Certifying Some Distributional Fairness with Subpopulation Decomposition

Certifying Some Distributional Fairness with Subpopulation Decomposition

Verify some distributional fairness using subgroup decomposition

Background : Extensive efforts have been made to understand and improve machine learning model fairness based on different fairness metrics, especially in high-risk domains such as health insurance, education, hiring decisions, etc.

Problems and flaws : The performance of end-to-end machine learning models lacks verifiable fairness .

Method and Model :

On a given data distribution, validation fairness of a trained ML model is formulated as an optimization problem based on model performance loss bounds on the distribution of fairness constraints within a bounded distribution distance of the training data distribution.

Contribute :

1. A general fairness verification framework is proposed, and instantiated verification with sensitive offsets and general offsets .
2. Decompose the source data distribution into analyzable subgroup distributions, solve the subproblems by proving their convexity, and thus solve the model optimization problem.
3. Experiments prove that the verifiability of the model is strict in the case of sensitive offsets, and it is strict in the case of general offsets.extraordinaryof.
4. The framework can flexibly integrate additionalunbiased constraint, and the result will be more stringent.
5. The proposed verifiable fairness bounds are compared with existingAdaptive Distribution Robust BoundsA comparison shows that the former is more stringent.

1 Introduction

Flaws in previous work :

1. Previous work mainly focuses on regularization training, disentanglement, duality, low-order matrix decomposition, distribution alignment and other methods to improve the fairness of ML.

2. Some work on verifiable fairness characterization has been done on ML, but there is a problem: training an end-to-end model on a randomly given data distribution, this model lacks verifiable fairness in predicting results.

3. The ML model that the existing literature on fairness focuses on is to train the model on a (non-)balanced data distribution, and evaluate the performance of the model through the existing fairness evaluation method in the measurable target domain, so the fairness The evaluation depends only on the choice of evaluation method, and does not take into account the validity of the method.

Verifiable Fairness : On a fairness-constrained test distribution Q, where Q is within a bounded distance from the training distribution P, define verifiable fairness as the worst-case upper bound on the model's prediction loss .

Base Rate Conditionsas a fairness constraint on the test distribution Q.

sensitive shifting Sensitive shift : The cascade distribution of sensitive attributes and labels can change.

general shifting : everything including the conditional distribution of non-sensitive attributes can be changed.

Group Fairness : Measuring the independence between sensitive features and model predictions. Separation [separation] indicates that given the target label, the sensitive feature is statistically independent of the model prediction. Sufficiency [sufficiency] indicates that a given model prediction, sensitive features are statistically independent of the target label. That is, group fairness requires sensitive attributes to be independent of target labels and model predictions.

Individual fairness : similar input features will produce similar output results.

This paper differs from previous work :

1. Validation fairness considers the performance of end-to-end ML models, not the level of representation learning.
2. Fairness is defined and verified based on a distribution of fairness constraints.
3. For any black-box model trained on a given random data distribution, verifiable fairness can be computed.

Question 1: How does the benchmark interest rate conditional constraint encode the fairness constraint distribution?

2. Verifiable fairness based on fair constraint distribution

Definition 1 Benchmark interest rate : Given a distribution $P$ , sensitive attribute value$s$ relative to label$The\; base\; rate\; of\; y$ is:$S$ is the sensitive attribute, [S] is the set of values ​​that the sensitive attribute can take,$s$ is a sensitive attribute value of a sensitive attribute. $Y$ is the prediction result of the model,$y$ is the sample label,$X$ is the sample feature. The benchmark interest rate of a test sample is [sensitive attribute characteristic$X_s$The value is $s$ , the predicted result is$Probability\; of\; y$ ]:

Definition 2. Fair benchmark interest rate distribution : if and only if in a distribution generated by the benchmark interest rate, for any two samples $i$ and$j$ , both have the same predicted label$y$ , and both have the sensitive attribute$For\; a\; certain\; attribute\; value\; in\; S$ , their corresponding benchmark interest rates are equal, and this distribution is called a fair benchmark interest rate distribution:

Demographic Parity : Group fairness evaluation index

2.1 Verifiable fairness

Data generation model : $X_o$Represents non-sensitive attribute features, $X_s$Represents sensitive attribute characteristics, $Y$ represents the sample label.

Verification fairness with general offset : $P$ is the training set distribution,$\rho$ is the test set distribution$Q$ andP \mathcal PThe distribution distance bound between$P.$For all generated and training set distributions at distribution distanceρ \rhoThe test set distribution within the range of$\rho$ , the fairness proof value is: the maximum value of the upper bound of the loss value in all test set distributions.

In real scenarios, the model training set is always limited by data management and data collection, so there is always inherent unfairness in the trained model. Assuming that the test set we construct is ideally fair, we hope that our model will not encode the bias generated during training when testing, so the performance of the model on the fair constraint distribution indicates its inherent unfairness.

Verifiable Fairness for Sensitive Offsets : In order to avoid the inherent unfairness of the model during training, a new constraint is added under the verification fairness of general offsets. $P_{yes,}$From $Q_{yes,}$is $P$ andQQThe subgroup of$Q$$s$与$y$ division].

To constrain the test set distribution $Q$ is not too skewed towards sensitive attributes$X_s$, so an additional constraint term is added to the sensitive offset:

$loss$ in each group and each category:

The above loss can be transformed into ε-DP and ε-EO :

Group parity (Demographic parity, DP): The difference in the predicted probability of predicting two different groups as positive classes.

Equalized odds: The difference between false-positive rates between groups or the difference between true-positive rates between groups. The smaller the difference, the fairer the model.

3. Fairness certification framework

3.1 Subgroup Decomposition [Core]

Hellinger distance : Measures the distance between two distributions. The value range is [0, 1], the larger the more relevant

Overall optimization problem :

Subgroup optimization problem :

3.2 Verifiable Fairness with Sensitive Offset

3.3 Verifiable Fairness with General Offset

Loss calculation after subgroup decomposition :

appendix

Scalar: A scalar is a single number (integer or real), unlike most other objects studied in linear algebra (usually arrays of numbers). Scalars are usually denoted by italic lowercase letters, for example: $x$ , a scalar is equivalent to the one defined in Python

x = 1

Vector (vector): A vector represents a set of ordered numbers. We can find each individual number through the index in the order. Vectors are usually represented by bold lowercase letters, for example: x \bf $x$ , each element in the vector is a scalar, the i-th element in the vector is represented by $x_i$, and the vector is equivalent to a one-dimensional array in Python

import numpy as np
#行向量
a = np.array([1,2,3,4])

Matrix (matrix): A matrix is ​​a two-dimensional array, each element of which is determined by two indices ( $A_{i,j}$), the matrix is ​​usually represented by bold italic capital letters, for example: $ \boldsymbol X$. We can think of the matrix as a two-dimensional data table, each row of the matrix represents an object, and each column represents a feature. Defined in Python as

import numpy as np
#矩阵
a = np.array([[1,2,3],[4,5,6],[7,8,9]])

Tensor: An array of more than two dimensions. Generally speaking, the elements in an array are distributed in a regular grid of several-dimensional coordinates, which is called a tensor. If a tensor is a three-dimensional array, then we need three indices to determine the position of the elements ( $A_{i,j,k}$), tensors are usually represented by bold capital letters, for example: $\mathbf{X}$

mport numpy as np
#张量
a = np.array([[[1,2],[3,4]],[[5,6],[7,8]]])

bound constraint to find the range

worst case boundary

How to calculate the r of each test sample, when r<bound is certified

Within the bound range, the prediction result must be greater than or equal to the worst case, so it has the nature of verification

Verifiable properties: In the prediction results, how many samples are in line with the user-specified bound prediction R.

data distribution