Exclusion-Based Balancing Techniques

• Exclusion-Based Balancing is a family of techniques that attains equilibrium by ruling out or de-emphasizing states, transitions, samples, or assignments to prevent over-concentration.
• It finds broad application across quantum resource theories, mixture-of-experts routing, simulation-based inference, clustering, metric search, and queueing, offering rigorous operational guarantees.
• Challenges arise in managing induced distortions and ensuring that balancing mechanisms maintain causal accuracy and unbiased optimization in diverse computational settings.

Exclusion-based balancing denotes a family of techniques in which balance is obtained by ruling out, suppressing, or selectively de-emphasizing states, assignments, transitions, or samples that would otherwise concentrate mass, load, or information. In the literature, the idea appears in several distinct forms: as an operational principle for quantum resource theories and exclusion games, as a routing controller for mixture-of-experts systems, as a regularization principle for conservative simulation-based inference, as a balancing penalty in clustering, as proxy-conditioned filtering for balanced cohort construction, as a partitioning principle in metric search, and as a structural mechanism in exclusion processes and balancing games (1908.10347).

1. Quantum operational formulations

In the quantum resource theory of measurement informativeness, a POVM $\mathbb{M}=\{M_a\}$ is uninformative when each effect is proportional to the identity, $M_a=q(a)\,\mathds{1}$ for all $a$. The central quantifier is the weight of informativeness, defined as the minimal informative weight needed to reproduce $\mathbb{M}$ as a convex mixture of an informative POVM and an uninformative one. It admits the closed form

$${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$

hence $0\le {\rm WoI}(\mathbb{M})\le 1$. Its operational meaning is furnished by the state exclusion game, in which a referee samples $x$ from $p(x)$, sends $\rho_x$, and the player must output a state label that was not sent; failure occurs when the output equals the actual label. For free or uninformative measurements the best error is $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$, whereas for a fixed measurement $M_a=q(a)\,\mathds{1}$0 the minimum exclusion error is

$M_a=q(a)\,\mathds{1}$1

The exact correspondence is

$M_a=q(a)\,\mathds{1}$2

with a tight lower bound for every $M_a=q(a)\,\mathds{1}$3. The same paper defines exclusion entropy and single-shot excludible information, and proves for quantum-to-classical channels $M_a=q(a)\,\mathds{1}$4 that

$M_a=q(a)\,\mathds{1}$5

This yields a three-way correspondence between a weight-based resource quantifier, the operational task of state exclusion, and an information-theoretic quantity, in parallel with the robustness/discrimination/accessible-information correspondence. The exclusion errors over all games also form a complete set of monotones for measurement simulability:

$M_a=q(a)\,\mathds{1}$6

The authors further conjecture that the weight-based/exclusion-based correspondence is generic across convex quantum resource theories (1908.10347).

A related operational distinction appears in parity-oblivious random exclusion codes. In POREC, Alice receives $M_a=q(a)\,\mathds{1}$7, Bob receives $M_a=q(a)\,\mathds{1}$8, and Bob must output $M_a=q(a)\,\mathds{1}$9 such that $a$0, under parity-obliviousness constraints forbidding information about multi-digit parities. For prime $a$1, the optimal classical and preparation-noncontextual bound is $a$2. For the first nontrivial case $a$3, the exact qubit optimum is $a$4, exceeding the noncontextual bound $a$5. The paper emphasizes that parity-oblivious exclusion displays a quantum advantage where parity-oblivious retrieval does not, making exclusion a distinct operational probe of preparation contextuality; for $a$6 it also yields the semi-device-independent witness $a$7 (Roy et al., 9 May 2026).

2. Routing control, conservative inference, and compensated sample exclusion

In large-scale Mixture-of-Experts training, exclusion-based balancing appears as a load-control mechanism that modifies routing decisions rather than the training objective. Loss-Free Balancing replaces the standard auxiliary balance loss

$a$8

with an expert-wise additive bias applied before top-$a$9 routing. For token $\mathbb{M}$0 and expert $\mathbb{M}$1, the router uses $\mathbb{M}$2 for selection, but the output weight remains $\mathbb{M}$3. The routing rule is

$\mathbb{M}$4

Biases are updated from previous-batch loads via

$\mathbb{M}$5

The method is designed to prevent routing collapse and avoid interference gradients. On DeepSeekMoE models, the reported results are: for 1B, perplexity $\mathbb{M}$6 and $\mathbb{M}$7 for loss-controlled balancing versus perplexity $\mathbb{M}$8 and $\mathbb{M}$9 for Loss-Free Balancing; for 3B, $${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$0 and $${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$1 versus $${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$2 and $${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$3. Under softmax gating, the reported comparison is $${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$4 and $${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$5 versus $${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$6 and $${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$7. The same work contrasts this controller with Expert Choice, which it criticizes for future token leakage and violation of causal language-modeling constraints (Wang et al., 2024).

In simulation-based inference, balancing is used to discourage posterior approximations from excluding plausible parameters too aggressively. The basic balance condition for binary neural ratio estimation is

$${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$8

with penalty

$${\rm WoI}(\mathbb{M}) = 1-\sum_a \lambda_{\min}(M_a),$$9

The paper extends this regularizer from NRE to any method that evaluates a posterior density by defining

$0\le {\rm WoI}(\mathbb{M})\le 1$0

This yields Balanced Neural Posterior Estimation and Balanced Contrastive Neural Ratio Estimation. The balance term is exactly a $0\le {\rm WoI}(\mathbb{M})\le 1$1-divergence between the target class marginal and the classifier marginal in the binary case. Empirically, the balanced variants tend to produce conservative posteriors on SLCP, Weinberg, Spatial SIR, Lotka–Volterra, and Two Moons, with Two Moons remaining overconfident for all methods; balancing usually does not significantly reduce informativeness at higher simulation budgets (Delaunoy et al., 2023).

Selective backpropagation furnishes a third learning-theoretic variant. K-ABENA excludes a fraction of low-loss “minor” observations from the backward pass, defining the minor set $0\le {\rm WoI}(\mathbb{M})\le 1$2 and sampling from it with the defensive-mixture design

$0\le {\rm WoI}(\mathbb{M})\le 1$3

The canonical estimator is Horvitz–Thompson:

$0\le {\rm WoI}(\mathbb{M})\le 1$4

which is design-unbiased, while the self-normalized practical variant has bias $0\le {\rm WoI}(\mathbb{M})\le 1$5. The paper proves an $0\le {\rm WoI}(\mathbb{M})\le 1$6 non-convex convergence guarantee for SGD, with bias floor $0\le {\rm WoI}(\mathbb{M})\le 1$7 for a biased estimator and the classical rate recovered for the unbiased HT form. Its main caution is that uncompensated loss-based selection, including OHEM and SBP, admits no stationary point at any minimizer where its selection bias is bounded away from zero. Under $0\le {\rm WoI}(\mathbb{M})\le 1$8 class imbalance, the reported test AUCs are $0\le {\rm WoI}(\mathbb{M})\le 1$9–$x$0 for uncompensated variants, $x$1 for full-batch SGD, and $x$2 for the compensated estimator at identical $x$3 compute savings (Bonbhel, 7 Jul 2026).

3. Balanced partitioning, filtering, and causal structure

In clustering, exclusion-based balancing is implemented through exclusive lasso. For an indicator matrix $x$4, the regularizer becomes

$x$5

which behaves like $x$6 when $x$7 is the size of cluster $x$8. The paper proves that $x$9 is minimized under $p(x)$0 exactly when $p(x)$1 for all $p(x)$2, so the exclusive-lasso term measures the balance degree of a clustering result. This yields the balanced $p(x)$3-means objective

$p(x)$4

and the balanced min-cut objective

$p(x)$5

The reported experiments on nine datasets state that balanced $p(x)$6-means achieves the best ACC and NMI on all datasets in the reported tables, and balanced min-cut consistently outperforms classical min-cut, Ratio Cut, Normalized Cut, and MinMax Cut; the parameter range $p(x)$7 is highlighted in the sensitivity discussion (Chang et al., 2014).

Balanced filtering via disclosure-controlled proxies addresses a different deployment constraint: sensitive group membership is unavailable or prohibited at deployment time. The method learns a proxy $p(x)$8 from a small labeled sample, forms proxy groups with conditional sensitive-group distributions $p(x)$9, and stacks them into $\rho_x$0. The fundamental condition is

$\rho_x$1

where $\rho_x$2 is the uniform group distribution. Sampling probabilities are then set by $\rho_x$3 and normalized by the largest value. The proxy must also be $\rho_x$4-disclosive:

$\rho_x$5

for all groups and proxy values. The paper gives an oracle-efficient decision-tree construction based on cost-sensitive classification and reports that QP Decision Tree and the relaxed $\rho_x$6-proxy tend to perform best out of sample, while larger $\rho_x$7 improves balance and small $\rho_x$8 budgets can still yield noticeably more balanced filtered sets (Deng et al., 2023).

A contrasting line of work shows that balancing is not, in general, equivalent to removing an undesired dependence. For fairness and robustness, the jointly balanced distribution is

$\rho_x$9

which makes $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$0 and $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$1 marginally independent in the idealized limit. The paper derives sufficient conditions under which the balanced risk minimizer is risk-invariant and optimal, but emphasizes that $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$2 does not imply $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$3, nor does balancing generally preserve the original causal factorization. It presents failure modes involving another confounder $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$4, entangled features $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$5, causal tasks in which balancing creates new dependencies, and cases where balancing one marginal worsens another. It also shows that balancing can interfere with regularization, because a regularizer justified under $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$6 may suppress informative features under $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$7 (Schrouff et al., 2024).

4. Geometric and combinatorial exclusion mechanisms

In exact metric search, exclusion-based balancing takes the form of partitioning rules that eliminate whole classes of data from consideration. The Ptolemaic partitioning mechanism starts from a pivot pair $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$8 and parameter $P_{\rm err}^{\rm C}(\mathcal E)=\min_x p(x)$9, and defines static data classes and dynamic query regions using the Ptolemaic lower bound. For the three-way partition, with $M_a=q(a)\,\mathds{1}$00, $M_a=q(a)\,\mathds{1}$01, and $M_a=q(a)\,\mathds{1}$02, the data are divided into $M_a=q(a)\,\mathds{1}$03 and excluded by query-time inequalities (Connor, 2022).

Subset	Definition	Query-time exclusion
$M_a=q(a)\,\mathds{1}$04	$M_a=q(a)\,\mathds{1}$05 and $M_a=q(a)\,\mathds{1}$06	$M_a=q(a)\,\mathds{1}$07 or $M_a=q(a)\,\mathds{1}$08
$M_a=q(a)\,\mathds{1}$09	$M_a=q(a)\,\mathds{1}$10 and $M_a=q(a)\,\mathds{1}$11	$M_a=q(a)\,\mathds{1}$12 or $M_a=q(a)\,\mathds{1}$13
$M_a=q(a)\,\mathds{1}$14	$M_a=q(a)\,\mathds{1}$15 and $M_a=q(a)\,\mathds{1}$16	$M_a=q(a)\,\mathds{1}$17 or $M_a=q(a)\,\mathds{1}$18

The paper states that this mechanism is always better than pivot or hyperplane partitioning, is weaker than Hilbert exclusion but cheaper to compute, and can be combined with Hilbert exclusion because the excluded sets are not nested. It also notes that $M_a=q(a)\,\mathds{1}$19 recovers traditional hyperplane exclusion and that values around $M_a=q(a)\,\mathds{1}$20 to $M_a=q(a)\,\mathds{1}$21 were often best empirically (Connor, 2022).

A more abstract geometric formulation appears in balancing games on unbounded sets. For a finite $M_a=q(a)\,\mathds{1}$22, a set $M_a=q(a)\,\mathds{1}$23 is $M_a=q(a)\,\mathds{1}$24-closed if $M_a=q(a)\,\mathds{1}$25 and $M_a=q(a)\,\mathds{1}$26 imply that either $M_a=q(a)\,\mathds{1}$27 or $M_a=q(a)\,\mathds{1}$28. The set

$M_a=q(a)\,\mathds{1}$29

is $M_a=q(a)\,\mathds{1}$30-closed, as are its translates. In the game $M_a=q(a)\,\mathds{1}$31, Pusher chooses $M_a=q(a)\,\mathds{1}$32, Chooser chooses $M_a=q(a)\,\mathds{1}$33, and $M_a=q(a)\,\mathds{1}$34; Chooser wins exactly when there exists a $M_a=q(a)\,\mathds{1}$35-closed set $M_a=q(a)\,\mathds{1}$36 with $M_a=q(a)\,\mathds{1}$37. The main theorem states that if $M_a=q(a)\,\mathds{1}$38 contains no parallel vectors, $M_a=q(a)\,\mathds{1}$39 is closed and $M_a=q(a)\,\mathds{1}$40-closed, and $M_a=q(a)\,\mathds{1}$41 is an extreme point of $M_a=q(a)\,\mathds{1}$42, then some translate of $M_a=q(a)\,\mathds{1}$43 contains $M_a=q(a)\,\mathds{1}$44 and lies in $M_a=q(a)\,\mathds{1}$45. This geometric rigidity determines the value of a special balancing game on an unbounded orthant and also yields a corollary: if $M_a=q(a)\,\mathds{1}$46 is not a power of $M_a=q(a)\,\mathds{1}$47, the $M_a=q(a)\,\mathds{1}$48-subsets of a $M_a=q(a)\,\mathds{1}$49-set can be colored Red and Blue so that complementary $M_a=q(a)\,\mathds{1}$50-sets have distinct colors and every point belongs to the same number of Red and Blue sets (Bárány et al., 2 Dec 2025).

5. Exclusion dynamics in statistical mechanics and queueing

In nonequilibrium statistical mechanics, balancing under exclusion is realized by networked exchange rules or by occupation-dependent exclusion counting. A balance network of ASEP subsystems connects equiprobable exclusion-process subsystems by bidirectional links and a reservoir. If subsystem $M_a=q(a)\,\mathds{1}$51 has particle number $M_a=q(a)\,\mathds{1}$52, the steady state is exactly

$M_a=q(a)\,\mathds{1}$53

with

$M_a=q(a)\,\mathds{1}$54

After identifying $M_a=q(a)\,\mathds{1}$55 and $M_a=q(a)\,\mathds{1}$56, the normalization becomes a grand partition function and the mean occupation is

$M_a=q(a)\,\mathds{1}$57

a Fermi–Dirac or Langmuir-isotherm form. The theorem is stated to be independent of network structure and to unify Langmuir kinetics, multiple lanes, and finite reservoirs under the same balance property (Ezaki et al., 2012).

Multiple exclusion statistics generalizes Haldane counting to spatially correlated states that can be excluded by more than one particle. The paper introduces the ansatz

$M_a=q(a)\,\mathds{1}$58

and defines the exclusion spectrum

$M_a=q(a)\,\mathds{1}$59

the average number of excluded states per particle at occupation $M_a=q(a)\,\mathds{1}$60. The limiting values are $M_a=q(a)\,\mathds{1}$61 and $M_a=q(a)\,\mathds{1}$62, so the spectrum interpolates between isolated-particle exclusion and dense-state exclusion. In the non-correlated limit $M_a=q(a)\,\mathds{1}$63, the theory reduces to Haldane fractional exclusion statistics and Wu’s distribution. For square-lattice $M_a=q(a)\,\mathds{1}$64-mers, $M_a=q(a)\,\mathds{1}$65, $M_a=q(a)\,\mathds{1}$66, and the reported multiple-exclusion constants increase from $M_a=q(a)\,\mathds{1}$67 at $M_a=q(a)\,\mathds{1}$68 to $M_a=q(a)\,\mathds{1}$69 at $M_a=q(a)\,\mathds{1}$70, with Monte Carlo agreement for occupation and exclusion spectrum across $M_a=q(a)\,\mathds{1}$71 to $M_a=q(a)\,\mathds{1}$72 (Riccardo et al., 2019).

Queueing theory provides a stochastic scheduling version. In the prioritising exclusion process, high-priority customers overtake low-priority customers by the bulk rule $M_a=q(a)\,\mathds{1}$73 at rate $M_a=q(a)\,\mathds{1}$74, with high-priority arrivals at rate $M_a=q(a)\,\mathds{1}$75, low-priority arrivals at rate $M_a=q(a)\,\mathds{1}$76, and service at rate $M_a=q(a)\,\mathds{1}$77. The queue behaves like an $M_a=q(a)\,\mathds{1}$78 system in length, but internally develops a jam of high-priority customers near service. In the unbounded phase, the service-frame density profile is exactly

$M_a=q(a)\,\mathds{1}$79

where $M_a=q(a)\,\mathds{1}$80 solves

$M_a=q(a)\,\mathds{1}$81

The paper distinguishes a finite-jam phase $M_a=q(a)\,\mathds{1}$82 from an infinite-jam phase $M_a=q(a)\,\mathds{1}$83, derives waiting-time formulas through Little’s law, and shows that the limiting cases interpolate between the ordinary $M_a=q(a)\,\mathds{1}$84 result and strict-priority-like behavior (Gier et al., 2014).

6. Structural themes and recurring limitations

Across these literatures, exclusion-based balancing repeatedly appears as a way of turning a balance requirement into an operational rule on what may be selected, simulated, routed, retained, or excluded. In the most explicit quantum formulation, balance is exact and dual: robustness corresponds to discrimination and accessible information, whereas weight corresponds to exclusion and excludible information. In POREC, exclusion and retrieval cease to be operationally equivalent once parity-obliviousness is imposed. In metric search, exclusion is a query-time elimination rule for whole partitions. In MoE routing, exclusion takes the milder form of suppressing overloaded experts through bias shifts before top-$M_a=q(a)\,\mathds{1}$85 selection rather than through an auxiliary loss (1908.10347).

A recurrent limitation is that exclusion is only benign when its induced distortion is understood or corrected. In sample exclusion, uncompensated selection can remain biased near the true minimizer, whereas Horvitz–Thompson correction restores the full gradient in expectation. In simulation-based inference, balancing is used precisely because unbalanced classifiers can yield overconfident posteriors that exclude plausible parameters. In data balancing for fairness or robustness, the warning is sharper: balancing $M_a=q(a)\,\mathds{1}$86 and $M_a=q(a)\,\mathds{1}$87 is not the same as removing an undesired path in the causal graph, and it can create new dependencies or interfere with regularization. In MoE, the main alternative criticized in the same balancing context is Expert Choice, which is described as achieving perfect load balance at the cost of future token leakage (Bonbhel, 7 Jul 2026).

This suggests a broad but technically consistent taxonomy. One class comprises exact correspondences, where exclusion determines an operational quantity or a stationary law. A second class comprises control mechanisms, where exclusion is implemented through biases, penalties, or sampling schemes to stabilize load or conservativeness. A third class comprises cautionary results, showing that exclusion or balancing can be misleading when it is treated as a heuristic surrogate for causal intervention, unbiased optimization, or stronger geometric tests. The surveyed literature therefore presents exclusion-based balancing not as a single algorithmic template, but as a recurrent design principle whose validity depends on whether the excluded mass, paths, or samples are handled by a provable operational, probabilistic, or geometric argument (Schrouff et al., 2024).