(674h) Mixed-Integer Formulations for Fair Classification

Classification models are typically used to separate datasets based on a set of descriptors; however, such models can also be used to make decisions (e.g., make go/no-go decisions in a project). In this context, fairness becomes a major issue because such decisions tend to affect multiple stakeholders and because descriptors might inadvertently introduce biases [1, 2, 3]. A recent application of fair classification models include allocation of financial loans (e.g., descriptor is credit score). In this work, we show how fair classification models can also be used to tackle problems of interest to the chemical engineering community. For instance, in a manufacturing facility, access to experimental lab equipment is simultaneously requested by different stakeholders to test a variety of products. The decision here is whether a sample should be tested or not (or when) given the importance of the sample (captured by descriptors) and given that there is limited lab equipment and budgets. Such decisions are often based on minimizing a loss function, which is an aggregate of the cost associated with false positives and false negatives. The model output is a threshold value (based on a linear combination of sample features) that activates the binary decision. The inherent degeneracy of such models can lead to decisions that prioritize testing in an unfair manner [4].

State of the art research in fairness in machine learning focuses on developing algorithms that do not violate state or federal anti-discrimination laws. This is achieved by adding either post-processing steps or additional constraints to the mathematical model to achieve properties such as demographic parity, equalized odds, and equal opportunity [1,2]. These conditions can be mutually exclusive and often lack a theoretical basis that connects them to the axiomatic view of fairness. In this work, we will present mixed-integer formulations that address the problem of fair classification from an axiomatic perspective [5]. In the field of game theory, the resource allocation problem has been viewed as a bargaining game between stakeholders. Nash [6] first provided an axiomatic approach to obtain solutions to the bargaining problem. These axioms include Pareto optimality, symmetry, affine invariance, and independence of irrelevant alternatives. Nash also proved that there exists an allocation scheme that satisfies these axioms (what is now known as the Nash solution). We observe that the ultimate goal of a fairness measure such as the Nash solution is to shape an allocation distribution in a desirable way. As such, the resource allocation problem can also be interpreted as a stochastic programming problem in which one seeks to find allocations that shape distribution of outcomes (in stochastic programming the outcome distribution is shaped by using a risk measure) [7, 8]. As in the case of fairness measures, axioms have been proposed in the stochastic programming literature to study the selection of suitable risk measures [9].

We present a theoretical analysis of the statistical properties of different fairness criteria introduced in binary classification models. We will also provide a case study of a manufacturing facility where a binary classification model is used to drive the decision of which samples should be tested. We will demonstrate the impact of using Nash solution on the allocation distribution of tests between the equipment and the overall testing efficiency (measured by false positives and false negatives). We will showcase how the Nash solution inherently captures this tradeoff between efficiency and fairness, and selects the optimal solution based on the axiomatic properties.

References:

[1] Gölz, P., Kahng, A. and Procaccia, A.D., 2019. Paradoxes in Fair Machine Learning. In Advances in Neural Information Processing Systems (pp. 8340-8350).

[2] Hardt, M., Price, E. and Srebro, N., 2016. Equality of opportunity in supervised learning. In Advances in neural information processing systems (pp. 3315-3323).

[3] Wattenberg, M., Viégas, F. and Hardt, M., 2016. Attacking discrimination with smarter machine learning. Google Research, 17.

[4] Sampat, A.M. and Zavala, V.M., 2019. Fairness measures for decision-making and conflict resolution. Optimization and Engineering, 20(4), pp.1249-1272.

[5] Moulin, H., 1991. Axioms of cooperative decision making (No. 15). Cambridge university press.

[6] Nash Jr, J.F., 1950. The bargaining problem. Econometrica: Journal of the Econometric Society, pp.155-162.

[7] Dowling, A.W., Ruiz-Mercado, G. and Zavala, V.M., 2016. A framework for multi-stakeholder decision-making and conflict resolution. Computers & Chemical Engineering, 90, pp.136-150.

[8] Hu, J. and Mehrotra, S., 2012. Robust and stochastically weighted multiobjective optimization models and reformulations. Operations research, 60(4), pp.936-953.

[9] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D., 1999. Coherent measures of risk. Mathematical finance, 9(3), pp.203-228.