In the stochastic linear contextual bandit setting there exist several minimax procedures for exploration with policies that are reactive to the data being acquired.

The unit ℓ2-spherein d-dimensions that is centered at the origin is denoted bySd 1. Additionally, given a pair of symmetric matricesA,B Rd, we write A B if and only if x (A B)x 0, x Rd. More linear algebra facts appear in AppendixE. Let V P be a subset of distributions indexed by the points in the hypercubeEd = { 1,1}d. For a number of facts from probability and statistics (both related and unrelated to exponential families),wereferthereadertoAppendixF.

We study the linear contextual bandit problem where an agent has to select one candidate from a pool and each candidate belongs to a sensitive group. In this setting,candidates' rewardsmaynotbedirectly comparable between groups,for example when the agent is an employer hiring candidates from different ethnic groups and some groups have a lower reward due to discriminatory bias and/or socialinjustice.

Variational Auto-Encoders (VAEs) and their extensions such asβ-VAEs have been shown to improve local alignment of latent variables with PCA directions, which can help to improve model disentanglement under some conditions.