Artificial-intelligence systems are becoming ubiquitous in society, yet their predictions typically inherit biases with respect to protected attributes such as race, gender, or age. Classical fairness notions, most notably Statistical Parity (SP), demand that predictions be independent of the protected attributes, but are overly restrictive when these attributes influence mediating variables that are considered business necessities. Recent causal formulations relax SP by distinguishing allowed from not-allowed causal paths and by complementing SP with Predictive Parity (PP), requiring the predictor to replicate the legitimate influence of business-necessities. Existing path-based definitions are mainly practical when applied to categorical attributes. This paper introduces a new framework for fairness in structural causal models that is tailored to continuous protected attributes. We formalize SP and PP through path-specific partial derivatives, establish conditions under which these criteria coincide with prior causal definitions, and characterize when a fair predictor, one that satisfies SP along not-allowed paths while achieving PP along allowed paths, exists. Building on this theory, we propose a fair tuning algorithm that either constructs such a predictor or, when not possible, allows for a trade-off between SP and PP. We present experiments on simulated and real data to evaluate our proposal, compare it with previously proposed methods, and show that it performs better when PP is considered.

Figure 1: A summary of several outlier statistics recorded from ImageNet validation set on ViT. We use zero-based indexing for dimensions. BERTRecall from Figure 1 that all the outliers are only present in hidden dimensions #123, #180,4 #225, #308, #381, #526, #720 (with the majority of them in #180, #720). In Figures 9 and 10 we show more6 examples of the discovered self-attention patterns for attention heads #3 and #12 ( hidden dim #1807 and #720, respectively). We also show self-attention patterns in attention heads and layers which are8 not associated with the outliers in Figures 11 and 12, respectively.9

Organization: This supplementary material is presented in a format parallel to the main paper. The section numbers and titles are consistent with the main paper. But, here we also add one new section: Section 10 where we describe the societal impacts and possible negative impacts of the paper. Similarly, the Theorem numbers are consistent with the main paper, but we also have several additional theorems and lemmas which were not included in the main paper. GAN-type Objective for KLEstimation Let f be a discriminator, f: X IR. Let p(x) and q(x) be two probability density functions defined over the space X.

In this section, we describe the details of the network architectures used in Sec. 4 and 5. We mainly used 4 GPUs (NVIDIAV100; 16GB) for the experiments in Sec. 4 and 5 and it took about 4 hours per seed (in the case of 3M steps). Actually, we conducted exhaustive evaluations through the enormous experiments, and we hope our empirical observations and recommendations help the practitioners to explore the explosive configuration space. Adam Adam Learning rate (policy) 1e-4 5e-5 3e-4 3e-4 Learning rate (value) 1e-4 1e-2 3e-4 3e-4 Weight initialization Uniform Xavier Uniform Xavier Uniform Xavier Uniform Initial output scale (policy) 1.0 1e-4 1e-2 1e-2 Target update Hard - Soft (5e-3) Soft (5e-3) Clipped Double QFalse - True True Table 7: Details of each network architecture. We refer the original implementations of each algorithm which is available online [23, 14, 48, 27, 42].

We have shown experimentally that our method is effective in a variety of domains; however, other problem domains may require additional hyperparameter tuning, which can be expensive.

Branch-and-bound Our work is Bbased on branch-and-boundBound (BaB), a powerful framework for -2.0 0.5 Lower bound = min(-2.0,