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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf,len=1293
+page_content='Aleatoric and Epistemic Discrimination in Classification  Hao Wang 1 Luxi He 2 Rui Gao 3 Flavio P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Calmon 4  Abstract  Machine learning (ML) models can underperform  on certain population groups due to choices made  during model development and bias inherent in  the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We categorize sources of discrimina-  tion in the ML pipeline into two classes: aleatoric  discrimination, which is inherent in the data distri-  bution, and epistemic discrimination, which is due  to decisions during model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We quan-  tify aleatoric discrimination by determining the  performance limits of a model under fairness con-  straints, assuming perfect knowledge of the data  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We demonstrate how to characterize  aleatoric discrimination by applying Blackwell’s  results on comparing statistical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We  then quantify epistemic discrimination as the gap  between a model’s accuracy given fairness con-  straints and the limit posed by aleatoric discrimi-  nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We apply this approach to benchmark ex-  isting interventions and investigate fairness risks  in data with missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our results indi-  cate that state-of-the-art fairness interventions are  effective at removing epistemic discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  However, when data has missing values, there is  still significant room for improvement in handling  aleatoric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Introduction  Algorithmic discrimination may occur in different stages of  the machine learning (ML) pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For example, histori-  cal biases in the data-generating process can propagate to  downstream tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' human biases can influence a ML model  through inductive bias;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' optimizing solely for accuracy can  lead to disparate model performance across groups in the  data (Suresh & Guttag, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Mayson, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The past  years have seen a rapid increase in algorithmic interventions  that aim to mitigate biases in ML models (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Zemel  1MIT-IBM  Watson  AI  Lab  2Harvard  College  3UT-  Austin  4Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Hao Wang <hao@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='com>,  Luxi  He  <luxihe@college.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='edu>,  Rui  Gao  <rui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='gao@mccombs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='edu>,  Flavio  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Calmon  <flavio@seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='edu>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Calmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Menon & Williamson, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Zafar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Friedler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Bellamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Celis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Jiang & Nachum,  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Lowy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Alghamdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' A recent survey (Hort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2022) found nearly 400 fairness-intervention algorithms,  including 123 pre-processing, 212 in-processing, and 56  post-processing algorithms introduced in the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Which sources of biases are (the hundreds of) existing fair-  ness interventions trying to control?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In order to create effec-  tive strategies for reducing algorithmic discrimination, it is  critical to disentangle where biases in model performance  originate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For instance, if the training set contains few sam-  ples from a given population group, then increasing sample  diversity is a more effective strategy than selecting a more  complex model class or training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Conversely, if a  model class does not accurately represent the underlying  distribution of a certain population group, then increasing  sample size for that group will not resolve performance  disparities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We divide algorithmic discrimination into two categories:  aleatoric and epistemic discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='1 Aleatoric discrimi-  nation captures inherent biases in the data distribution that  can lead to unfair decisions in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Epistemic  discrimination, in turn, is due to algorithmic choices made  during model development and lack of knowledge about the  optimal “fair” predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  In this paper, we provide methods for measuring aleatoric  and epistemic discrimination in classification task for group  fairness metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Since aleatoric discrimination only de-  pends on properties of the data distribution and the fairness  measure of choice, we quantify it by asking a fundamental  question:  For a given data distribution, what would be the best achiev-  able performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', accuracy) under a group fairness  constraint?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  1We borrow this notion from ML uncertainty literature (see  H¨ullermeier & Waegeman, 2021, for a survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Therein, aleatoric  uncertainty refers to the variability in the outcome of an experiment  resulting from inherently random effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' epistemic uncertainty  refers to uncertainty caused by a lack of knowledge about the best  predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='11781v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='LG]  27 Jan 2023  Aleatoric and Epistemic Discrimination in Classification  We refer to the answer as the fairness Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This  frontier delineates the optimal performance achievable by  a classifier when unlimited data and computing power are  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For a fixed data distribution, the fairness Pareto  frontier represents the ultimate, information-theoretic limit  for accuracy and group fairness beyond which no model can  achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Characterizing this limit enables us to (i) separate  sources of discrimination and create strategies to control  them accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (ii) evaluate the effectiveness of existing  fairness interventions for reducing epistemic discrimination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  and (iii) inform the development of data collection methods  that promote fairness in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  At first, computing the fairness Pareto frontier can appear to  be an intractable problem since it requires searching over all  possible classifiers—even if the data distribution is known  exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our main technical contribution is to provide a pre-  cise characterization of this frontier by solving a sequence of  optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our main proof technique is based  on Blackwell’s seminal results (Blackwell, 1953), which  proposed the notion of comparisons of statistical experi-  ments and inspired a line of works introducing alternative  comparison criteria (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Shannon, 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Cam, 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Torgersen, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Raginsky, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Here,  we apply these results to develop an algorithm that itera-  tively refines the achievable fairness Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We also  prove convergence guarantees for our algorithm and demon-  strate how it can be used to benchmark existing fairness  interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We quantify epistemic discrimination by comparing a clas-  sifier’s performance with the information-theoretic optimal  given by the fairness Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our experiments indi-  cate that given sufficient data, state-of-the-art fairness inter-  ventions are effective at reducing epistemic discrimination  as their gap to the information-theoretic limit is small (see  Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Existing interventions do not eliminate aleatoric  discrimination as this type of discrimination is not caused  by choice of learning algorithm or model class, and is due  to the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We further analyze the fairness Pareto frontier to show that  factors such as data missing values can significantly con-  tribute to aleatoric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We observe that when  population groups have disparate missing patterns, aleatoric  discrimination escalates, leading to a sharp decline in the  effectiveness of fairness intervention algorithms (see Fig-  ure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Related Work  There is significant work analyzing the tension between  group fairness measures and model performance metrics  (Kleinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Chouldechova, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Corbett-  Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Dutta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We recast the fairness Pareto frontier  in terms of the conditional distribution PˆY|Y,S of predicted  outcome ˆY given true label Y and group attributes S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This  conditional distribution is related to confusion matrices2  conditioned on each subgroup (see Remark 1 for detailed  discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In this regard, our work is related to Verma &  Rubin (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Alghamdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2021), which observed that  many group fairness metrics can be written in terms of the  confusion matrices for each subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Among them, the  closest work to ours is Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020), which optimized  accuracy and fairness objectives over these confusion matri-  ces and proposed a post-processing technique for training  fair classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' However, they only imposed marginal sum  constraints for the confusion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We demonstrate  that the feasible region of confusion matrices can be much  smaller (see Remark 2 for an example), leading to a tighter  approximation of the fairness Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Recently, many strategies have been proposed to reduce  the tension between group fairness and model performance  by investigating properties of the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For ex-  ample, Blum & Stangl (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Suresh & Guttag (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Fogliato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Mehrotra &  Celis (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Fernando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Wang & Singh (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Zhang & Long (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Kallus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (2022) studied how noisy or missing data affect fairness and  model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Ustun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2021) considered training a separate classifier  for each subgroup when their data distributions are differ-  ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Another line of research introduces data pre-processing  techniques that manipulate data distribution for reducing its  bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Calmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Kamiran & Calders, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Among all these works, the closest one to ours is Chen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2018), which decomposed group fairness measures  into bias, variance, and noise (see their Theorem 1) and pro-  posed strategies for reducing each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The main difference  compared with Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2018) is that we characterize  a fairness Pareto frontier that depends on not only fairness  metrics but also on a performance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This effort gives  a complete picture of how data distribution influences the  fairness-accuracy tension and is more technically involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Also, the fairness Pareto frontier only depends on the data  distribution and fairness metrics of choice so it cannot be  improved by adding more data or altering the model class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Preliminaries  Next, we introduce notation, overview the key results in  Blackwell (1953) on comparisons of experiments, and out-  line the fair classification setup considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  2A confusion matrix (Kulkarni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2020) is a table that  measures the performance of a given ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In binary classi-  fication, a confusion matrix reports the number of true positives,  false negatives, false positives, and true negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Aleatoric and Epistemic Discrimination in Classification  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  For a positive integer n, let [n] ≜ {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We denote all probability distributions on the set X by  P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Moreover, we define the probability simplex ∆m ≜  P([m]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' When random variables A, X, Z form a Markov  chain, we write A → X → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We write the mutual infor-  mation between A, X as I(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' X) ≜ EPA,X  �  log  PA,X(A,X)  PA(A)PX(X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Since I(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' X) is determined by the marginal distribution  PA and the conditional distribution PX|A, we also write  I(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' X) as I(PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' PX|A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' When A, X are independent, we  write A  |=  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  If a random variable A ∈ [n] has finite support, the condi-  tional distribution PX|A : [n] → P(X) can be equivalently  written as P ≜ (P1, · · · , Pn) where each Pi = PX|A=i ∈  P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Additionally, if X is a finite set [m], then PX|A  can be fully characterized by a transition matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use  T (m|n) to denote all transition matrices from [n] to [m]:  �  �  �P ∈ Rn×m ��� 0 ≤ Pi,j ≤ 1,  m  �  j=1  Pi,j = 1, ∀i ∈ [n]  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Comparisons of Experiments  Given two statistical experiments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', conditional distribu-  tions) P and Q, is there a way to decide which one is more  informative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Here P and Q have the common input alpha-  bet [n] and potentially different output spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Blackwell  gave an answer in his seminal work (Blackwell, 1953) from  a decision-theoretic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We review these results  next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Let A be a closed, bounded, convex subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' A deci-  sion function f(x) = (a1(x), · · · , an(x)) is any mapping  from X to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' It is associated a loss vector:  v(f) =  ��  a1(x)dP1(x), · · · ,  �  an(x)dPn(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (1)  The collection of all v(f) is denoted by B(P , A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Black-  well defined that P is more informative than Q if for every  A, B(P , A) ⊇ B(Q, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Intuitively, this result means any  risk achievable with Q is also achievable with P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Moreover,  Blackwell considered the standard measure P ∗ which is the  probability distribution of p(¯X) where p(x) : X → ∆n is a  function defined as  �  dP1  dP1 + · · · + dPn  , · · · ,  dPn  dP1 + · · · + dPn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (2)  and ¯X follows the probability distribution P1+···+Pn  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' One  of the most important findings by Blackwell in his paper is  to discover the following equivalent conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Lemma 1 (Blackwell (1951;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' 1953)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The following three  conditions are equivalent:  P is more informative than Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  for any continuous and convex function φ : ∆n → R,  �  φ(p)dP ∗(p) ≥  �  φ(p)dQ∗(p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  there is a stochastic transformation T such that TPi =  Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In other words, there exists a Markov chain A →  X → Z for any distributions on A such that P = PX|A  and Q = PZ|A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Additionally, if P and Q can be characterized by transition  matrices, the above conditions are also equivalent to:  there exists a transition matrix M such that Q =  P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  If P = PX|A is more informative than Q = PZ|A, by the  third condition of Lemma 1 and the data processing inequal-  ity,  I(PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' PX|A) ≥ I(PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' PZ|A)  (3)  holds for any marginal distribution PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' However, the con-  verse does not hold in general—even if (3) holds for any PA,  P is not necessarily more informative than Q (see Rauh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (2017) for a counter-example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In this regard, Blackwell’s  conditions are “stronger” than mutual information-based  form of the data processing inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Fair Classification  Consider a multi-class classification task, where the goal is  to train a probabilistic classifier h : X → ∆C that uses input  features X to predict their true label Y ∈ [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Additionally,  assume the classifier produces a predicted outcome ˆY ∈ [C]  and let S ∈ [A] represent group attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', race and  sex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our framework can be easily extended to the setting  where multiple subgroups overlap (Kearns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Throughout this paper, we focus on three standard group  fairness measures: statistical parity (SP) (Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2015), equalized odds (EO) (Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Pleiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2017), and overall accuracy equality (OAE) (Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2021) (see Table 1 for their definitions) but our analysis can  be extended to many other group fairness metrics, including  the ones in Table 1 of Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use Blackwell’s conditions to provide a precise charac-  terization of fairness Pareto frontier in multi-class classifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In brief, Blackwell’s conditions allow us to approxi-  mate the set of achievable joint distributions PˆY|S,Y across  all classifiers h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Since both accuracy and the group fairness  criteria in Table 1 can be cast in terms of PˆY|S,Y, this ap-  proximation can then be used to bound the best achievable  accuracy under a group fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We will develop  this procedure in detail in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Aleatoric and Epistemic Discrimination in Classification  FAIRNESS METRIC  ABBR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  DEFINITION  EXPRESSION W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' P  Statistical Parity  SP ≤ αSP  | Pr(ˆY = ˆy|S = s) − Pr(ˆY = ˆy|S = s′)| ≤ αSP  ���  �C  y=1  �  µs,y  µs P(s,y),ˆy −  µs′,y  µs′ P(s′,y),ˆy  ���� ≤ αSP  Equalized Odds  EO ≤ αEO  | Pr(ˆY = ˆy|S = s, Y = y) − Pr(ˆY = ˆy|S = s′, Y = y)| ≤ αEO  ��P(s,y),ˆy − P(s′,y),ˆy  �� ≤ αEO  Overall Accuracy Equality  OAE ≤ αOAE  | Pr(ˆY = Y |S = s) − Pr(ˆY = Y |S = s′)| ≤ αOAE  ����C  y=1  �  µs,y  µs P(s,y),y −  µs′,y  µs′ P(s′,y),y  ���� ≤ αOAE  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Standard group fairness metrics under multi-group and multi-class classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Here αSP, αEO, αOAE, ∈ [0, 1] are threshold  parameters, ˆy, y ∈ [C], s, s′ ∈ [A], and µs,y, µs are defined in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our analysis can be extended to many other group fairness  metrics (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Table 1 in Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Fairness Pareto Frontier  In this section, we introduce our main concept—fairness  Pareto frontier (FairFront).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use it to measure aleatoric  discrimination and quantify epistemic discrimination by  comparing a classifier’s performance to the FairFront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We  recast FairFront in terms of the conditional distribution  PˆY|S,Y and apply Blackwell’s conditions to characterize the  feasible region of this conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This effort  converts a functional optimization problem into a convex  program with a small number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' However, this  convex program may involve infinitely many constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Hence, we introduce a greedy improvement algorithm that  iteratively computes FairFront and tightens the feasible re-  gion of PˆY|S,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We end this section by establishing a con-  vergence guarantee for our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Recall that we refer to aleatoric discrimination as the inher-  ent biases of the data distribution that can lead to an unfair  or inaccurate classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' As its definition suggests, aleatoric  discrimination only relies on properties of the data distri-  bution and fairness metric of choice—it does not depend  on the hypothesis class nor optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Below  we introduce FairFront that delineates a curve of optimal  accuracy over all probabilistic classifiers under certain fair-  ness constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use FairFront to quantify aleatoric  discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For αSP, αEO, αOAE  ∈  [0, 1], we define  FairFront(αSP, αEO, αOAE) as the solution of the following  optimization problem:  max  h  E  �  IˆY=Y  �  (4a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE  (4b)  where ˆY is produced by applying the classifier h to X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' the  maximum is taken over all measurable h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' and the definitions  of SP, EO, and OAE are in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Solving this functional optimization problem is difficult  since it optimizes over all measurable classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' There is a  line of works that proposed different fairness-intervention al-  gorithms for training group-fair classifiers (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Menon  & Williamson, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Zafar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Celis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Al-  ghamdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' They restrict the model class and  vary loss functions and optimizers to find classifiers that  approach FairFront as close as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' However, these  algorithms only describe a lower bound for FairFront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' They  do not determine what is the best achievable accuracy for a  given fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We circumvent the above-mentioned challenges by rewrit-  ing FairFront in terms of the conditional distribution PˆY|S,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  The caveat is that although each classifier yields a PˆY|S,Y,  not every conditional distribution corresponds to a valid  classifier (see an example in Remark 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Hence, we intro-  duce the following definition which characterizes all feasible  PˆY|S,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Given PX|S,Y, we define C as the set of all  conditional distributions PˆY|S,Y where ˆY is produced by  some probabilistic classifier h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In other words,  C ≜  �  PˆY|S,Y | (S, Y) → X → ˆY  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (5)  Throughout this paper, we write PˆY|S,Y or its corresponding  transition matrix P interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We demonstrate the connection between the  conditional distribution PˆY|S,Y and confusion matrices in the  setting of binary classification with binary subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We  define ˆC as the counterpart of C when we replace PX|S,Y with  an empirical distribution ˆPX|S,Y computed from a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  The confusion matrix for group s ∈ {0, 1} consists of four  numbers: True Positive (TPs), False Positive (FPs), False  Negative (FNs), True Negative (TNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Assume that the num-  ber of positive-label data n+  s = TPs + FNs and negative-  label data n−  s = TNs + FPs are given—these numbers do  not depend on the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Then there is a one-to-one  Aleatoric and Epistemic Discrimination in Classification  mapping from each element in ˆC to a confusion matrix:  ˆPˆY|S=s,Y=+(+) = 1  n+  s  TPs,  ˆPˆY|S=s,Y=−(+) = 1  n−  s  FPs,  ˆPˆY|S=s,Y=+(−) = 1  n+  s  FNs,  ˆPˆY|S=s,Y=−(−) = 1  n−  s  TNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Hence, ˆC essentially characterizes all feasible confusion  matrices and C is the population counterpart of ˆC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Note  that C is determined by the data distribution while ˆC (and  confusion matrices) are tailored to a specific dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We  establish  basic  properties  of  C  and  FairFront(αSP, αEO, αOAE) in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Then we  demonstrate how to use C for characterizing the fairness  Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' C is a convex subset of T (C|AC) and  FairFront(αSP, αEO, αOAE) is a concave function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  αSP, αEO, αOAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' FairFront(αSP, αEO, αOAE) in (4) is equal to  the solution of the following convex optimization:  max  P ∈RAC×C  A  �  s=1  C  �  y=1  µs,yP(s,y),y  (6a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE  (6b)  P ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (6c)  Here the constants µs,y ≜ Pr(S = s, Y = y) and µs ≜  Pr(S = s) for s ∈ [A], y ∈ [A] and P(s,y),ˆy denotes the  (C(s − 1) + y)-th row, ˆy-th column of P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  For example, in the setting of binary classification with  binary group attribute, the above optimization only has 8  variables, 14 linear constraints + a single convex constraint  P ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Hence, its optimal value can be directly computed  by standard convex optimization solvers as long as we know  how to characterize C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Next, we discuss two special cases—  X is independent of (S, Y) or X is discrete—under which C  has a simple characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Note that Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020) investigated fairness  Pareto frontiers via confusion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The main difference  is that Definition 1 in Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2020) relaxed the constraint  (6c) to P ∈ T (C|AC) where T (C|AC) represents all  transition matrices from [AC] to [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This leads to a loose  approximation of the frontier because C is often a strict  subset of T (C|AC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' To demonstrate this point, consider  the scenario where X  |=  (S, Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Then ˆY  |=  (S, Y) by data  processing inequality so  C = {P ∈ T (C|AC) | each row of P is the same} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (7)  Optimizing over C rather than T (C|AC) can significantly  tighten the fairness Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' If X is a discrete variable with a finite sup-  port [D], we can write PX|S,Y as a transition matrix Φ ∈  T (D|AC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' By introducing an auxiliary variable M ∈  T (C|D), we can write P ∈ C equivalently as linear con-  straints: P = ΦM by using the last condition of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Consequently, Proposition 1 boils down to a linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  However, this characterization fails to generalize to continu-  ous data because Φ and M will have an infinite dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  for categorical data, this characterization suffers from the  curse of dimensionality since the support size of X grows  exponentially fast w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' the number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  The above two remarks provide precise characterizations of  C under specific assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In what follows, we consider  a more general setting by leveraging Blackwell’s conditions  (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Before diving into the analysis, we first intro-  duce a function g : X → ∆AC defined as  g(x) =  �  PS,Y|X(1, 1|x), · · · , PS,Y|X(A, C|x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (8)  To obtain this function in practice, one can train a probabilis-  tic classifier that uses input features X to predict (S, Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We  use this classifiers’ output probability as an approximation  of the function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  The following theorem is the main theoretical result in this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' It provides a precise characterization of the set C  through a series of convex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The set C is the collection of all transition  matrices P ∈ T (C|AC) such that the following condition  holds:  For any k ∈ N and any {ai | ai ∈ [−1, 1]AC, i ∈ [k]},  C  �  ˆy=1  max  i∈[k]  �  aT  i ΛΛΛµpˆy  �  ≤ E  �  max  i∈[k]{aT  i g(X)}  �  ,  (9)  where pˆy  is the  ˆy-th column of P  and ΛΛΛµ  =  diag(µ1,1, · · · , µA,C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Intuitively, (9) uses piece-wise linear functions to approx-  imate the boundary of C where k represents the num-  ber of linear pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Unfortunately, replacing P ∈ C  with this series of constraints in (6) may result in an in-  tractable problem since standard duality-based approaches  will lead to infinitely many dual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  To resolve  this issue, we first fix k and let Ck be the set of P such  that (9) holds under this fixed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Accordingly, we define  FairFrontk(αSP, αEO, αOAE) as the optimal value of (6) when  replacing C with Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Since C1 ⊇ C2 ⊇ · · · ⊇ C, we have  FairFront1(αSP, αEO, αOAE) ≥ FairFront2(αSP, αEO, αOAE) ≥  · ·  ≥  FairFront(αSP, αEO, αOAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' However, computing  FairFrontk(αSP, αEO, αOAE) still involves infinitely many con-  straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Next, we introduce a greedy improvement algorithm that  consists of solving a sequence of tractable optimization  Aleatoric and Epistemic Discrimination in Classification  Algorithm 1 Approximate the fairness Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Input: D = {(xi, yi, si)}N  i=1, maximum number of iterations  T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' maximum pieces k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' classifier g(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' threshold parameters  αSP, αEO, αOAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Initialize: set A = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' µs,y = |{i|si=s,yi=y}|  N  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Repeat:  Solve a convex program:  max  P  A  �  s=1  C  �  y=1  µs,yP(s,y),y  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' P ∈ T (C|AC)  SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE  C  �  ˆy=1  max  i∈[k]  �  aT  i ΛΛΛµpˆy  �  ≤ E  �  max  i∈[k]{aT  i g(X)}  �  ∀(a1, · · · , ak) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Let vt and P t be the optimal value and optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Solve a DC program:  min  ai∈[−1,1]AC  i∈[k]  E  �  max  i∈[k]{aT  i g(X)}  �  −  C  �  ˆy=1  max  i∈[k]  �  aT  i ΛΛΛµpt  ˆy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  If the optimal value is ≥ 0 or t = T,  stop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  otherwise,  add the optimal (a1, · · · , ak) to A and t = t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  return: vt, P t, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  problems for approximating FairFrontk(αSP, αEO, αOAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We  use A to collect the constraints of P and set A = ∅ initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  At each iteration, our algorithm solves a convex program  to find an optimal P that maximizes the accuracy while  satisfying the desired group fairness constraints and the con-  straints in A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' then we verify if this P is within the set Ck by  solving a DC (difference of convex) program (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Horst & Thoai, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' If P ∈ Ck, then the algo-  rithm stops;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' otherwise, the algorithm will find the constraint  that is mostly violated by P and add this constraint to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We describe our algorithm in Algorithm 1 and establish a  convergence guarantee in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Let T = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' If Algorithm 1 stops, its output P t  is an optimal solution of FairFrontk(αSP, αEO, αOAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Other-  wise, any convergent sub-sequence of {P t}∞  t=1 converges  to an optimal solution of FairFrontk(αSP, αEO, αOAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Note that the output vt from Algorithm 1 is always an up-  per bound for FairFront(αSP, αEO, αOAE), assuming the es-  timation error is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The tightness of this  upper bound is determined by k (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', how well the piece-  wise linear functions approximate the boundary of C), T  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', the total number of iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' On the other hand,  running off-the-shelf in-processing and post-processing  fairness interventions can only yield lower bounds for  FairFront(αSP, αEO, αOAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In the next section, we compare  our upper bound given by Algorithm 1 with the lower  bounds given by some state-of-the-art methods to demon-  strate the tightness of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Numerical Experiments  Recall that FairFront in Definition 1 characterizes the high-  est achievable accuracy under a fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use  it to quantify aleatoric discrimination and measure epistemic  discrimination by comparing a classifier’s accuracy and fair-  ness violation with FairFront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In this section, we apply  FairFront to analyze the performance of existing fairness in-  terventions and how data biases, specifically missing values,  impact their effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We find that given sufficient data,  state-of-the-art fairness interventions are successful at reduc-  ing epistemic discrimination as their gap to the FairFront is  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' However, we also discover that when different popu-  lation groups have varying missing data patterns, aleatoric  discrimination increases, which diminishes the performance  of fairness intervention algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We provide additional  experimental results and details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Benchmark Fairness Interventions  Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We evaluate our results on the UCI Adult dataset  (Bache & Lichman, 2013), the ProPublica COMPAS dataset  (Angwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2016), and the German Credit dataset (Bache  & Lichman, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='3 We consider a binary classification  problem with binary groups since most existing fairness  interventions are designed for this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For the Adult  dataset, we choose sex (female or male) as the group at-  tribute and income (> 50K or <= 50K) as the target for pre-  diction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' for the COMPAS dataset, we choose race (African-  American or Caucasian) as the group attribute and is recid  (recid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' or no recid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=') as the target for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The details  about how we pre-process these datasets are deferred to Ap-  pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We measure fairness violations via Max equalized  odds:  max | Pr(ˆY = ˆy|S = s, Y = y) − Pr(ˆY = ˆy|S = s′, Y = y)|  where the max is taken over y, ˆy, s, s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We run Algorithm 1  with k = 6 pieces, 20 iterations, and varying αEO to get  FairFront on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We compute the expectations  and the g function from the empirical distributions and solve  the DC program by using the DCCP package provided by  Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Fairness  interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We  consider  five  existing  fairness-intervention algorithms:  Reduction (Agar-  wal  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2018),  EqOdds  (Hardt  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2016),  CalEqOdds (Pleiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2017), LevEqOpp (Chzhen  3We defer the experimental results on the German Credit  dataset to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Aleatoric and Epistemic Discrimination in Classification  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Comparing existing fairness interventions with FairFront on the Adult (Left) and COMPAS (Right) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use FairFront  to quantify aleatoric discrimination and measure epistemic discrimination by comparing a classifier’s accuracy and fairness violation  with FairFront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' As shown, SOTA fairness interventions are effective at reducing epistemic discrimination as their gap to the FairFront is  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019), and FairProjection (Alghamdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Among them, Reduction is an in-processing  method and the rest are all post-processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For  the first three benchmarks, we use the implementations  from IBM AIF360 library (Bellamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' for  LevEqOpp and FairProjection, we use the Python  implementations from the Github repo in Alghamdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For Reduction and FairProjection, we  can vary their tolerance of fairness violations to produce a  fairness-accuracy curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' for EqOdds, CalEqOdds, and  LevEqOpp, each of them produces a single point since  they only allow hard equality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We note that  FairProjection is optimized for transforming prob-  abilistic classifier outputs (see also Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2021), but  here we threshold the probabilistic outputs to generate bi-  nary predictions which may limit its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Finally,  we train a random forest as the Baseline classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In  order to emulate the setting where the data distribution is  known exactly, we train models using the entire dataset and  resample 30% data as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We benchmark fairness interventions against  FairFront in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' First, we observe that if we run  Algorithm 1 for a single iteration, which is equivalent to  solving Proposition 1 without (6c), its solution is very close  to 1 for all αEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This demonstrates the benefits of incorporat-  ing Blackwell’s conditions into the fairness Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Second, we observe that fairness-accuracy curves given  by state-of-the-art (SOTA) fairness interventions are very  close to the fairness Pareto frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This result not only  demonstrates the tightness of our approximation (recall  that Algorithm 1 gives an upper bound of FairFront and  benchmarks give a lower bound) but also shows that SOTA  fairness interventions have already achieved near-optimal  fairness-accuracy curves—their epistemic discrimination  is small since they approach the FairFront limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In what  follows, we demonstrate how missing values in data can  increase aleatoric discrimination and dramatically reduce  the effectiveness of fairness interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Fairness Risks in Missing Values  Real-world data often have missing values and the missing  patterns can be different across different protected groups  (see Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2022, for some examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' There is a grow-  ing line of research (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Fernando  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Wang & Singh, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Subramonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Caton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Zhang & Long, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Schelter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=',  2019) studying the fairness risks of data with missing val-  ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In this section, we apply our result to demonstrate how  disparate missing patterns influence the fairness-accuracy  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We choose sex (group 0: female, group 1: male) as  the group attribute for the Adult dataset, and race (group 0:  African-American, group 1: Caucasian) for the COMPAS  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' To investigate the impact of disparate missing pat-  terns on aleatoric discrimination, we artificially generate  missing values in both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This is necessary as the  datasets do not contain sufficient missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The miss-  ing values are generated according to different probabilities  for different population groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For each data point from  group 0, we erase each input feature with a varying proba-  bility p0 ∈ {10%, 50%, 70%}, while for group 1, we erase  Adult  Aleatoric  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  Epistemic  discrimination,  discrimination  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  (%)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  Accuracy (  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  FairFront  Baseline  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  FairProjection  Reduction  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  LevEqOpp  CalEqOdds  EqOdds  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='50  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  Max equalized odds (%)COMPAS  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  (%)  Accuracy  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  FairFront  Baseline  FairProjection  Reduction  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  LevEqOpp  CalEqOdds  EqOdds  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We demonstrate the fairness risks of disparate missing patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We vary missing probability of group 0 (female in Adult/African-  American in COMPAS) among {10%, 50%, 70%} and let the missing probability of group 1 (male in Adult/Caucasian in COMPAS) be  10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use mode imputation to pre-process missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We apply Reduction to the imputed data and plot its fairness-accuracy  curve against the FairFront with the level of transparency representing the degree of disparity in the missing patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  each input feature with a fixed probability p1 = 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We  then apply mode imputation to the missing values, replacing  them with the mode of non-missing values for each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Finally, we apply Algorithm 1 along with Reduction and  Baseline to the imputed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The experimental results  are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  As we increase the missing probability of  group 0, FairFront decreases since it becomes more difficult  to accurately predict outcomes for group 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This in turn  affects the overall model performance, since the fairness  constraint requires that the model performs similarly for  both groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We also observe the fairness-accuracy curves  of Reduction decrease as the missing data for group 0  become more prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In other words, as the missing data  for group 0 increase, it becomes more difficult to maintain  both high accuracy and fairness in the model’s prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Final Remarks and Limitations  The past years have witnessed a growing line of research in-  troducing various fairness-intervention algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Most of  these interventions focus on optimizing model performance  subject to group fairness constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Though comparing  and benchmarking these methods on various datasets is valu-  able (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', see benchmarks in Friedler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Bellamy  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2021), this does not reveal if there is  still room for improvement in their fairness-accuracy curves,  or if existing methods approach the information-theoretic  optimal limit when infinite data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our results  address this gap by introducing the fairness Pareto frontier,  which measures the highest possible accuracy under a group  fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We precisely characterize the fairness  Pareto frontier using Blackwell’s conditions and present a  greedy improvement algorithm that approximates it from  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our results show that the fairness-accuracy curves  produced by state-of-the-art fairness interventions are very  close to the fairness Pareto frontier on standard datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Additionally, we demonstrate that when data are biased  due to missing values, the fairness Pareto frontier degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Although existing fairness interventions can still reduce per-  formance disparities, they come at the cost of significantly  lowering overall model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The methods we present  for computing the fairness Pareto frontier can also be ap-  plied to analyze other sources of aleatoric discrimination,  such as when individuals may misreport their data or when  there are measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Overall, the fairness Pareto  frontier can serve as a valuable framework for guiding data  collection and cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Our results indicate that existing fairness interventions can  be effective in reducing epistemic discrimination, and there  are diminishing returns in developing new fairness inter-  ventions focused solely on optimizing accuracy for a given  group fairness constraint on pristine data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' However, existing  fairness interventions have yet to effectively provide both  fair and accurate classification when additional sources of  aleatoric discrimination are present (such as missing values  in data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' This suggests that there is still significant need for  research on handling aleatoric sources of discrimination that  appear throughout the data collection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Adult  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Accuracy (%)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  FairFront  Baseline  Reduction  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Max equalized odds (%)COMPAS  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  %)  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Accuracy (  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='9  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='9  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  FairFront  Baseline  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Reduction  0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification  References  Agarwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Beygelzimer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Dud´ık, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Langford, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', and  Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' A reductions approach to fair classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  60–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Alghamdi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Asoodeh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Calmon, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content=', and Ramamurthy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Model projection: Theory  and applications to fair machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In 2020 IEEE  International Symposium on Information Theory (ISIT),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' 2711–2716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Alghamdi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Hsu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content=', Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content=' In Advances in Neural Information Processing  Systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content=', Hoffman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Ai fairness 360: An extensible toolkit  for detecting, understanding, and mitigating unwanted  algorithmic bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content='  Bellamy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content=', Hind, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Hoffman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Houde,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content=', Mehta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Mo-  jsilovi´c, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content=' Ai fairness 360: An extensible toolkit  for detecting and mitigating algorithmic bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' IBM Jour-  nal of Research and Development, 63(4/5):4–1, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Berk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content='  Fairness in criminal justice risk assessments: The state of  the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Sociological Methods & Research, 50(1):3–44,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Blackwell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Comparison of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
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+page_content='  Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' and Singh, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Analyzing the impact of missing val-  ues and selection bias on fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' International Journal  of Data Science and Analytics, 12(2):101–119, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Wei, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Ramamurthy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', and Calmon, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Optimized  score transformation for consistent fair classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 22:258–1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Cisse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', and Koyejo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Fairness with overlap-  ping groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' a probabilistic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In Advances in  Neural Information Processing Systems, volume 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  4067–4078, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Zafar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Valera, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Gomez-Rodriguez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', and Gum-  madi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Fairness constraints: A flexible approach  for fair classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The Journal of Machine Learning  Research, 20(1):2737–2778, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Zemel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Swersky, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Pitassi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', and Dwork, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Learning fair representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In International conference  on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' 325–333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' PMLR, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', Lemoine, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', and Mitchell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Mitigating un-  wanted biases with adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In Proceedings  of the 2018 AAAI/ACM Conference on AI, Ethics, and  Society, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' 335–340, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' and Long, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Assessing fairness in the presence of  missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Advances in neural information processing  systems, 34:16007–16019, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Aleatoric and Epistemic Discrimination in Classification  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Technical Background  In this section, we extend some results in Blackwell (1951;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' 1953) to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For a random variable X, we denote  its probability distribution by L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' A conditional distribution PX|A : [n] → P(X) can be equivalently written as  P ≜ (P1, · · · , Pn) where each Pi = PX|A=i ∈ P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Let A be a closed, bounded, convex subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' A decision  function is a mapping f : X → A, which can also be written as f(x) = (a1(x), · · · , an(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' A decision function is  associated a loss vector:  v(f) =  ��  a1(x)dP1(x), · · · ,  �  an(x)dPn(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (11)  The collection of all v(f) is denoted by B(PX|A, A) or B(P , A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  For a vector λλλ ∈ ∆n such that λλλ > 0, we define a function pλλλ(x) : X → ∆n:  pλλλ(x) =  �  λ1dP1  λ1dP1 + · · · + λndPn  , · · · ,  λndPn  λ1dP1 + · · · + λndPn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (12)  Note that pλλλ(X) is a sufficient statistic for X, considering A as the parameter (it can be proved by Fisher-Neyman factorization  theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' In other words, two Markov chains hold: A → pλλλ(X) → X and A → X → pλλλ(X) for any distribution on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Consider a new set of probability distributions P ∗  λλλ ≜ (L(pλλλ(X1)), · · · , L(pλλλ(Xn))) where L(Xi) = Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Here P ∗  λλλ can be  viewed as a conditional distribution from [n] to P(∆n) since each L(pλλλ(Xi)) is a probability distribution over ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The  following lemma follows from the sufficiency of pλλλ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Lemma 3 (Adaptation of Theorem 3 in Blackwell (1951)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For any A, B(P , A) = B(P ∗  λλλ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Suppose that f ∗(p) = (a∗  1(p), · · · , a∗  n(p)) is a decision function for (P ∗  λλλ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Accordingly, we define f(x) =  (a∗  1(pλλλ(x)), · · · , an(pλλλ(x))) where the function pλλλ is defined in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Then it is clear that f is a decision function for  (P , A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' By the law of unconscious statistician, we have  �  a∗  i (p)dP ∗  λ  λ  λ,i(p) = E [a∗  i (pλλλ(Xi))] =  �  a∗  i (pλλλ(x))dPi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (13)  Hence, v(f ∗) = v(f), which implies B(P ∗  λλλ, A) ⊆ B(P , A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For the other direction, suppose f(x) = (a1(x), · · · , an(x))  is a decision function for (P , A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Let f ∗(p) = (a∗  1(p), · · · , a∗  n(p)) where a∗  i (p) ≜ E [ai(Xi) | pλλλ(Xi) = p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Since pλλλ(X)  is a sufficient statistics, for any i ∈ [n]  L(Xi|pλλλ(Xi) = p) = L(X1|pλλλ(X1) = p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (14)  Therefore, f ∗(p) = E [f(X1)|pλλλ(X1) = p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Since A is a convex set, f ∗ is a decision function for (P ∗, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' By the law of  total expectation, we have  �  a∗  i (p)dP ∗  λλλ,i(p) =  �  ai(x)dPi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (15)  Hence, v(f) = v(f ∗), which implies B(P , A) ⊆ B(P ∗  λλλ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  For a vector λλλ ∈ ∆n such that λλλ > 0, the condition distribution PX|A induces a weighted standard measure P ∗  λλλ ≜ L (pλλλ(¯X))  where L(¯X) = λ1P1 + · · · + λnPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Theorem 3 (Adaptation of Theorem 4 in Blackwell (1951)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For any two conditional distributions PX|A and QY|A, let  P ∗  λλλ and Q∗  λ��λ be their weighted standard measures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Then B(PX|A, A) ⊇ B(QY|A, A) for any closed, bounded,  convex set A if and only if for any continuous convex φ : ∆n → R,  �  φ(p)dP ∗  λλλ(p) ≥  �  φ(p)dQ∗  λλλ(p)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' First, by Lemma 3, we know B(PX|A, A) = B(P ∗  λλλ, A) and B(QY|A, A) = B(Q∗  λλλ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We denote ΛΛΛ =  diag(λ1, · · · , λn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Consider any A = conv(a1, · · · , ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Let  f ∗(p) = argmin  a∈A  pTΛΛΛ−1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (16)  Aleatoric and Epistemic Discrimination in Classification  Note that f ∗(p) ∈ {a1, · · · , ak} since this set contains all the extreme points of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='4 By definition, for any decision function  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (P ∗  λλλ, A), we have  pTΛΛΛ−1f(p) ≥ pTΛΛΛ−1f ∗(p),  ∀p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (17)  Let v = v(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' By the same reason with (13), we have  vj =  �  aj(pλλλ(x))dPj(x)  (18)  = 1  λj  �  aj(pλλλ(x))  λjdPj  λ1dP1 + · · · + λndPn  (x)(λ1dP1 + · · · + λndPn)(x)  (19)  = 1  λj  �  aj(pλλλ(x))[pλλλ(x)]j(λ1dP1 + · · · + λndPn)(x)  (20)  = 1  λj  E [aj(pλλλ(¯X))[pλλλ(¯X)]j]  (21)  = 1  λj  �  aj(p)pjdP ∗  λλλ(p),  (22)  where the last step is due to the law of unconscious statistician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Therefore,  n  �  j=1  vj =  �  pTΛΛΛ−1f(p)dP ∗  λλλ(p)  (23)  ≥  �  pTΛΛΛ−1f ∗(p)dP ∗  λλλ(p)  (24)  =  �  min  i {pTΛΛΛ−1ai}dP ∗  λλλ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (25)  The equality is achieved by v(f ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Hence, for any A = conv(a1, · · · , ak)  min  v∈B(PX|A,A)  n  �  j=1  vj =  �  min  i {aT  i ΛΛΛ−1p}dP ∗  λλλ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (26)  Recall that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (3) in Blackwell (1951) states  B(PX|A, A) ⊇ B(PY|A, A)  for every closed, bounded, convex A  ⇔  min  v∈B(PX|A,A)  n  �  j=1  vj ≤  min  v∈B(PY|A,A)  n  �  j=1  vj  for every closed, bounded, convex A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  By approximation theory, the second condition can be relaxed to any A that is a convex hull of a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' By (26), this  relaxed condition is equivalent to  �  φ(p)dP ∗  λλλ(p) ≥  �  φ(p)dQ∗  λλλ(p)  (27)  for all φ(p) that are the maximum of finitely many linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' By approximation theory again, the above condition is  equivalent to the one holding for any continuous convex function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Omitted Proofs  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Proof of Lemma 2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Clearly, C is a subset of T (C|AC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Let λ ∈ (0, 1) and PˆY0|S,Y, PˆY1|S,Y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Now we introduce a Bernoulli random  variable B such that Pr(B = 0) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Finally, we define ˆYλ = BˆY1 +(1−B)ˆY0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' By definition, we have (S, Y) → X → ˆYλ  4If (16) has multiple optimal solutions, we always choose the one from {a1, · · · , ak}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Aleatoric and Epistemic Discrimination in Classification  so PˆYλ|S,Y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Moreover,  PˆYλ|S,Y = λPˆY0|S,Y + (1 − λ)PˆY1|S,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Hence, C is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Let λ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Assume P and ¯P achieve the maximal values of Proposition 1 under (αSP, αEO, αOAE) and (¯αSP, ¯αEO, ¯αOAE),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We define Pλ = λP + (1 − λ) ¯P , which satisfies the constraints of Proposition 1 with thresholds (λαSP + (1 −  λ)¯αSP, λαEO + (1 − λ)¯αEO, λαOAE + (1 − λ)¯αOAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Finally, since the objective function of Proposition 1 is a linear function, it  is equal to λFairFront(αSP, αEO, αOAE) + (1 − λ)FairFront(¯αSP, ¯αEO, ¯αOAE) under Pλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Proof of Theorem 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The proof relies on Theorem 3 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For simplicity, we write the conditional PˆY|S,Y as its corresponding  transition matrix P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Let µµµ = (Pr(S = 1, Y = 1), · · · , Pr(S = A, Y = C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The function (12) in our setting can be written  as:  pµµµ(ˆy) =  �  µ1,1P(1,1),ˆy  �  s,y µs,yP(s,y),ˆy  , · · · ,  µA,CP(A,C),ˆy  �  s,y µs,yP(s,y),ˆy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (28)  pµµµ(x) =  �  µ1,1dPX|S=1,Y=1  �  s,y µs,ydPX|S=s,Y=y  (x), · · · ,  µA,CdPX|S=A,Y=C  �  s,y µs,ydPX|S=s,Y=y  (x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (29)  Note that pµµµ(x) = g(x) due to Bayes’ rule (see (8) for the definition of g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' By Lemma 1, we can rewrite C in Definition 2 as  C =  �  P | PˆX|S,Y is more informative than P  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (30)  By Lemma 1 and Theorem 3, the above set is further equivalent to all transition matrices P ∈ T (C|AC) satisfying  C  �  ˆy=1  φ  �  µ1,1P(1,1),ˆy  �  s,y µs,yP(s,y),ˆy  , · · · ,  µA,CP(A,C),ˆy  �  s,y µs,yP(s,y),ˆy  � �  s,y  µs,yP(s,y),ˆy ≤ E [φ(g(X))]  (31)  for any function φ : ∆AC → R which is the maximum of finitely many linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Now we can write φ(p) =  maxi∈[k]  �  aT  i p  �  —we ignore the bias term because aT  i p+bi = (ai +bi1)T p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Then the inequality in (31) can be simplified  as  C  �  ˆy=1  max  i∈[k]  �  aT  i ΛΛΛµpˆy  �  ≤ E  �  max  i∈[k]{aT  i g(X)}  �  ,  (32)  where pˆy is the ˆy-th column of P and ΛΛΛµ = diag(µ1,1, · · · , µA,C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Finally, we can always normalize the above inequality  so that each ai ∈ [−1, 1]AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Proof of Theorem 2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We denote  f(P ) ≜  A  �  s=1  C  �  y=1  µs,yP(s,y),y,  g(P ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' a1, · · · , ak) ≜  C  �  ˆy=1  max  i∈[k]  �  aT  i ΛΛΛµpˆy  �  − E  �  max  i∈[k]{aT  i g(X)}  �  ,  F ≜ Ck ∩ {P ∈ T (C|AC) | SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Let Ft be the constraint set of P at the t-th iteration of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Note that F ⊆ Ft by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' If the algorithm  stops at the t-th iteration, then for any {ai | ai ∈ [−1, 1]AC, i ∈ [k]}, P t satisfies  g(P t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' a1, · · · , ak) ≤ 0,  Aleatoric and Epistemic Discrimination in Classification  which implies P t ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Consequently,  f(P t) = max  P ∈Ft f(P ) ≥ max  P ∈F f(P ) ≥ f(P t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  As a result, f(P t) = maxP ∈F f(P ) so P t is an optimal solution of FairFrontk(αSP, αEO, αOAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  If the algorithm never stops, consider any convergent sub-sequence of P t that converges to a limit point P ∗ ∈ T (C|AC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  To simplify our notation, we assume P t → P ∗ as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Since {Ft}t≥1 is non-increasing and they all contain F, there  exists a set F∗ such that  lim  t→∞ Ft = F∗,  F ⊆ F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Therefore, we have  f(P ∗) = lim  t→∞ f(P t) = lim  t→∞ max  P ∈Ft f(P ) = max  P ∈F∗ f(P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Since F ⊆ F∗, we have  f(P ∗) = max  P ∈F∗ f(P ) ≥ max  P ∈F f(P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  If P ∗ ̸∈ F, then there exists a (¯a1, · · · , ¯ak), such that g(P ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' ¯a1, · · · , ¯ak) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Let (a1,t, · · · , ak,t) be the output of Step  2 at t-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Since P ∗ ∈ Ft for all t, we have  g(P ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' a1,t, · · · , ak,t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (33)  By the optimality of (a1,t, · · · , ak,t), we have  g(P t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' a1,t, · · · , ak,t) ≥ g(P t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' ¯a1, · · · , ¯ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  (34)  Suppose that some sub-sequence of (a1,t, · · · , ak,t) converges to a vector (a∗  1, · · · , a∗  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For the sake of simplicity, we  assume (a1,t, · · · , ak,t) → (a∗  1, · · · , a∗  k) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' On the one hand, taking limit of t → ∞ on both sides of (34) leads to  g(P ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' a∗  1, · · · , a∗  k) ≥ g(P ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' ¯a1, · · · , ¯ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  On the other hand, taking limit of t → ∞ on both sides of (33) leads to  g(P ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' a∗  1, · · · , a∗  k) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Therefore,  0 ≥ g(P ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' a∗  1, · · · , a∗  k) ≥ g(P ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' ¯a1, · · · , ¯ak) > 0,  which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Therefore, P ∗ ∈ F and, as a result, we have  f(P ∗) = max  P ∈F∗ f(P ) ≥ max  P ∈F f(P ) ≥ f(P ∗) =⇒ max  P ∈F f(P ) = f(P ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Details on the Experimental Results  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Additional Experiments  In this section, we present additional experimental results to further support our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' First, we reproduce our ex-  perimental results on the German Credit dataset (Bache & Lichman, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We compare existing fairness interventions  with FairFront in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Our observation is consistent with those on the previous two datasets—the fairness-  accuracy curves given by SOTA fairness interventions, such as Reduction and FairProjection, are close to the  information-theoretic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  As previously demonstrated in Figure 1 and 3, the fairness-accuracy curves generated by Reduction and  FairProjection are close to FairFront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' To further evaluate the performance of these methods, we train a classifier that  approximates the Bayes optimal and feed it to Reduction and FairProjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' The results are shown in Figure 4,  which demonstrates that the (training) accuracy-fairness curves generated by Reduction and FairProjection can  approach FairFront when using the Bayes optimal baseline classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Aleatoric and Epistemic Discrimination in Classification  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Comparing existing fairness interventions with FairFront on the German Credit dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Dataset  Adult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use sex (female or male) as the group attribute and income (> 50K or <= 50K) as the target for  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use sex, hours-per-week, education-num, age, marital status, relationship status (husband or wife)  as the input features—we include the group attribute as an input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We group age into a total of 12 dis-  joint intervals: [0, 20), [20, 25), · · · , [65, 70), [70, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' we group hours-per-week into a total of 14 disjoint intervals:  [0, 10), [10, 15), · · · , [65, 70), [70, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  COMPAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use race (African-American or Caucasian) as the group attribute and is recid (recid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' or no recid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=') as the  target for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use race, age, c charge degree, sex, priors count, c jail in, c jail out as the input features—we  include the group attribute as an input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use the last two features by taking their difference to be their length of stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We remove entries where COMPAS case could not be found (is recid = -1) and entries with inconsistent arrest information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We also binarize sex and remove traffic offenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We quantize age the same way we do in the Adult dataset and quantize  length of stay by every 30 days and let 0 be a separate category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  German Credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use age (below or above 25 years old) as the group attribute and the credit column, which represents  whether the loan was a good decision, as the target for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use loan duration in month, credit amount, age,  number of existing credits at this bank, sex, credit history, savings, and length of present employment as input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We  include the group attribute age as an input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We group credit amount into three disjoint intervals: [0, 5000), [5000,  10000),[10000,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We group duration of loan into two categories: under 36 months and over 36 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Benchmark  Each benchmark method’s hyper-parameter values are provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Each point in Figure 1 for Baseline, EqOdds,  CalEqOdds, Reduction, LevEqOpp, and FairProjection is obtained by applying the obtained classifier to 10  different test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For the Adult dataset, we use Random Forest with n estimators=15, min samples leaf=3, criterion =  log loss, bootstrap = False as our baseline classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' for the COMPAS dataset, we use Random Forest with n estimators = 17  as our baseline classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' For the German Credit dataset, we use Random Forest with n estimators=100,min samples split  =2,min samples leaf=1 as our baseline classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' They are all implemented by Scikit-learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  EqOdds (Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use AIF360 implementation of EqOddsPostprocessing and the default hyper-  parameter setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  CalEqOdds (Pleiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use AIF360 implementation of CalibratedEqOddsPostprocessing and  the default hyper-parameter setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  German Credit  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  %)  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  Accuracy (  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  FairFront  Baseline  FairProjection  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Reduction  LevEqOpp  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  CalEqOdds  EqOdds  0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' Comparing Reduction and FairProjection with FairFront on the Adult (Left), COMPAS (Middle), and German Credit  (Right) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We train a baseline classifier that approximates the Bayes optimal and feed it into the two fairness interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' As  shown, their fairness-accuracy curves are very close to the FairFront in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Reduction (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use AIF360 implementation of ExponentiatedGradientReduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We vary the allowed fairness constraint violation ϵ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5, 1, 2, 5, 10, 15} for Adult dataset and ϵ ∈  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5, 1, 2, 5, 10, 15} for Adult with missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We vary ϵ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='001, 2, 5, 10, 15, 20, 25, 30, 35, 40} for  COMPAS to obtain a fairness-accuracy curve, and ϵ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5, 1, 2, 7, 8, 10, 15, 20, 25, 30} for COMPAS with 50%  missing values in the minority group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use ϵ ∈ {20, 50, 80, 95} for German Credit dataset and ϵ ∈ {5, 8, 10, 20, 23}  when using Bayes Optimal classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  LevEqOpp (Chzhen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use the Python implementation of LevEqopp from the Github repo in Alghamdi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We follow the same hyperparameters setup as in the original method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  FairProjection (Alghamdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  We use the implementation from the Github repo in Alghamdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' (2022)  and set use protected = True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We use Random Forest with n estimators = 17 as the baseline classifier to predict S from  (X, Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We set the list of fairness violation tolerance to be {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='075, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='085, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='09, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='095, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0} for  Adult dataset and {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0} for COMPAS dataset to obtain a fairness-accuracy  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content=' We set the list of fairness violation tolerance to be {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='15} on the German Credit dataset  experiment, and {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='05} when using a Bayes optimal baseline classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='  Adult  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='2  %)  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Accuracy  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='7  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='2  FairFront  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Baseline  FairProjection  Reduction  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='7  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='00  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='50  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='5  Max equalized odds (%)COMPAS  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='8  Accuracy  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='6  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='4  FairFront  Baseline  FairProjection  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='2  Reduction  0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Max equalized odds (%)German Credit  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='9  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='9  (%)  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='8  Accuracy (  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='8  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='7  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='7  FairFront  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='6  Baseline  FairProjection  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='6  Reduction  0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}
+page_content='0  Max equalized odds (%)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'}