Entropy-Regularized Fairness Metric

• The metric family uses entropy to normalize probability distributions and weight vectors to quantify dispersion, inequality, or deviation from fairness benchmarks.
• It is applied across various domains such as TU-cooperative games, secure ISAC, and fair ERM, each incorporating specific benchmark and normalization strategies.
• The methods leverage entropy measures—like Rényi divergence, generalized entropy, and Shannon entropy—to regularize, constrain, or evaluate fairness in optimization problems.

Searching arXiv for the cited papers to ground the article in current arXiv metadata. “Entropy-regularized fairness metric” denotes several distinct constructions rather than a single canonical object. In the cited literature, entropy appears as a direct fairness functional, as a regularizer on auxiliary weight variables, as an inequality index inside constrained empirical risk minimization, and as a centered discrepancy between group distributions. In one line of work on TU-cooperative games, fairness is a worst-case Rényi-divergence gap over core allocations relative to a benchmark solution concept (Bonchis et al., 2012). In fairness-aware secure ISAC, the phrase refers more specifically to an entropy-regularized Jain-fairness formulation over user weights $\boldsymbol{\mu}$, where fairness is assessed on weighted legitimate-user SINRs rather than on entropy itself (Boroujeni et al., 15 Jul 2025). In fair empirical risk minimization, generalized entropy is used as a hard fairness constraint on classifier-induced benefits (Jin et al., 2022). Related work on entropy-regularized optimal transport provides a statistical foundation for Sinkhorn-based group-discrepancy measures, although it does not discuss fairness explicitly (Bigot et al., 2017). A further network-science formulation defines fairness as the complement of normalized Shannon entropy over QoS-transformed all-pairs accessibility scores (Ren et al., 9 Aug 2025). Taken together, these formulations show that the term is best understood as a family of entropy-centered fairness notions whose mathematical role depends on the optimization domain and the object being normalized.

1. Conceptual scope and recurring design pattern

A recurring pattern across the literature is: choose an object that can be interpreted as a probability distribution or normalized weight vector, apply an entropy or entropy-related functional, and use the result to quantify dispersion, inequality, or deviation from a fairness benchmark. In the cooperative-game formulation, any nonnegative vector $X$ is viewed as a probability distribution by normalization, and fairness is evaluated on normalized core imputations relative to a reference distribution such as the uniform benchmark $U$ or the normalized Shapley value $Sh$ (Bonchis et al., 2012). In secure ISAC, entropy is computed on the simplex-constrained user-weight vector $\boldsymbol{\mu}$, while the Jain-style fairness term is applied to the weighted SINR vector $\{\mu_k\rho_k^L\}_{k=1}^K$ (Boroujeni et al., 15 Jul 2025). In generalized-entropy fair ERM, the underlying normalized object is the distribution of nonnegative benefits $b_i$ across individuals (Jin et al., 2022). In the network-imbalance construction, normalized pairwise QoS scores $p_{u,v}$ are fed into Shannon entropy (Ren et al., 9 Aug 2025).

This diversity is accompanied by a terminological caution. Several of the cited works explicitly do not use “entropy-regularized” in the modern sense of a primary objective of the form “main term plus entropy penalty.” The TU-cooperative-game paper instead defines an entropy-based fairness evaluation metric and studies its worst-case value over feasible core allocations (Bonchis et al., 2012). The network-imbalance paper likewise defines an entropy-derived fairness/imbalance functional rather than an optimization regularizer (Ren et al., 9 Aug 2025). By contrast, the secure ISAC formulation does place an entropy term inside a composite objective, but that entropy acts on the user-weight allocation rather than directly on rates, secrecy, or beamforming powers (Boroujeni et al., 15 Jul 2025). This suggests that the phrase is structurally descriptive rather than semantically uniform.

2. Rényi-divergence worst-case fairness in TU-cooperative games

In the TU-cooperative-game setting, the game is $\Gamma=(N,v)$, with $N$ the player set and $X$0 the coalition value function. Fairness is assessed over core imputations, and the central construction is a pessimistic fairness index: choose a benchmark solution concept $X$1, compare each feasible allocation $X$2 to $X$3 via Rényi divergence, and then maximize this discrepancy over the feasible set (Bonchis et al., 2012). The main definition is

$X$4

where, for distributions $X$5, $X$6, and $X$7,

$X$8

The Rényi entropy of order $X$9 is

$U$0

with the Shannon limit

$U$1

The paper treats rationality as core membership and discusses several benchmarks: the uniform distribution $U$2 as a strict egalitarian reference, the normalized Shapley value as a marginalist benchmark, and the egalitarian solution of Dutta–Ray / Arin–Iñarra as a possible reference concept (Bonchis et al., 2012). When the benchmark is uniform,

$U$3

so worst-case fairness against $U$4 is equivalent to minimum-entropy core allocation. This is the clearest entropy-based interpretation in the paper: $U$5 The abstract states that this parametric family is related to the Cowell–Kuga generalized entropy indices in welfare economics, and the paper explicitly frames the construction as a fairness analogue of price of anarchy: benchmark a designated fair solution, optimize over rational outcomes, and quantify the worst deterioration (Bonchis et al., 2012).

The paper also gives a bridge between entropy and divergence for nonuniform benchmarks. For a distribution $U$6, define

$U$7

Lemma 3 states

$U$8

When $U$9 is uniform, $Sh$0, so entropy differences and divergence differences coincide exactly (Bonchis et al., 2012). That identity is central to the paper’s interpretation of fairness as entropy deficit relative to complete equality.

Computationally, the paper proves NP-completeness for deciding whether $Sh$1 for induced subgraph games with nonnegative weights, for any $Sh$2 (Bonchis et al., 2012). To approximate the worst-case value, it analyzes the ReverseGreedy algorithm and derives additive guarantees in terms of packing constants $Sh$3 and $Sh$4. For induced subgraph games, a structural lemma yields $Sh$5, making the additive constants explicit. The paper also studies an alternative algorithm based on biased orientations and reports a three-player example in which strictly egalitarian worst-case fairness exceeds marginalist worst-case fairness relative to the Shapley benchmark (Bonchis et al., 2012).

3. Entropy-regularized Jain-fairness in secure ISAC

The secure ISAC formulation is narrower and more literal in its use of entropy regularization. The baseline problem optimizes beamformers $Sh$6 and artificial noise $Sh$7 to maximize secrecy-oriented performance under communication, sensing, and power constraints. The fairness issue identified by the authors is that conventional QoS constraints of the form $Sh$8 tend to be “just satisfied,” allowing the optimizer to exploit stronger channels while weaker users remain marginalized (Boroujeni et al., 15 Jul 2025). Fairness is therefore defined across users’ legitimate communication quality, measured through the legitimate SINRs $Sh$9.

The fairness score itself is Jain-style rather than entropic: $\boldsymbol{\mu}$0 with

$\boldsymbol{\mu}$1

and equality $\boldsymbol{\mu}$2 iff the effective SINRs are equal across users (Boroujeni et al., 15 Jul 2025). The entropy term enters through the composite outer objective

$\boldsymbol{\mu}$3

where

$\boldsymbol{\mu}$4

The paper states that $\boldsymbol{\mu}$5 “encourages diversity in the weight allocation,” but the formula is exactly $\boldsymbol{\mu}$6, so the prose and the algebraic sign are somewhat in tension; reimplementation must preserve the printed objective (Boroujeni et al., 15 Jul 2025).

A central technical point is that entropy is not taken over rates or SINRs. Entropy acts on the simplex-normalized weight vector $\boldsymbol{\mu}$7, whereas fairness acts on the weighted SINR vector $\boldsymbol{\mu}$8 (Boroujeni et al., 15 Jul 2025). The legitimate-user SINR is

$\boldsymbol{\mu}$9

and for PSK symbols $\{\mu_k\rho_k^L\}_{k=1}^K$0. A normalization constant

$\{\mu_k\rho_k^L\}_{k=1}^K$1

rescales the throughput term to the unit interval so that it can be combined with the fairness score (Boroujeni et al., 15 Jul 2025).

Fairness enters the optimization at the outer weight-update level, not directly in the inner beamforming/AN subproblem. The thresholded fairness requirement is

$\{\mu_k\rho_k^L\}_{k=1}^K$2

and the penalty reformulation is

$\{\mu_k\rho_k^L\}_{k=1}^K$3

The paper identifies three fairness-related coefficients: the tradeoff parameter $\{\mu_k\rho_k^L\}_{k=1}^K$4, the entropy coefficient $\{\mu_k\rho_k^L\}_{k=1}^K$5, and the penalty coefficient $\{\mu_k\rho_k^L\}_{k=1}^K$6, together with the fairness threshold $\{\mu_k\rho_k^L\}_{k=1}^K$7 (Boroujeni et al., 15 Jul 2025). The edge cases are explicit: for $\{\mu_k\rho_k^L\}_{k=1}^K$8, all weight is assigned to the user with highest SINR; for $\{\mu_k\rho_k^L\}_{k=1}^K$9, fairness is maximized by equalizing effective weighted SINRs, yielding

$b_i$0

Algorithmically, the fairness term is handled in an outer simplex-constrained gradient update, while the inner weighted sum-rate beamforming and AN problems are solved via an accelerated quadratic transform with a non-homogeneous bound (Boroujeni et al., 15 Jul 2025). The paper gives the gradient

$b_i$1

followed by simplex projection and trust-region interpolation (Boroujeni et al., 15 Jul 2025). The paper reports simulation values $b_i$2 and $b_i$3, and claims improvements in average secrecy rate, average data rate, and beam gain, but does not isolate fairness-specific ablations such as a sweep over $b_i$4 or a direct comparison with and without entropy regularization (Boroujeni et al., 15 Jul 2025).

4. Generalized entropy as a fairness constraint in empirical risk minimization

In fair ERM, generalized entropy is not an auxiliary smoothing term but the fairness criterion itself. For a nonnegative benefit vector $b_i$5 with mean $b_i$6, the generalized entropy index is

$b_i$7

The paper notes that $b_i$8 is the Theil index and states that “the more positive $b_i$9, the more sensitive $p_{u,v}$0 to the inequalities of high income distribution” (Jin et al., 2022).

The fairness application depends on a benefit mapping from prediction outcomes to nonnegative values. Extending the mapping of Speicher et al., the paper uses

$p_{u,v}$1

This yields

$p_{u,v}$2

Fairness is then inequality in the classifier-induced benefit vector $p_{u,v}$3 (Jin et al., 2022). The main constrained learning problem is

$p_{u,v}$4

where

$p_{u,v}$5

The paper solves a randomized version over distributions $p_{u,v}$6, with Lagrangian

$p_{u,v}$7

but presents generalized entropy primarily as a hard fairness constraint rather than as a standalone regularizer-only objective (Jin et al., 2022).

A key reason generalized entropy is attractive in this context is additive decomposability. For a partition into groups $p_{u,v}$8,

$p_{u,v}$9

where

$\Gamma=(N,v)$0

and $\Gamma=(N,v)$1 is the between-group term with explicit formulas for $\Gamma=(N,v)$2, $\Gamma=(N,v)$3, and $\Gamma=(N,v)$4 (Jin et al., 2022). This decomposition supports the paper’s claim that generalized entropy can simultaneously represent individual fairness and group fairness. The paper further proves that if a classifier satisfies equal prediction, then the between-group term $\Gamma=(N,v)$5, while also emphasizing that controlling generalized entropy does not generally control equalized odds or equal opportunity (Jin et al., 2022).

The theoretical contribution is a set of fairness generalization bounds. Extending generalized entropy to the population distribution,

$\Gamma=(N,v)$6

the paper proves that if $\Gamma=(N,v)$7 is bounded, then the extended index satisfies additive decomposability and the Pigou–Dalton transfer principle for all $\Gamma=(N,v)$8 (Jin et al., 2022). It then gives explicit $\Gamma=(N,v)$9 bounds for

$N$0

with constants depending on $N$1, $N$2, and $N$3, and a sharper result that additionally depends on VC dimension $N$4 and empirical risk $N$5 (Jin et al., 2022). Practically, the paper draws two lessons: smaller $N$6 is often preferable, and larger $N$7 improves fairness generalization bounds, although excessively large $N$8 may flatten the unfairness signal.

5. Entropy-regularized optimal transport as a fairness discrepancy foundation

The optimal-transport paper does not discuss fairness explicitly, but it provides a rigorous finite-space framework for building an entropy-regularized discrepancy between group distributions. On a finite metric space $N$9, with $X$00, transport polytope

$X$01

and cost matrix $X$02, the entropy-regularized transport cost is

$X$03

where

$X$04

The centered version, called the Sinkhorn loss in the paper, is

$X$05

The paper proves that

$X$06

and that $X$07 as $X$08 (Bigot et al., 2017).

The fairness-oriented interpretation, which the paper itself does not claim normatively, is that group-conditional empirical distributions can be compared through the centered Sinkhorn loss: $X$09 This provides a full-distribution discrepancy rather than a scalar parity gap, and the cost matrix $X$10 encodes the geometry of the outcome space (Bigot et al., 2017). The paper’s principal value for such use is statistical. It establishes differentiability of the regularized OT cost,

$X$11

where $X$12 is an optimal dual pair, and derives CLTs for both the raw regularized OT cost and the centered Sinkhorn loss in one-sample and two-sample settings (Bigot et al., 2017).

The null behavior of the centered loss is especially important if it is used as a fairness-audit statistic for equality of group distributions. At equality, the gradient vanishes,

$X$13

so the null asymptotics are second order: $X$14 and $X$15 converge to chi-square mixtures rather than Gaussian limits (Bigot et al., 2017). The paper also proposes bootstrap procedures, noting that the ordinary bootstrap is valid under the alternative but fails under first-order degeneracy, where a corrected Babu bootstrap is required in practice. This makes the paper a statistical foundation for entropy-regularized OT discrepancy measurement on discretized fairness problems rather than a fairness framework by itself.

6. Entropy-derived imbalance and cross-framework issues

The network-imbalance framework provides a different entropy-based fairness construction. For an undirected graph $X$16, the shortest-path hop count is

$X$17

This raw QoS variable is mapped through a tunable sigmoid

$X$18

with threshold $X$19 and steepness $X$20. Summing over ordered node pairs gives

$X$21

Shannon entropy is then

$X$22

The paper states $X$23, defines $X$24 when $X$25, and interprets low imbalance as high functional fairness in the sense of uniform end-to-end connection experiences (Ren et al., 9 Aug 2025).

This formulation is explicitly not an entropy-regularized optimization objective. Entropy is the aggregation mechanism, and unfairness is the complement of normalized entropy (Ren et al., 9 Aug 2025). The paper proves boundedness, continuity and differentiability with respect to the parameters $X$26 and $X$27, and characterizes $X$28 for connected graphs with finite nonzero $X$29 as equivalent to equality of all shortest-path lengths across ordered node pairs; for nontrivial connected graphs, this yields the complete graph $X$30 as the unique topology with zero imbalance (Ren et al., 9 Aug 2025). It also proves that imbalance is not monotonic under edge addition or deletion, providing a counterexample on $X$31, and argues that low imbalance may arise either from topological symmetry or from extreme path compression in structurally heterogeneous networks such as BA scale-free graphs (Ren et al., 9 Aug 2025).

Across the cited literature, three recurrent issues emerge. First, entropy-based fairness is benchmark-sensitive or representation-sensitive: the fairness value depends on the chosen benchmark distribution $X$32 in TU games (Bonchis et al., 2012), on the benefit encoding $X$33 in fair ERM (Jin et al., 2022), on the cost matrix $X$34 and regularization level $X$35 in OT (Bigot et al., 2017), on the simplex weights $X$36 and fairness threshold $X$37 in secure ISAC (Boroujeni et al., 15 Jul 2025), and on the QoS parameters $X$38 in network imbalance (Ren et al., 9 Aug 2025). Second, “entropy-regularized” and “entropy-based” should not be conflated. The cooperative-game and network papers are entropy-based but not regularized in the conventional Lagrangian sense [(Bonchis et al., 2012); (Ren et al., 9 Aug 2025)], whereas the ISAC paper is more accurately described as an entropy-regularized Jain-fairness formulation (Boroujeni et al., 15 Jul 2025). Third, these entropy-centered notions are not interchangeable with standard parity constraints. The fair-ERM paper is explicit that controlling generalized entropy has no specific impact on equalized odds or equal opportunity (Jin et al., 2022), and the OT paper likewise does not claim that distributional discrepancy alone resolves the normative content of fairness (Bigot et al., 2017).

In that sense, an entropy-regularized fairness metric is best understood not as a single metric but as a methodological family. Its common feature is the use of entropy, Rényi divergence, generalized entropy, or relative entropy to quantify concentration, inequality, or discrepancy after an application-specific normalization step. Its differences arise from what is normalized, where entropy is inserted in the optimization pipeline, and whether fairness is treated as worst-case deviation, outer-loop regularization, hard constraint, or centered distributional discrepancy [(Bonchis et al., 2012); (Boroujeni et al., 15 Jul 2025); (Jin et al., 2022); (Bigot et al., 2017); (Ren et al., 9 Aug 2025)].