Fair Multiple Decision Making Through Soft Interventions

subject ： Make fair multiple decisions through soft intervention

Soft intervention ： Make X=g(z)

Hard intervention ： Make X=x

Research background ： Previous studies mostly focused on a single decision-making model . However , In reality , There are usually multiple decision models in a system , All these models may contain a certain amount of discrimination , These discrimination may have been introduced by themselves , It may also be transferred by the upstream model .

for example , Consider two decision tasks Y1 and Y2, among Y1 It is used by the municipal government to allocate police resources to different locations ,Y2 Used by a local bank to make personal loan decisions . Due to the historical isolation of housing , The ethnic composition of the community varies according to geographical location , stay y1 There may also be direct racial discrimination . therefore , Some locations will allocate more police resources than others , So as to produce more criminal arrest records . therefore , When Y2 When using criminal arrest records , Some ethnic groups will be unfairly disadvantaged when obtaining loans .

If decision models interact , Even if we know how to build a fair model for each task , This is not a simple problem . Deploying a new equity model will change the distribution of attribute variables affected by its decisions , And inherited discrimination . therefore , Subsequent models based on the original distribution may not perform well in terms of accuracy and fairness . On the other hand , If we build a fair model one by one in chronological order , Deploy one model at a time and collect output data before building the next model , Then the time required to build all the models may not be acceptable for some applications .

Ideally , We hope to establish a fair model for all decision-making tasks . therefore , This paper carries out relevant research , A method of learning multiple fair classifiers at the same time and only one static training data set is proposed ..

Research questions ： Problems arising from the interaction of decision models

Research methods ： In our approach , The deployment of the new decision-making model is considered as soft intervention in decision-making , Its influence can be inferred as the distribution after intervention . By quantifying fairness as the causal impact of protected attributes on all decisions , In the case of hard intervention on protected attributes and soft intervention on decisions , We formulate multiple decisions as a fair classification of single constrained optimization problems .

We propose a method to learn multiple fair classifiers from static training data sets . We regard the deployment of each classifier as soft intervention , The distribution after deployment is inferred as the distribution after intervention . We use proxy function to smooth the loss function and fairness constraint , The fair classification problem is expressed as a constrained optimization problem . Besides , We prove theoretically that combining multiple decision models in optimization will not introduce additional substitution errors . Experiments using synthetic and real datasets show , Our method is better than the single training method .

contribution ：   1. This is the first work to study fair multiple decision making , The feature distribution may change due to the deployment of the decision model .

              2. Our method provides a general way to incorporate fairness constraints into the general classification formula , In this way, we can easily use ready-made classification models and optimization algorithms .

              3. Causal reasoning allows us to jointly train all decision models from one data set .

              4. Because our method is based on SCM Of , All based on SCM The concept of fairness , Including the total effect 、 Direct and indirect discrimination 、 Counterfactual fairness and pc fair , Can be naturally applied to our problem formula .

              5. The theoretical results show that , We don't need to worry about the extra losses caused by multiple proxy functions . Through experiments on synthetic data sets and real data sets , We prove that our method is always better than the method of building a fair classifier for each decision .

Classification error             

Proxy functions         

Fair constraint         

experiment ： For synthetic data , We manually define a causal relationship , Pictured 2S、X1、X2、Y1、y2 Graph of five variables . Then define a conditional probability table on its parent attribute for each attribute , According to the conditional probability, each attribute is sampled in topological order to generate data . For real-world data , We use adult data sets , And use tetrads [25] Implemented in PC Algorithm constructs cause and effect diagram .

Data sets ：

This paper designs an evaluation process that simulates the deployment process of the actual model . The data set is randomly divided into training and test data sets . We deploy and evaluate the learning classifiers according to their topological order . First deploy the first classifier h1, And evaluate on the original test data set . Then generate Y1 Prediction decision of , Use the cause and effect diagram of prediction decision to regenerate the sequential values of subsequent variables and the next classifier h2 The true value of . Last , Based on the regenerated data h2 To assess the .

In terms of training , Our approach ( It is called joint method ) Formulate optimization problems for training data , Learn all classifiers at the same time . We also considered a simplified version of our method ( It is called serial method ), It learns classifiers in topological order , Similar to the deployment process . Each classifier uses only the immediate parent of the tag . After learning each classifier , Think of it as soft intervention , Thus infer the distribution after the intervention and use it to train the subsequent classifier . We compare our method with the baseline method ( Refers to a separate method ), Each classifier uses direct parents to learn from the training data respectively .

All classifiers are implemented as empirical risk minimization classifiers , It uses logistic Proxy functions . For unconstrained 、 Separation and serial methods , Each classifier is studied separately as a convex optimization problem . therefore , We use CVXPY package [7] Directly solve unconstrained / Constrained convex optimization problems . For joint law , Because the objective function and constraint are non convex , We add constraints as penalty terms to the objective function , And USES the PyTorch[22] adopt Adam The optimizer optimizes it . In this paper, we study the non convex optimization Adam The convergence of the algorithm .

result ： In the test , The performance of serial method and joint method is consistent , But a separate method cannot guarantee the right h2 The fairness of .

We observed that , Even if we use training data for testing to avoid any generalization errors , But in these pairs produced by a separate method , Yes 71.43% Of h2 Beyond the range [-0.05,0.05], Thus violating the fairness criterion . contrary , The classifiers generated by serial and joint methods are fair in the test .

The advantage of the joint method is that all classifiers can be adjusted at the same time to obtain better overall performance . This is not shown in the current experiment , It may be due to the small scale of the problem . In the future work , We will study whether the joint method is better than the serial method on a larger problem .

summary ： In this paper , Let's assume Markov model . When Markov hypothesis is not satisfied , Causal models are called semi Markov models , It faces the problem of identifiability , That is, the causal effect may not be uniquely identified from the observed data . lately ,[6] The author introduced calculus to systematically identify the causal effects of soft intervention . It will be our further work to extend our method to the recognizable cases in semi Markov models .

Our research can benefit any organization or system that uses computer algorithms to make important decisions , Especially for large-scale systems with multiple decision tasks . By adopting our method , Decision makers can simultaneously establish multiple decision models from a historical data set , And ensure that all decision models are fair after deployment . Our research can also help users involved in the system , Especially those disadvantaged users , Prevent them from accepting biased decisions .