Fair Projections as a Means Towards Balanced Recommendations

The goal of recommender systems is to provide to users suggestions that match their interests, with the eventual goal of increasing their satisfaction, as measured by the number of transactions (clicks, purchases, etc.). Often, this leads to providing recommendations that are of a particular type. For some contexts (e.g., browsing videos for information) this may be undesirable, as it may enforce the creation of filter bubbles. This is because of the existence of underlying bias in the input data of prior user actions.

Reducing hidden bias in the data and ensuring fairness in algorithmic data analysis has recently received significant attention. In this paper, we consider both the densest subgraph and the \(k\)-clustering problem, two primitives that are being used by some recommender systems. We are given a coloring on the nodes, respectively the points, and aim to compute a fair solution \(S\), consisting of a subgraph or a clustering, such that none of the colors is disparately impacted by the solution.

Unfortunately, introducing fair solutions typically makes these problems substantially more difficult. Unlike the unconstrained densest subgraph problem, which is solvable in polynomial time, the fair densest subgraph problem is NP-hard even to approximate. For \(k\)-clustering, the fairness constraints make the problem very similar to capacitated clustering, which is a notoriously hard problem to even approximate.

Despite such negative premises, we are able to provide positive results in important use cases. In particular, we are able to prove that a suitable spectral embedding allows recovery of an almost optimal, fair, dense subgraph hidden in the input data, whenever one is present, a result that is further supported by experimental evidence.

We also show a polynomial-time, \(2\)-approximation algorithm to the problem of fair densest subgraph, assuming that there exist only two colors and both colors occur equally often in the graph. This result turns out to be optimal assuming the small set expansion hypothesis. For fair \(k\)-clustering, we show that we can recover high quality fair clusterings effectively and efficiently. For the special case of \(k\)-median and \(k\)-center, we offer additional, fast and simple approximation algorithms as well as new hardness results.

The above theoretical findings drive the design of heuristics, which we experimentally evaluate on a scenario based on real data, in which our aim is to strike a good balance between diversity and highly correlated items from Amazon co-purchasing graphs and facebook contacts. We additionally evaluated our algorithmic solutions for the fair \(k\)-median problem through experiments on various real-world datasets.