: We use STL-10 and Tiny-ImageNet  as pre-training datasets to pre-train encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We adopt these  two datasets because they contain more images than CIFAR10  and SVHN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, we use the unlabeled data of STL10  to pre-train an encoder when STL10 is used as a pre-training  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' When Tiny-ImageNet is used as a pre-training dataset,  we use its training dataset to pre-train an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Unless  otherwise mentioned, we adopt MoCo [1] as the pre-training  algorithm in SEaaS, while we adopt MoCo with our spectral-  norm regularization as the pre-training algorithm in REaaS,  since they only need unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, we adopt the  public implementation of MoCo [26] in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' When  calculating the spectral norm of the encoder during pre-training,  we run 10 iterations of power iteration in each mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  architecture of the encoder can be found in Table XIII in  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We pre-train an encoder for 500 epochs with a  learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='06 and a batch size 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Training downstream classifiers: As we have four datasets,  we use the other three datasets as downstream datasets when  a dataset is used as a pre-training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, when a  dataset is used as a downstream dataset, we adopt its training  dataset as the downstream training dataset and testing dataset as  the downstream testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We use the downstream training  dataset to train a base downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, in  BC based certification, we use standard supervised learning to  train a base downstream classifier on the feature vectors of the  training inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We note that some works [27], [28] proposed  new methods to train a base classifier to improve its certified  robustness in BC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' These methods are also  applicable in our REaaS, but we do not evaluate them since our  focus is to show the applicability of BC based certification in  REaaS instead of its optimal certified robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For SC based  certification, we train a base downstream classifier via adding  Gaussian noise N(0, σ2I) to the training inputs in SEaaS and  the feature vectors of the training inputs in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We use a fully connected neural network with two hidden  layers as a base downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We respectively adopt  8  ReLU and Softmax as the activation functions in the two hidden  layers and the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The number of neurons in both  hidden layers is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We train a base downstream classifier  for 25 epochs using cross-entropy loss, a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='06,  and a batch size of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Certification methods: For BC based certification, we adopt  CROWN [9] to derive the certified radius of a base downstream  classifier for a testing input in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We adopt the public  implementation of CROWN [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For SC based certification,  we adopt Gaussian noise based randomized smoothing [13] to  build a smoothed classifier and derive its certified radius for  a testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In SEaaS, a client treats the composition of  the encoder and its downstream classifier as a base classifier,  while a client treats its downstream classifier alone as a base  classifier in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We use the public code [30] for Gaussian  noise based randomized smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Appendix B and C show  the technical details of CROWN and Gaussian noise based  randomized smoothing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Evaluation metrics: Recall that REaaS aims to achieve three  design goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We can evaluate the generality goal by showing  that REaaS supports both BC and SC based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For  the efficiency goal, we use #Queries per training (or testing)  input to measure the communication cost between a client and  the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, we use running time per testing  input on the cloud server to measure its computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We  do not consider running time on a client as it is the same in  SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that #Queries per training input also  characterizes the computation cost per training input for the  cloud server as it is linear to the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For the  robustness goal, we use average certified radius (ACR) of the  correctly classified testing examples to measure the certified  robustness of a base or smoothed classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Note that there often exists a trade-off between robustness  and accuracy for a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, we further consider  accuracy under adversarial perturbation as an evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In particular, we consider the widely adopted certified accuracy  @ a perturbation size, which is the fraction of testing inputs  in a downstream testing dataset whose labels are correctly  predicted and whose certified radii are no smaller than the given  perturbation size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Certified accuracy @ a perturbation size is  the least testing accuracy that a classifier can achieve no matter  what adversarial perturbation is added to each testing input once  its ℓ2-norm is at most the given perturbation size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The certified  accuracy @ a perturbation size decreases as the perturbation  size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' ACR is the area under the certified accuracy  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' perturbation size curve (details are shown in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Therefore, ACR can also be viewed as a metric to measure  the robustness-accuracy trade-off of a classifier, where a larger  ACR indicates a better trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Parameter settings: F2IPerturb-API has the following three  parameters: ρL  1 and ρU  1 which specify the range of R in the  first round of binary search, and β which characterizes the  binary-search precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We set ρL  1 to be 0 and set ρU  1 to be  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that they do not impact experimental results once  ρL  1 is set to 0 and ρU  1 is set to a large value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We  set the default value of β as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We note that β has a  negligible impact on certified accuracy and ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular,  the absolute difference between the certified accuracy (or ACR)  TABLE II: ACR and #Queries in SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (a) Pre-training dataset is Tiny-ImageNet  Service Certification  method  Downstre-  am dataset  ACR  #Queries  Per train-  ing input  Per test-  ing input  SEaaS  BC  CIFAR10  N/A  SVHN  STL10  SC  CIFAR10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='157  25  1 × 105  SVHN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='226  STL10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='134  REaaS  BC  CIFAR10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='138  1  2  SVHN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='258  STL10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='090  SC  CIFAR10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='171  SVHN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='275  STL10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='143  (b) Pre-training dataset is STL10  Service Certification  method  Downstre-  am dataset  ACR  #Queries  Per train-  ing input  Per test-  ing input  SEaaS  BC  CIFAR10  NA  SVHN  Tiny-  ImageNet  SC  CIFAR10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='155  25  1 × 105  SVHN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='244  Tiny-  ImageNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='016  REaaS  BC  CIFAR10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='139  1  2  SVHN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='272  Tiny-  ImageNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='027  SC  CIFAR10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='173  SVHN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='278  Tiny-  ImageNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='033  when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='001 and that when β is an arbitrarily small value  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', 10−50) is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Randomized smoothing has the following three parameters:  the number of Gaussian noise N, standard deviation σ of  the Gaussian noise, and error probability α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Following prior  work [13], unless otherwise mentioned, we set N = 100, 000,  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='5, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We set the default value of the  hyperparameter λ in our pre-training method as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='00075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We  normalize pixel values to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Experimental Results  We first show that REaaS achieves our three design goals,  but SEaaS does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then, we show the impact of relevant  factors on REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, we consider 1) different ways  to pre-train an encoder, 2) image scaling, and 3) different  hyperparameters of certification methods such as N, σ, and α  for randomized smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that we fix all other parameters  to their default values when studying the impact of one  parameter on REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  9  TABLE III: Comparing the running time per testing input  for the cloud server in SC for SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  pre-training dataset is Tiny-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Service Downstream  dataset  Running time (s) per testing input  SEaaS  CIFAR10  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='77  SVHN  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='65  STL10  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='48  REaaS  CIFAR10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='05  SVHN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='06  STL10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='04  TABLE IV: Training without noise vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' training with  noise for SC in SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-training dataset is Tiny-  ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Downstream  dataset  ACR  Training with noise Training without noise  CIFAR10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='157  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='106  SVHN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='226  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='155  STL10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='134  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='088  TABLE V: Impact of N on ACR for SC in SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  pre-training dataset is Tiny-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Downstream  dataset  N  100  1,000 10,000 100,000  CIFAR10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='132  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='148  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='157  SVHN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='186  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='211  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='226  STL10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='111  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='127  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='134  REaaS achieves the generality, efficiency, and robustness  goals:  In SC based certification, a client respectively adds  Gaussian noise to images and their feature vectors to train a base  downstream classifier in SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Thus, the certified  robustness of the smoothed classifiers are not comparable even  if we use the same standard deviation σ of Gaussian noise in  SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, we try multiple values of σ and  report the largest ACR for each service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, we select σ  values such that the largest ACR is not reached at the smallest  or largest value of σ, to ensure the largest ACR is found for  each service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, we try σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='75, 1  for both SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We note that σ controls a tradeoff  between certified accuracy without attacks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', perturbation  size is 0) and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, a smaller σ can achieve  a larger certified accuracy without attacks but also make the  curve drop more quickly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', less robust).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' ACR measures such  trade-off, and thus we adopt the σ that achieves the largest  ACR for each method when comparing the certified accuracy  of SC based certification in SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Table II compares ACR and #Queries per training/testing  input in SEaaS and REaaS, while Table III compares the running  time per testing input for the server in SC for SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We have the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' First, REaaS supports both  BC and SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, REaaS achieves the generality goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In contrast, SEaaS only supports SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Second, REaaS achieves  the efficiency goal as it is much more efficient than SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  TABLE VI: Comparing the ACRs in REaaS for different  downstream datasets when the encoders are pre-trained  by different self-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-  training dataset is Tiny-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (a) CIFAR10  Certification Method Pre-training Method ACR  BC  Non-robust MoCo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='012  RoCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='016  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='138  SC  Non-robust MoCo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='020  RoCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='024  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='171  (b) SVHN  Certification Method Pre-training Method ACR  BC  Non-robust MoCo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='009  RoCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='015  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='258  SC  Non-robust MoCo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='011  RoCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='014  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='275  (c) STL10  Certification Method Pre-training Method ACR  BC  Non-robust MoCo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='011  RoCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='014  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='090  SC  Non-robust MoCo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='015  RoCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='020  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='143  Specifically, #Queries per training/testing input in REaaS is  orders of magnitude smaller than that in SEaaS for SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We  note that a client using SEaaS could choose to train a base  downstream classifier without adding noise to its training inputs  to reduce the #Queries per training input to 1 or use a small N  to reduce the #Queries per testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, the smoothed  classifier achieves (much) smaller ACRs in such cases as shown  in Table IV and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Base on Table III, REaaS also incurs a  much lower computation cost for the server than SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Third, REaaS achieves the robustness goal as it achieves  larger ACRs than SEaaS for SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The reason is that, in SEaaS,  the base classifier is the composition of an encoder and a  base downstream classifier, but the client can only train the  base downstream classifier with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In contrast, a client can  build a smoothed classifier upon a base downstream classifier  alone which can be trained with noise and the encoder is pre-  trained in a robust way in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Figure 7 in Appendix further  compares the certified accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' perturbation size of SC in  SEaaS and REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We find that REaaS can achieve a better  trade-off between accuracy without attacks and robustness than  SEaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Specifically, REaaS achieves larger certified accuracy  than SEaaS when the perturbation size is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, the  gap between the certified accuracy of SEaaS and REaaS is  much larger when the perturbation size is small than that when  the perturbation size is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Impact of methods to pre-train encoders:  We can use  different methods to pre-train an encoder in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Table VI  and VII show ACRs in REaaS when different self-supervised  10  TABLE VII: Comparing the ACRs in REaaS for different  downstream datasets when the encoders are pre-trained  by different self-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-  training dataset is STL10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (a) CIFAR10  Certification Method Pre-training Method ACR  BC  Non-robust MoCo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='010  RoCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='012  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='139  SC  Non-robust MoCo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='014  RoCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='017  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='7  Certified accuracy  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='00025  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='001  (a) BC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='001  (b) SC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' 2: Impact of λ on certified accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' perturbation  size for BC and SC in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-training dataset is  Tiny-ImageNet and the downstream dataset is CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  learning methods are used to pre-train encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular,  we consider non-robust MoCo [1], RoCL [14], and our robust  pre-training method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', MoCo with our spectral-norm regu-  larization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Table VIII shows ACRs of REaaS when different  supervised learning methods are used to pre-train encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  In particular, we consider a standard, non-robust supervised  learning method, adversarial training [16] (we use the default  parameter settings in the authors’ public implementation),  and our robust pre-training method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', standard supervised  learning with our spectral-norm regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We only show  results when the pre-training dataset is Tiny-ImageNet for  supervised pre-training methods, as STL10 dataset only has a  small number of labeled training images which are insufficient  to pre-train high-quality encoders using supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We try σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='75, 1 and report the largest  ACR for each pre-training method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' As the results show, our  TABLE VIII: Comparing the ACRs in REaaS for different  downstream datasets when the encoders are pre-trained  by different supervised learning (SL) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-  training dataset is Tiny-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (a) CIFAR10  Certification Method Pre-training Method  ACR  BC  Non-robust SL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='019  Adversarial Training 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='035  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='174  SC  Non-robust SL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='022  Adversarial Training 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='008  Adversarial Training 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='011  Adversarial Training 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content=' 3: Impact of λ on ACR for BC and SC in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  pre-training dataset is Tiny-ImageNet and the downstream  dataset is CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  robust pre-training method achieves substantially larger ACRs  than existing methods for both supervised learning and self-  supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our method is better than RoCL and  adversarial training because they aim to train empirically robust  rather than certifiably robust encoders, and is better than MoCo  and standard supervised learning because the encoders pre-  trained by them are non-robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Impact of hyperparameter λ: Figure 2 shows the impact of λ  on certified accuracy in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We find that λ measures a trade-  off between accuracy without attacks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content=', perturbation size is  0) and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, when λ is smaller, the accuracy  without attacks is larger, but the certified accuracy decreases  11  100  1,000  10,000  100,000  N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='18  ACR  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' 4: Impact of N, σ, and α on ACR of SC in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-training dataset is Tiny-ImageNet and the downstream  dataset is CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='0  Perturbation size  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='6  Certified accuracy  16x16  32x32  64x64  (a) BC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='6  Certified accuracy  16x16  32x32  64x64  (b) SC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' 5: Impact of downstream input size on certified accu-  racy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' perturbation size for BC and SC in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  pre-training dataset is Tiny-ImageNet and the downstream  dataset is (or created from) CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The input size of  the pre-trained encoder is 32x32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  more quickly as the perturbation size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Figure 3 shows  the impact of λ on ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our results show that, for both BC  and SC, ACR first increases as λ increases and then decreases  after λ is larger than a certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The reason is that a  larger or smaller λ leads to a worse trade-off between accuracy  without attacks and robustness as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Impact of image rescaling: To study the impact of image  rescaling, we create downstream datasets with different input  image sizes via resizing images in CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Table IX shows  the results on ACR and Figure 5 shows the results on certified  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We find that, when the size of the images in a  downstream dataset is larger (or smaller) than the input size  of the encoder, the downstream input-space ACR is larger (or  smaller) for both BC and SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The reason is that down-scaling  (or up-scaling) the downstream input images to be the same  size as the input size of the encoder reduces (or enlarges) the  perturbation in the downstream image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Impact of N, σ, and α for SC: Figure 4 and 8 (in Appendix)  shows the impact of N, σ, and α on ACR and certified accuracy  of SC in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We have the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' First,  both ACR and certified accuracy increase as N or α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The reason is that the estimated certified radii are larger when  N or α is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Second, we find that σ achieves a trade-  off between accuracy without attacks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', perturbation size  is 0) and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, a smaller σ can achieve a  larger accuracy without attacks, but the curve drops faster as  the perturbation size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Third, ACR first increases and  TABLE IX: Impact of image rescaling on ACR in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The pre-training dataset is Tiny-ImageNet and the down-  stream dataset is (or created from) CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The input  size of the encoder is 32x32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Certification Method  Size of Images in  Downstream Dataset ACR  BC  16x16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='082  32x32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='138  64x64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='303  SC  16x16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='068  32x32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='153  64x64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='305  then decreases as σ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The reason is that a smoothed  classifier is less accurate without attacks when σ is larger and  is less robust when σ is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  REaaS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' white-box access to the encoder: In REaaS, a  client has black-box access to the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We compare REaaS  with the scenario where a client has white-box access to the  encoder, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', the cloud server shares its encoder with a client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Specifically, with white-box access to the encoder, a client  can use either BC or SC by treating the composition of the  encoder and its downstream classifier as a base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For  BC, the client can use CROWN [29] to derive the certified  radius of its base classifier for a testing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For SC, the  client can train/fine-tune the base classifier (both the encoder  and downstream classifier) using training inputs with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The white-box scenario represents the upper-bound robustness  a client can achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, comparing with the robustness  in the white-box scenario enables us to understand how close  our REaaS with the two APIs is to such upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Table X  compares the ACRs of REaaS and such white-box scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We find that REaaS can achieve comparable ACRs with the  white-box scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  DISCUSSION  Extension to ℓp-norm adversarial perturbations: We focus  on certified robustness against ℓ2-norm adversarial perturbation  in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The certified robustness can be extended to other  ℓp-norms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', via leveraging the relationship between ℓ2-norm  and other ℓp-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For instance, suppose the certified radius  is R for an image in ℓ2-norm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' the certified radius in ℓ1-norm  12  TABLE X: Comparing the ACRs of REaaS and the white-  box scenario for different downstream datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-  training dataset is Tiny-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (a) CIFAR10  Certification Method  Service  ACR  BC  White-box 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='157  REaaS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='138  SC  White-box 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='188  REaaS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='171  (b) SVHN  Certification Method  Service  ACR  BC  White-box 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='286  REaaS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='258  SC  White-box 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='302  REaaS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='275  (c) STL10  Certification Method  Service  ACR  BC  White-box 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='102  REaaS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='090  SC  White-box 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='151  REaaS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='143  and ℓ∞-norm can respectively be computed as R and  R  √  dim,  where dim is the product of the number of pixels and the  number of channels in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Figure 6 shows the certified  accuracy of SC in REaaS for ℓ1-norm and ℓ∞-norm adversarial  perturbations, where the ℓ1-norm and ℓ∞-norm certified radii  are obtained from ℓ2-norm certified radius with N = 100, 000,  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='5, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Extending REaaS to natural language processing (NLP)  domain: An attacker can make a text classifier predict an  incorrect label for a text by substituting a small number of  words as their synonyms [31], [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our REaaS can  also be applied to enable adversarially robust downstream  text classifiers against those attacks by slightly adapting our  F2IPerturb-API (please refer to Appendix E for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given a  text and a feature-space certified radius, our adapted F2IPerturb-  API returns an input-space certified radius, which is the  maximum number of words that can be substituted such that  the downstream classifier’s predicted label for the text is  unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Table XI shows our experimental results (please  refer to Appendix E for details of the experimental setup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our  results show that our REaaS is also applicable to NLP domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Encoder stealing: Our REaaS introduces a new F2IPerturb-  API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A natural question is whether the new F2IPerturb-API  makes the encoder more vulnerable to stealing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We  argue that the answer is probably no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The reason is that our  new API returns a certified radius for a query image, which  can also be obtained by an attacker via calling the existing  Feature-API many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However, an attacker may obtain  such certified radius with less queries using our new API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We explore whether certified radii can be exploited to assist  encoder stealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, we extend StolenEncoder [34],  TABLE XI: ACR and #Queries of REaaS in NLP domain,  where BC is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-training dataset is SST-2 [35]  and the downstream dataset is IMDB [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  ACR  #Queries  Per training input Per testing input  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='517  1  2  TABLE XII: Comparing StolenEncoder and its extended  version  using  our  F2IPerturb-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The  pre-training  dataset is Tiny-ImageNet and the downstream dataset is  CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  StolenEncoder  Extended StolenEncoder  γ  0  10−4 10−3 10−2 10−1  Stolen Accuracy (%)  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='3  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='6  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='4  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='5  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='0  Perturbation size  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='7  Certified accuracy  (a) ℓ1-norm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='025  Perturbation size  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='7  Certified accuracy  (b) ℓ∞-norm  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' 6: Certified accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' perturbation size of SC in  REaaS under ℓ1-norm and ℓ∞-norm adversarial perturba-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-training dataset is Tiny-ImageNet and the  downstream dataset is CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  which uses Feature-API to steal encoder, to steal encoders using  both Feature-API and F2IPerturb-API (see Appendix F for the  details of StolenEncoder and its extended version as well as the  experimental setup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Table XII shows our experimental results,  where γ is a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that the total number of  queries to the APIs made by the extended StolenEncoder is  twice of StolenEncoder in our comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our results show that  the downstream classifiers built upon stolen encoders obtained  by StolenEncoder and its extended version achieve comparable  accuracy, which implies that certified radii may not be able to  assist encoder stealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Privacy-preserving encoder as a service: In both SEaaS and  REaaS, a client sends his/her raw images to the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Therefore, an untrusted service provider may compromise the  privacy of the client, especially when the downstream datasets  contain sensitive images such as facial and medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We believe it is an interesting future work to develop privacy-  preserving encoder as a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For instance, we can leverage  (local) differential privacy [37], [38], [39], secure hardware [40],  and cryptography [41], [42] based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Other attacks to pre-trained encoders: In this work, we  focus on adversarial examples [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Some recent studies [43],  [44], [45] show that pre-trained encoders are also vulnerable  to poisoning and backdoor attacks, which are orthogonal to  our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We believe it is an interesting future work to extend  our framework to defend against those attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  13  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  CONCLUSION AND FUTURE WORK  In this work, we show that, via providing two APIs, a cloud  server 1) makes it possible for a client to certify robustness of  its downstream classifier against adversarial perturbations using  any certification method and 2) makes it orders of magnitude  more communication efficient and more computation efficient to  certify robustness using smoothed classifier based certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Moreover, when the cloud server pre-trains the encoder via  considering our spectral-norm regularization term, it achieves  better certified robustness for the clients’ downstream classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Interesting future work includes extending REaaS to poisoning  and backdoor attacks as well as designing both robust and  privacy-preserving encoder as a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We thank the anonymous reviewers for the constructive  comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' This work was supported by NSF under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  2131859, 2112562, and 1937786, as well as ARO under grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' W911NF2110182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  REFERENCES  [1]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content=' Kailkhura,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Lin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content=' Hsieh, “Automatic perturbation analysis for scalable  certified robustness and beyond,” in NeurIPS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content=' Perelygin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Chuang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Manning, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Ng,  and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Potts, “Recursive deep models for semantic compositionality  over a sentiment treebank,” in EMMLP, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  [49]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Jia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Raghunathan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' G¨oksel, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Liang, “Certified robustness to  adversarial word substitutions,” arXiv preprint arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='00986, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  14  TABLE XIII: Architecture of the neural network for the  encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Layer Type  Layer Parameters  Input 32 × 32  Convolution  16 × 3 × 3, strides=(1, 1), padding=1  Activation  ReLU  Convolution  16 × 4 × 4, strides=(2, 2), padding=1  Activation  ReLU  Convolution  32 × 3 × 3, strides=(1, 1), padding=1  Activation  ReLU  Convolution  32 × 4 × 4, strides=(2, 2), padding=1  Activation  ReLU  Fully Connected  256  Output  APPENDIX A  ACR IS THE AREA UNDER THE CERTIFIED ACCURACY VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  PERTURBATION SIZE CURVE  Suppose the largest certified radius of a testing input is  Rmax and the certified radius of the testing inputs follows  a probability distribution in the interval [0, Rmax], whose  probability density function is p(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The certified accuracy  (CA) at a perturbation size R can be formally defined as  follows:  CA(R) =  ∫︂ Rmax  R  p(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (16)  By definition, the area under the certified accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' pertur-  bation size curve is calculated as follows:  Area =  ∫︂ Rmax  0  CA(R)dR  (17)  = CA(R) · R|Rmax  0  −  ∫︂ Rmax  0  R · dCA(R)  (18)  = 0 +  ∫︂ Rmax  0  R · p(R) · dR  (19)  = ACR,  (20)  where ACR is the average certified radius of the testing inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  APPENDIX B  TECHNICAL DETAILS OF CROWN [9]  Suppose F is a base classifier that maps an input x to  one of c classes {1, 2, · · · , c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For instance, the composition  of the encoder and downstream classifier g ◦ f is the base  classifier F in SEaaS, and a client can treat the downstream  classifier g as the base classifier F in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We use  H(x) to denote the base classifier’s last-layer output vector  for x, where Hl(x) represents the lth entry of H(x) and  l = 1, 2, · · · , c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' F(x) denotes the predicted label for x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=',  F(x) = argmaxl=1,2,··· ,c Hl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Suppose the base classifier  predicts label y for x when there is no adversarial perturbation,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', F(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' A base classifier based certification method  derives a certified radius R such that F(x + δ) = y for all  ∥δ∥2 < R, where δ is adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Next, we discuss  how to derive the certified radius R for CROWN [9], a state-  of-the-art base classifier based certification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The key idea of CROWN is to bound each entry of the  output vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', Hl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, when the output of  the activation function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', ReLU) can be bounded by two  linear functions, CROWN shows that each entry Hl(x) can be  bounded by two linear functions HL  l (x) and HU  l (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Formally,  we have HL  l (x) ≤ Hl(x) ≤ HU  l (x), where l ∈ {1, 2, · · · , c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Suppose an adversarial perturbation δ that satisfies ∥δ∥2 < r  is added to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then, we have the following:  min  ∥δ∥2<r HL  l (x + δ) ≤ Hl(x + δ) ≤ max  ∥δ∥2<r HU  l (x + δ),  (21)  where l = 1, 2, · · · , c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The base classifier F still predicts label  y for x + δ when the lower bound of the yth entry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=',  min∥δ∥2<r HL  y (x + δ)) is larger than the upper bounds of all  other entries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', maxl̸=y max∥δ∥2<r HU  l (x + δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore,  CROWN derives the certified radius R as follows:  R = max  r  r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  min  ∥δ∥2<r HL  y (x + δ) > max  l̸=y  max  ∥δ∥2<r HU  l (x + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Note that CROWN shows that min∥δ∥2<r HL  l (x + δ) and  max∥δ∥2<r HU  l (x + δ) have closed-form solutions for l =  1, 2, · · · , c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  APPENDIX C  TECHNICAL DETAILS OF SC BASED CERTIFICATION [13]  Given a base classifier F (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', g ◦ f in SEaaS and g in  REaaS), randomized smoothing builds a smoothed classifier  h via adding random Gaussian noise to the testing input of  the base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The smoothed classifier h provably predicts  the same label for the testing input when the ℓ2-norm of the  perturbation add to the testing input is less than a threshold  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', certified radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Next, we respectively discuss how to  build a smoothed classifier, how to derive the certified radius  for a testing input, as well as how to train the base classifier  to improve the certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Building a smoothed classifier: For simplicity, we use  N(0, σ2I) to denote a zero-mean isotropic Gaussian noise,  where σ is the standard deviation and I is an identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Given an input x, a base classifier F, and a zero-mean  Gaussian noise N(0, σ2I), we define the label probability  pl, l ∈ {1, 2, · · · , c}, as follows:  pl = Pr(F(x + N(0, σ2I)) = l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (22)  Roughly speaking, pl is the probability that the base classifier  F predicts label l when taking x + N(0, σ2I) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The  smoothed classifier h predicts the label with the largest label  probability for the input x:  h(x) =  argmax  l∈{1,2,··· ,c}  pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (23)  In practice, it is usually computationally intractable to compute  the exact label probabilities due to the complexity of the base  classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In response, we sample N noise n1, n2, · · · , nN  from N(0, σ2I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For each l ∈ {1, 2, · · · , c}, we denote Nl =  ∑︁N  j=1 I(F(x + nj) = l) (called label frequency), where I is  an indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The label (denoted as y) with the largest  label frequency is predicted as the label of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Computing certified radius for an input: Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' [13]  proved that the smoothed classifier predicts the label y for the  input x when the ℓ2-norm of the adversarial perturbation is less  15  than σ  2 · (Φ−1(py) − Φ−1(pu)), where py is a lower bound of  py and pu is an upper bound of maxl̸=y pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In other words, we  have h(x+δ) = y when ∥δ∥2 < R = σ  2 ·(Φ−1(py)−Φ−1(pu)),  where R is the certified radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  To compute the certified radius, we need to estimate a lower  bound of py and an upper bound of maxl̸=y pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  proposed to estimate such lower or upper bounds using the  standard one-sided Clopper-Pearson method [46] based on label  frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, Ny follows a binomial distribution  with parameters N and py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Based on the Clopper-Pearson  method, we have the following lower bound for py:  py = Beta(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Ny, N − Ny + 1),  (24)  where 1−α is the confidence level and Beta(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' ς, ϑ) is the αth  quantile of the Beta distribution with shape parameters ς and ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Intuitively, with probability at least 1 − α over the randomness  of the sampling of Gaussian noise, we have py ≥ py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given  the lower bound py and the fact that the summation of label  probabilities is 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', ∑︁c  l=1 pl = 1), we can derive 1 − py as  an upper bound for maxl̸=y pl, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', maxl̸=y pl ≤ 1−py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given  the lower bound of py and the upper bound of maxl̸=y pl, we  have the certified radius R = σ · Φ−1(py).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In other words,  with probability at least 1 − α over the randomness of the  sampling of Gaussian noise, we have g(x + δ) = y when  ∥δ∥2 < Φ−1(py).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Training a base classifier with Gaussian noise: The certified  radius for an input depends on whether the base classifier F  can correctly classify the input with Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' However,  when the base classifier F is trained using the normal training  examples, it is very likely that the base classifier cannot  correctly predict the label for an input with Gaussian noise due  to the difference between training and testing data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Therefore, Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' [13] proposed to add Gaussian noise  to the training inputs when training the base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In  particular, for each mini-batch of training examples, we add  random Gaussian noise to each training input and then use them  to update the base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Such Gaussian noise based training  method can significantly improve the certified radius [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  APPENDIX D  ON THE TIGHTNESS OF EQUATION 5  We show that the upper and lower bounds in (5) are tight  when the encoder consists of one linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Suppose f(x) =  Wx + b, where W and b are the weight matrix and bias  vector of the linear encoder f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For simplicity, we respectively  use W[i, :] and b[i] to denote the ith row of W and ith  element of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Based on Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='2 in CROWN [9], we have  f U  i (x) = W[i, :]x + b[i] and f L  i (x) = W[i, :]x + b[i] when  f(x) = Wx + b (we omit the details for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Next, we  prove the right-hand side of Equation 5 is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' To reach the  goal, our idea is to show there exists δ′ satisfying ∥δ′∥2 = ρk  such that fi(x + δ′) > max∥δ∥2<ρk f U  i (x + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We first derive  an upper bound of max∥δ∥2<ρk f U  i (x + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  max  ∥δ∥2<ρk f U  i (x + δ)  (25)  = max  ∥δ∥2<ρk(W[i, :]x + W[i, :]δ + b[i])  (26)  =W[i, :]x + b[i] +  max  ∥δ∥2<ρk W[i, :]δ  (27)  ≤W[i, :]x + b[i] +  max  ∥δ∥2<ρk ∥W[i, :]∥2 ∥δ∥2  (28)  <W[i, :]x + b[i] + ∥W[i, :]∥2 ρk  (29)  We have Equation 28 from 27 based on Cauchy-Schwartz  inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Our derived upper bound is the same as the  one obtained from the closed-form solution of f U  i  based on  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='3 in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We let δ′ = ρkW[i,:]T  ∥W[i,:]∥2 , where T represents  the transpose operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We can verify that ∥δ′∥ = ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then,  we have:  fi(x + δ′)  (30)  =W[i, :]x + b[i] + W[i, :]ρkW[i, :]T  ∥W[i, :]∥2  (31)  =W[i, :]x + b[i] + ∥W[i, :]∥2 ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (32)  Combining the above equations with Equation 25 and  29,  we know that there exist δ′ satisfying ∥δ′∥2 = ρk such that  fi(x + δ′) > max∥δ∥2<ρk f U  i (x + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Thus, we prove the  tightness of the right-hand side of Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Similarly, we  can prove the tightness of the left-hand side of Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  APPENDIX E  APPLYING REAAS TO NLP DOMAIN  Our REaaS provides two APIs: Feature-API and F2IPerturb-  API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The Feature-API can be directly applied to the NLP  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, given a text as input, the Feature-API  returns the feature vector produced by a text encoder for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Next, we discuss how to adapt our F2IPerturb-API to the NLP  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Adapting our F2IPerturb-API to NLP domain: Given a text  input and a feature-space certified radius, we can adapt our  F2IPerturb-API to provide the input-space certified radius for  the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, we consider an attacker can replace  a small number of words in a text with their synonyms such that  the downstream classifier of a client makes incorrect prediction  for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' For simplicity, given a text input x, we use x[i] to denote  the ith word in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, we use δ to denote an adversarial  text perturbation whose length is the same as x, where δ[i]  is an empty string if we don’t perturb x[i];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' we use x ⊕ δ to  denote the perturbed text obtained by replacing x[i] as δ[i] if  δ[i] is not an empty string;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' we use ∥δ∥0 to denote the number  of words in δ that are not empty string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The input-space certified radius returned by our adapted  F2IPerturb-API is the maximum number of words that can be  replaced by their synonyms such that the predicted label by  the downstream classifier is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Formally, given a text  input x and a feature-space certified radius RF , our adapted  F2IPerturb-API aims to solve the following optimization  problem:  R = max  r  r  (33)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  max  ∥δ∥0<r ∥f(x ⊕ δ) − f(x)∥2 < RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (34)  Similar to the image domain, we can use binary search to solve  the above optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The key difference with the  image domain is how to estimate the lower or upper bounds of  fi(x ⊕ δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We note that Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' [47] extend CROWN [9] for  general perturbations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' which can be applied to derive lower and  16  Algorithm 2: F2IPerturb-API (NLP)  Input from client: text x and feature-space certified  radius RF  Output for client: text-space certified radius R  ρL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' ρU ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' 10  while ρU − ρL > 1 do  ρk = ⌈ ρL+ρU  2  ⌉  for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' d do  f L  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' f U  i ← EXTENDEDCROWN(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' f)  Li = min∥δ∥0≤ρk f L  i (x + δ) − fi(x)  Ui = max∥δ∥0≤ρk f U  i (x + δ) − fi(x)  end for  R′  F =  √︂∑︁d  i=1 max(L2  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' U 2  i )  if R′  F < RF then  ρL = ρk  else  ρU = ρk  end if  end while  return ρL  upper bounds (denoted as f L  i (x⊕δ) and f U  i (x⊕δ)) of fi(x⊕δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Moreover, they also provide a computation-efficient method  to compute min∥δ∥0<r f L  i (x ⊕ δ) and max∥δ∥0<r f U  i (x ⊕ δ)  (please see Theorem 2 in [47] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given those bounds,  our adapted F2IPerturb-API can be used to obtain the certified  radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Algorithm 2 shows our complete algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The function  EXTENDEDCROWN uses the extended CROWN [47] to obtain  lower or upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Experimental setup: We use SST-2 dataset [48] as the pre-  training dataset and train a text encoder using supervised  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Following [47], we use LSTM as the encoder  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover, we use public implementation from [47]  in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We consider the same word substitutions as  previous work [33], [49] when deriving the input-space certified  radius for our F2IPerturb-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We use IMDB dataset [36] as  the downstream dataset, which is a benchmark dataset used  for sentiment analysis in the NLP domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We train a logistic  regression classifier as the downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover,  we use CROWN [9] as BC based certification to derive the  certified radius of the downstream classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  APPENDIX F  EXTENDING STOLENENCODER [34] TO STEAL ENCODERS  WITH OUR NEW F2IPERTURB-API  We first introduce StolenEncoder [34] and then discuss how  to extend it to steal encoders with our F2IPerturb-API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  StolenEncoder: Suppose an attacker has an unlabeled surrogate  dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given black-box access to an encoder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', an  attacker can query the Feature-API in our terminology), the  attacker can use StolenEncoder to steal a target encoder f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Suppose fs is the stolen encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' StolenEncoder aims to solve  the following optimization problem:  min  fs L(D) =  1  |D|  ∑︂  x∈D  [d(f(x), fs(x)) + κd(f(x), fs(A(x)))],  (35)  where κ is a hyperparameter, d is a distance metric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=', ℓ2-  distance), and A is the data augmentation component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=',  RandomHorizontalFlip, ColorJitter, and RandomGrayScale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The total number of queries to the Feature-API is |D| as a  single query is made for each input in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' StolenEncoder uses  standard stochastic gradient descent (SGD) to solve the above  optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Extending  StolenEncoder  with  our  new  F2IPerturb-  API: An attacker can also query F2IPerturb-API in our REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  We extend StolenEncoder to leverage our F2IPerturb-API to  steal the target encoder f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, given an input image x  and a feature-space certified radius RF for x, our F2IPerturb-  API returns an input-space certified radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Therefore, we  know the following based on the output of F2IPerturb-API:  ∥f(x) − f(x + δ)∥2 < RF for ∀δ, ∥δ∥2 < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  (36)  We can train the stolen encoder fs such that it also satisfies  Equation 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Suppose we have a surrogate dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Given  an input x ∈ D, we first randomly sample a radius from  [Rmin  F  , Rmax  F  ] and view it as the feature-space certified radius  Rx  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then, we query F2IPerturb-API to obtain the input-  space certified radius Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' To make the stolen encoder fs  satisfy Equation 36 for x, we randomly sample an input-space  perturbation δx that satisfies ∥δx∥2 = Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Then, we define the  following loss:  L′(D) =  1  |D|  ∑︂  x∈D  d(fs(x), fs(x + δx))·  (37)  I(d(fs(x), fs(x + δx)) > Rx  F ),  where I is an indicator function whose output is 1 if the  condition is satisfied and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Intuitively, the loss term  in L′(D) for an input x is 0 if d(fs(x), fs(x+δx)) ≤ Rx  F and is  non-zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Combining with the loss of StolenEncoder,  our final optimization problem is as follows:  min  fs L1(D) = L(D) + γL′(D),  (38)  where γ is a hyperparameter and L is defined in Equation 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  The total number of queries is 2|D| as we respectively make  one query to the Feature-API and F2IPerturb-API for each input  in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Similarly, we use stochastic gradient descent to solve the  optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Note that our extended StolenEncoder  reduces to StolenEncoder when γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  Experimental setup: We use Tiny-ImageNet as the pre-training  dataset and use CIFAR10 as the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Moreover,  we randomly sample 10,000 images from STL10 dataset as  the surrogate dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We adopt the same κ, d, and A as  StolenEncoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' We set Rmin  F  and Rmax  F  to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='5,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' As we use the same surrogate dataset, we give  advantages to the extended StolenEncoder since its total number  of queries is twice of StolenEncoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' Following [34], we use  Stolen Accuracy as evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' In particular, given a  stolen encoder and a downstream task, Stolen Accuracy is the  classification accuracy of the downstream classifier built upon  the stolen encoder for the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='7  Certified accuracy  N = 100  N = 1, 000  N = 10, 000  N = 100, 000  (a) N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
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+page_content='1  (c) α  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' 8: Impact of N, σ, and α on certified accuracy of SC in REaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content=' The pre-training dataset is Tiny-ImageNet and the  downstream dataset is CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}
+page_content='  18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfFwM8/content/2301.02905v1.pdf'}