---
title: 'Fairness Discrepancy Rate: Cross-Domain Insights'
url: https://www.emergentmind.com/topics/fairness-discrepancy-rate
type: topic
---

# Fairness Discrepancy Rate: Cross-Domain Insights

Fairness Discrepancy Rate (FDR) is a task-dependent discrepancy metric used to summarize how unevenly a system’s performance is distributed across clients, demographic groups, or competing agents. In federated unlearning, it denotes the normalized gap between the best-off and worst-off clients after an unlearning operation, computed from per-client post-unlearning performance on retained data [2510.07022]. In biometric verification, face morphing attack detection, and adaptive streaming, the same label refers to mathematically distinct quantities, including worst-case demographic error-rate gaps and the complement of Jain’s fairness index [2011.02395] [2111.12115] [1704.08535]. The term therefore does not identify a single canonical fairness functional; its meaning is fixed by the evaluation object, the operating point, and the fairness notion adopted in a given paper.

## 1. Terminological scope and cross-domain usage

The expression “Fairness Discrepancy Rate” is reused across several literatures, but the underlying fairness object varies substantially. In some settings it is a disparity measure, so lower values indicate greater fairness; in others it is a parity score, so higher values indicate greater fairness. This polarity inversion is a recurrent source of confusion.

| Setting | Definition sketch | Fairness direction |
|---|---|---|
| Federated unlearning | $\mathrm{FDR}=\frac{F_{\max}-F_{\min}}{F_{\max}}$ over clients | $0$ is perfect fairness |
| Biometric verification / S-MAD | $\mathrm{FDR}=1-[\alpha A+(1-\alpha)B]$ over group error gaps | $1$ is perfect fairness |
| DASH rate control | $\mathrm{FDR}(t)=\sqrt{1-J(t)}$ from Jain fairness index | $0$ is perfect fairness |

In federated unlearning, the quantity measures cross-client disparity after deleting information associated with a forgotten class. In biometric verification and morph attack detection, it measures how much group-specific error rates diverge at a common threshold. In DASH, it measures instantaneous bitrate inequality among competing clients [2510.07022] [2011.02395] [2111.12115] [1704.08535].

A direct implication is that FDR values are not portable across domains. An FDR of \(0.06\) in federated unlearning and an FDR of \(0.94\) in biometrics can both indicate comparatively fair behavior, because the metrics are normalized in opposite directions and summarize different observables.

## 2. Formal definition in federated unlearning

In “Federated Unlearning in the Wild: Rethinking Fairness and Data Discrepancy,” FDR is introduced for a federated unlearning session with \(K\) clients, where each client \(k\) has some post-unlearning performance or fairness measure \(F_k\) [2510.07022]. The paper defines
\[
F_{\max}=\max_{1\le k\le K} F_k,\qquad
F_{\min}=\min_{1\le k\le K} F_k,
\]
and then
\[
\mathrm{FDR}
=
\frac{F_{\max}-F_{\min}}{F_{\max}}
=
1-\frac{F_{\min}}{F_{\max}}.
\]

By dividing by \(F_{\max}\), the metric is a unit-less fraction in \([0,1]\). A value of \(0\) means perfect client-level fairness, because all clients have equal performance. A value near \(1\) means extreme disparity. The paper states that \(F_k\) can be a performance or fairness measure, such as accuracy on the remaining classes or a more sophisticated group-fairness score, but its experiments focus on per-client accuracy on the retained classes rather than the forgotten class [2510.07022].

This formulation operationalizes fairness as worst-case client spread. It does not average disparities across the federation; instead, it isolates the gap between the best-off and worst-off clients. That design makes the metric sensitive to whether an unlearning intervention disproportionately harms a small subset of uninvolved clients.

## 3. Computation and role in the experimental protocol

The paper specifies a direct post-unlearning workflow for computing FDR. First, one identifies the evaluation metric \(F_k\). In the reported experiments, this is per-client accuracy on the retained classes. After the unlearning procedure has converged, the final global model is tested on each client’s local private data, and
\[
F_k=\mathrm{Accuracy}_k(\text{retained classes})
\]
is computed for every client. The protocol then takes the clientwise maximum and minimum and forms
\[
\mathrm{FDR}=1-\frac{F_{\min}}{F_{\max}}.
\]
A low FDR indicates that unlearning did not make some clients much worse than others; a high FDR indicates a disproportionate drop for at least one client [2510.07022].

The metric is embedded in the paper’s Cross-Domain (Real-NonIID) benchmark. The benchmark includes Handwriting clients from MNIST10, SVHN, and USPS, and image-recognition settings based on CIFAR10\(\Leftrightarrow\)ImageNet with \(9\) shared classes and CIFAR100\(\Leftrightarrow\)ImageNet with \(65\) shared classes. Only the forgetting client retrains; other clients keep previous model parameters. The evaluation reports per-client accuracy on the forgotten class, per-client accuracy on the retained classes, global accuracy, and FDR computed on retained-class accuracies [2510.07022].

The experimental role of FDR is therefore not ancillary. It is part of the core protocol for deciding whether an unlearning method simultaneously removes the forgotten information and preserves equitable utility across the federation.

## 4. Motivation, fairness semantics, and related variants

The motivation for introducing FDR in federated unlearning is that prior work had focused on whether the forgotten class was removed and whether overall accuracy stayed high, but not on whether the burden of unlearning was distributed fairly across clients. The paper identifies two sources of unfairness. Exact schemes such as Delete-Retrain force all clients, including those who never held the forgotten data, to do a full retrain. Approximate interventions such as Neuron-Zeroing or weight-pruning can wipe out the forgotten class but also damage features on which other clients relied, causing steep drops in retained-class accuracy for some of them [2510.07022].

FDR was introduced to quantify precisely that cross-client disparity. Instead of collapsing the federation to a single global accuracy number, it flags the largest client-level gap after unlearning. The paper makes the corresponding design principle explicit:

> “When we unlearn for client i, we must ensure that no other client j sees a degradation that is disproportionately large, i.e. keep FDR below some tolerance threshold \(\delta\).” [2510.07022]

The same paper also mentions an average-pairwise version, sometimes called Average Pairwise Discrepancy (APDR), defined by
\[
\mathrm{APDR}
=
\frac{2}{K(K-1)}
\sum_{i<j}
\frac{|F_i-F_j|}{\max(F_i,F_j)}.
\]
Unlike the max–min FDR, APDR averages over all \(K(K-1)/2\) client pairs. The paper describes it as giving a smoother view of global fairness, but notes that it was used only in a brief ablation [2510.07022].

A useful conceptual distinction follows. FDR is a worst-case federation fairness summary, whereas APDR is an all-pairs smoothness summary. The former is more sensitive to outlier harm; the latter is more descriptive of overall spread.

## 5. Quantitative behavior in the federated unlearning benchmark

For the CIFAR10\(\Leftrightarrow\)ImageNet setting with \(9\) shared classes, averaged over \(10\) clients, the paper reports the following distilled comparisons among baseline methods and FedCCCU [2510.07022]:

| Method | Forgotten-class acc | Retained-class acc / FDR |
|---|---|---|
| Delete-Retrain | \(82.8\%\) \(\rightarrow\) partial | \(90.1\%\) / \(0.12\) |
| Relabel-Poison | \(83.7\%\) \(\rightarrow\) partial | \(90.4\%\) / \(0.15\) |
| Neuron-Zeroing | \(7.9\%\) | \(80.2\%\) / \(0.48\) |
| FedCCCU | \(16.6\%\) | \(89.2\%\) / \(0.06\) |

These numbers are used to support three distinct empirical points. First, Delete-Retrain and Relabel-Poison achieve only modest forgetting while still producing nontrivial client disparity, with FDR values on the order of \(0.12\) to \(0.15\). Second, Neuron-Zeroing drives forgotten-class accuracy close to zero but induces very large collateral damage, yielding an FDR near \(0.5\). Third, FedCCCU reduces forgotten-class accuracy to \(16.6\%\), described as a \(78\%\) drop from the original, while keeping FDR at \(0.06\), the lowest gap among the reported methods [2510.07022].

The paper further states that across all three benchmark splits—Handwriting, Image(9), and Image(65)—FedCCCU consistently achieves the smallest FDR while still providing strong unlearning. Within the paper’s framing, this is the central empirical significance of FDR: it exposes the trade-off between forgetting precision and fairness across the federation, and it distinguishes methods that preserve average utility from methods that preserve utility equitably.

## 6. Other formalizations and broader fairness-evaluation context

Outside federated unlearning, FDR has been defined differently. In biometric verification, Pereira and Marcel define
\[
\mathrm{FDR}(\tau)=1-[\alpha A(\tau)+(1-\alpha)B(\tau)],
\]
where \(A(\tau)\) is the maximum absolute difference in False Match Rate across demographic groups at threshold \(\tau\), \(B(\tau)\) is the corresponding maximum difference in False Non-Match Rate, and \(\alpha\in[0,1]\) weights the two error types. Here \(\mathrm{FDR}=1\) denotes perfect fairness, and \(\mathrm{FDR}=0\) denotes maximal discrepancy [2011.02395]. The face morphing attack detection literature adopts the same template, replacing FMR and FNMR by APCER and BPCER at a chosen operating point \(\tau\) [2111.12115].

In adaptive video streaming, TFDASH defines fairness discrepancy as
\[
\mathrm{FDR}(t)=\sqrt{1-J(t)},
\]
where \(J(t)\) is Jain’s fairness index over the instantaneous bitrates \(v_i(t)\) of \(N\) clients. Again, \(0\) denotes perfect fairness and values closer to \(1\) denote stronger disparity [1704.08535]. In fairness-aware maximum coverage, discrepancy is expressed through covered color counts \(p_i(T)\), especially the ratio
\[
g_2(T)=\max_{i,j}\frac{p_i(T)}{p_j(T)},
\]
with exact fairness at \(g_2(T)=1\) [2007.08069]. A different line of work on fairness-aware classification proposes Maximal Cumulative-ratio Disparity along varying prediction neighborhoods (MCDP), a worst-case local CDF-gap metric that the provided summary characterizes as a fairness discrepancy rate [2406.03255].

Broader fairness-evaluation research complicates any attempt to treat an FDR value as self-sufficient evidence. “When Fairness Metrics Disagree” shows that fairness assessments can vary significantly depending on the choice of metrics and introduces the Fairness Disagreement Index (FDI) to quantify inconsistency across fairness measures [2604.15038]. This suggests that FDR, in any of its domain-specific forms, is best interpreted as one component of a multi-metric fairness audit rather than as a universally decisive scalar.

A common misconception is therefore to regard “Fairness Discrepancy Rate” as a standardized quantity. The literature instead supports a narrower statement: FDR is a reusable label for discrepancy-based fairness summaries whose formal definition, operating range, and fairness polarity must be read from the surrounding task formulation.

Source: https://www.emergentmind.com/topics/fairness-discrepancy-rate