Proving Algorithmic Fairness I  Introduction
Overview
This semester (Fall 2016) I’m beginning an independent study course with Dr. Aws Albarghouthi of UWMadison’s CS department. His research focuses on the “art and science of program analysis.”
During this course, we plan to investigate an idea called “algorithmic fairness.” It is interesting for both its technical content and cultural relevance. Here’s an example illustrating algorithmic fairness’s place in the big picture:
Suppose you’re hiring employees for your business, and you’ve decided to use a fancy machine learning algorithm to sift through piles of application documents and select candidates for interviews^{1}. In hiring, it is ethically correct (and legally necessary) to be unbiased with respect to protected classes—e.g., race or religion. Can you guarantee that your algorithm is unbiased?
In order to get me acquainted with this topic, Dr. Albarghouthi pointed me to a paper he’s been collaborating on, titled “Proving Algorithmic Fairness”^{2}. There is already a body of work devoted to the topic of algorithmic fairness^{3}, but this paper introduces the following innovations:

It presents the notion of fairness with respect to a population model. A population model can be thought of as a “random person generator”, drawing from a joint distribution over the space of demographic traits. In our hiring example, it would describe the population of possible applicants. The population model is a useful construct in that it provides a standard by which to judge fairness. For example, if the population of possible applicants has a certain ethnic composition, we can judge the fairness of an algorithm by comparing the ethnic composition of its output with the population’s ethnic composition. Previous literature on the subject judges fairness with respect to particular datasets instead; it is argued that these datasets may possibly be biased themselves. It is envisioned that social scientists in government agencies or NGOs would prepare these models.

The paper introduces a newfangled integration method for its computation of probabilities. It’s described as a symbolic volumecomputation algorithm that uses an SMT solver. The paper’s new method is preferred over more typical Markov Chain Monte Carlo integrators because it guarantees a lower bound for the integral; this guarantee allows us to prove fairness or unfairness, rather than express a mere statistical confidence in fairness or unfairness.

The concepts of the paper are packaged into a fairness verification tool called FairSquare, which is tested against a set of benchmark population models and classifier algorithms.
Proving Fairness
Suppose we have a binary classifier algorithm \(\mathcal{P}\), whose input is a person \(\vec{v}\) (a vector of that person’s traits) and whose output is “true” or “false”, “hired” or “not hired”, etc. A proof of fairness for algorithm \(\mathcal{P}\) consists of showing that the following inequality holds:
where \(v_s \) is an entry of \(\vec{v}\) indicating whether the person belongs to a protected class—e.g., a particular religion or ethnicity—and \(\epsilon < 1\) is some agreedupon or mandated standard of fairness, with smaller \(\epsilon))s implying stricter fairness requirements. In English: if we have two people who are equal in every way except their status in a protected class, we must show that the algorithm is equally likely to approve both people (within some threshold).
Note that it isn’t sufficient for the algorithm to simply ignore protected class data; correlations between protected class data and hiring criteria can lead to unbalanced outcomes even if ethnicity or religion are simply “left out” of a hiring algorithm.
In proving this inequality, it is useful to eliminate the conditional probabilities via the identity \(P[A  B] = \frac{P[A \land B]}{P[B]}\), giving the inequality
The probabilities of intersections are easier to directly compute than conditional ones.
In order to prove the inequality, it suffices to find a lower bound \(> 1  \epsilon\) on the LHS; hence it suffices to find lower bounds on the probabilities in the numerator, and upper bounds on the probabilities in the denominator. Similarly, in order to disprove the inequality (i.e. prove unfairness), it suffices to find an upper bound on the numerator and a lower bound on the denominator. This pursuit of upper and lower bounds lends itself to the paper’s Symbolic Volume Integration scheme, which is proven to converge to exact integrals in a monotonically increasing manner.
In upcoming posts, I will dig into the content of this paper in more detail. Topics will include the paper’s Symbolic Volume Integration scheme and a description of the FairSquare tool’s performance on some benchmarks. \( \blacksquare\)

Using an algorithm is a good idea not only for the obvious speed considerations, but also for consistency; Daniel Kahneman’s research in behavioral psychology has shown that even simple classifier algorithms are more reliable for candidate selection (among other things) than human judgment. See Thinking, Fast and Slow, Part III, chapter 21: Intuitions vs. Formulas. ↩

Albarghouthi, D’Antoni, Drews, Nori; in submission as of this writing (20160901). ↩

See these for example: Fairness Through Awareness; Certifying and Removing Disparate Impact ↩