The Geometry of Algorithmic Stability: A Hodge Theoretic View on Structural vs. Statistical Instability
Authors: Karen Sargsyan
Venue: Accepted by TMLR ()

Abstract: Algorithmic stability—the robustness of predictions to training data perturbations—is fundamental to reliable machine learning. We propose a unified mathematical framework that rigorously distinguishes between two sources of instability: structural inconsistency and statistical variance. We formalize structural inconsistency using Combinatorial Hodge Theory, characterizing it as cyclical flows (Condorcet cycles) on a graph of hypotheses. This framework reveals that methods like inflated operators and regularization specifically target these structural obstructions, while methods like bagging primarily address statistical variance. We provide direct empirical validation through three key experiments. First,  in a controlled setting with engineered Condorcet cycles (pure structural instability), inflated operators achieve perfect stability while bagging fails, confirming the core distinction. Second, we validate on a standard digit classification task that structural obstructions are negligible ($||C_{cycle}|| \approx 2.3 \times 10^{-16}$, machine precision), explaining the empirical dominance of variance-reduction methods. Third, we demonstrate that significant structural obstructions naturally emerge in fairness-constrained model selection on real-world data ($||C_{cycle}|| = 0.857$, approximately $10^{15}$ times larger), providing a topological characterization of the instability arising from incompatible objectives.

Published in Transactions on Machine Learning Research (01/2026)
The Geometry of Algorithmic Stability: A Hodge Theoretic View on Structural vs. Statistical Instability
Karen Sargsyan
karen.sarkisynn@@aal.oo
Institute of Chemistry, Academia Sinica
Reviewed on OpenReview: https: // openreview. net/ forum? id rd=
Abstract
Algorithmic stability the robustness predictions to training data perturbations fun- damental to reliable machine learning. We propose a unified mathematical framework that rigorously distinguishes between two sources of instability: structural inconsistency and statistical variance. We formalize structural inconsistency using Combinatorial Hodge The- ory, characterizing it as cyclical flows (Condorcet cycles) framework reveals that methods like inflated operators and regularization specifically tar- get these structural obstructions, while methods like bagging primarily address statistical variance. We provide direct empirical validation through three key experiments. First, in a controlled setting with engineered Condorcet cycles (pure structural instability), inflated op- erators achieve perfect stability while bagging fails, confirming the core distinction. Second, validate standard digit classification task that structural obstructions are negligible (l]Ceyctel R x 10-16, machine precision), explaining the empirical dominance of variance- reduction methods. Third, we demonstrate that significant structural obstructions naturally emerge in caineee-constraineed model selection on real-world data (1|Ceyetel 0.857, approx- imately 10t5 times larger), providing a topological characterization the instability arising from incompatible objectives.
graph of hypotheses. This
1 Introduction
The generalization performance of machine learning models is deeply connected to their stability the ro- bustness of outputs to perturbations in the training data (Bousquet Elisscoff 2002). Stability matters not only for training algorithms but also for any data-driven selection procedure: choosing among pre-trained models, selecting from ensembles, or aggregating predictions. Methods to enhance stability have been devel- oped across these contexts, including bagging (Breiman 1996; Soloff et al. 2024). regularization (Bousquet
& Elisseeff 2002), and more recently, 'inflated' operators (Adrian et al. 2024 Liang et al. 2025
While effective, these mothods appear disconnected, and it is often unclear which method is appropriato for a given source of instability. Moreover, the classical stability framework assumes i.i.d. data from a single distribution, but modern applications face distribution shift, domain adaptation, and multi-objective learning where this assumption fails. This fragmentation raises fundamental question: Can rigorously distinguish the sources of instability and unify the approaches to address them across both single- and multi-distribuutinna settings?
We argue the answer yes, provided by the tools Combinatorial Hodge Theory applied to the geometry of preferences. We propose that instability arises from two distinct sources: statistical variance (sensitivity of the learning process to noise or near-ties) and structural inconsistency (fundamental conflicts in the data preferences). We can couceptualize structural inconsistency by viewing subsets of the data as "voters' and the potential models as 'candidates.'
Example 1.1 (Running Example: The Ranking Cycle). Consider selecting the best model among three candidates (A,,C)). Suppose subset of the data (voter group 1) strongly suggests A is better than B

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(A > B). A second subset suggests B C, and a third suggests C A. This creates cycle: A> B> C> B>C> A. No single global ranking can satisfy all local preferences. An algorithm forced to choose single winner will be highly unstable in this scenario.
This scenario is the well-known Condorcet Paradox. We demonstrate that these paradoxes, which can be rigorously characterized by the cyclical component in Hodge decomposition (Lim 2020 Jiang et al. 2011]. represent structural instability. This leverages ostablished connections between social choice theory and
algebraic topology (Chichilnisky 1980: Baryshnikov 1993).
While we find that structural inconsistencies are rare in standard supervised learning tasks explaining the empirical success of mothods like bagging-we demonstrate that they arise naturally in domains with conflicting objectives, such as fairness-aware machine learning. Our framework provides mathematical characterization of the inherent. instability in these scenarios.
Our framework formalizes the stability problem as one of preference aggregation under metric constraints. We show that:
The Hodge dccomposition of ordinal preferences characterizes the mathematical conditions that determine the source of instability. A non-zero cyclical component (Cayote 0) indicates structural inconsistency, while Coyele 0 indicates instability is primarily due to statistical variance.
2. Structural inconsistency requires structural solutions. Inflated operators provide obstruction reso- lution (target space enlargement), while regularization can provide obstruction prevention.
Statistical variance is effectively addressed by variance reduction techniques like bagging. Crucially, we demonstrate that bagging ineffective against pure structural instability.
4. We demonstrate that the framework naturally generalizes beyond the standard single-distribution setting to encompass distribution shift, domain adaptation, and multi-objective learning. The stan- dard algorithmic stability framework (i.id. samples from single distribution) emerges special case-the restriction of the preference sheaf to a single point in distribution space. This gener- alization provides a rigorous mathematical foundation for understanding instability arising from heterogeneous data sources or conflicting distributional objectives.
This framework provides mathematical unification, revealing deep connections between algorithmic stabil- ity, social choice theory, and algebraic topology, and provides theoretical insight into selecting appropriate stabilization methods.
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Related Work
Our work sits at the intersection of algorithmic stability, topological data analysis, and social choice theory.
2.1
Foundations of Algorithmic Stability
The foundational work [Bousquct & Elisscoff (2022) established the link between algorithmic stability and generalization. The standard paradigm often addresses stability through statistical variance reduction, canonically exemplified by bagging (Breiman 1996), whose assumption stability properties have been recently formalized (Soloff et al. 2024).
More recently, research has locused on the instability inherent in selection procedures like the argmax. Inflated operators (Adrian et al.) 2024 Liang et al.) 2025) address this by returning a stable set of near- optimal solutions, motivated largely by the metric scnsitivity of the loss function under near-ties. Our work provides a unifying theoretical framework that reinterprets these methods as strategies for handling topological obstructions in the underlying data preferences.
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2.2
Topological Data Analysis in Machine Learning
The application of Topological Data Analysis (TDA) to machine learning is growing field ([Wasserman, 2018 Hensel et al. 2021 Zia et al. 2024)). Much of this work focuses on the geometry of the parameter space or the loss landscape. For instance, persistent homology and Morse theory have been used to identify obstructions to optimization, such as the structure local minima and saddle points [Barannikov et al. 2020).
Crucially, our work analyzes the geometry of the preference space how data induces rankings over the hypothesis space. The obstructions identify not features of the optimization landscape, but rather fundamental inconsistencies in the problem definition itself (conflicting data preferences). which exist inde- pendently of the optimization procedure.
2.3 Cohomology and Hodge Theory in ML
The mathematical tools we employ have precursors in related ML domains, though their application and interpretation differ significantly from ours.
Fairness. The foundational theory of topological social choice (Chichilnisky 1980 Baryshnikov 1993 has been applied to diagnose impossibility results in fairness. Prior work demonstrates that sinullaneeuug Laneously satisfying disparate fairness criteria can be impossible due to Condorcet-like cycles across different metrics. Our work generalizes this insight from fairness to the foundational property of algorithmic stability.
Rank Aggregation. Combinatorial Hodge theory was introduced to the problem of rank aggregation by Lim (2020]; Jiang et al.] (2011). They utilized the Hodge decomposition to extract global ranking (the gradient component) from pairwise comparison data and quantify the remaining inconsistencies (the cyclical components). While this provides a powerful tool for analyzing ranking data, our work makes the crucial connection between this mathematical structure and algorithmic stability. We reinterpret the cyclical component as the obstruction that causes instability, thereby providing a unifying framework for understanding stability-enhancing methods like bagging and inflated operators.
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Stability as Constrained Preference Aggregation
We formalize the learning problem and stability constraints, analyzing how local data preferences can conflict.
3.1
The Learning Setup and Stability
Let X' be the data space and let P denote family of probability distributions over X. We define the dataset space D as the space of all training datasets of fixed size N that may be encountered in practice, potentially arising from different distributions in P.
Distributional Structure. In the most general sctting, D encompasses datascts sampled from a mixture or lamily of distributions, representing scenarios such as distribution shift, domain adaptation, or heterogeneous data sources. We equip D with an adjacency structure: two datasets S, S' are adjacent if they are sufficiently similar in the distributional sense. This can be formalized via:
Point-wise adjacency: S and S' differ by exactly one data point (standard leave-onc-out).
Distributional adjacency: The empirical measures P3 and P& satisfy Wp(Ps, Ps:) 8 for some Wasserstein distance W, and tolerance 8 0.
The Standard Case as Specialization. When p= {Po} consists of a single distribution and datasets are sampled i..d. from Po, point-wise adjacency coincides with the classical leave-one-out perturbation. The algorithmic stability literature (Bousquet & Elissceff] 2002) primarily focuses on this special case. Our framework naturally generalizes to accommodate the milti-distributionall setting, treating the standard case the restriction of the sheaf to single point (see Section 4.5
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Throughout, we write D denote the support of the dataset distribution, equipped with prescribed probability measure j over this distributional mixture. Expectations are taken with respect to
Let H. be the hypothesis space, equipped with a metric dx. Let L H x D - R be the loss function. A learning algorithm A: + H. aims to select A(S) xe arg mingen Lhh,5)
Definition 3.1 (Algorithmic Stability). An algorithm A is e-stable if for any two adjacent datasets S, E D:
A(()))),A(5)) < c.
3.2 Local Preferences and the Accuracy-Stabiliy Tradeoff
The loss function induces preferences over the hypothesis space for cach dataset.
Definition 3.2 (Inducod Preference Relation). For a dataset S, we say hy preferred over h2 (denotod h1 >s h2) if L(b1, < S).
To capture the set of viable hypotheses, we define the set of near-optimal solutions.
Definition 3.3 (Near-Optimal Hypothesis Set). For accuracy tolerance 8 2 0, the 8-optimal hypothesis set for dataset S is: Ox(S) = (h E Hl, S) S minwer L(h',5) 6].
The fundamental challenge is finding an algorithm that is simultaneously accurate and stable. Definition 3.4 (The Stable Selection Problem). Given e, 6 2 0, the Stable Selection Problem is to find a map A: : D - H such that for all S, S" E D:
A(S) Os(S) (Accuracy)
2. If S, S' are adjacent, du (((()))) S (Stability)
To ensure the problem is well-posed, we assume standard regularity conditions. Assumption 3.5. The dataset space D (as defined by adjacency) is connected.
This assumption is necessary to analyze the global implications of local stability constraints. Tf D were disconnected, an algorithm could be locally stable within cach component but globally unstable across components. For the definition of 'D given above (fixed size N, differing by one point), this connectivily generally holds.
4 Cyclical Obstructions to Stability
We introduce the mathematical framework used to characterize inconsistencies in preferences. While moti- vated by deep connections between social choice theory and abstract algebraic topology (Chichilnisky 1980; Baryshnikov 1993 see Appendix A); we utilize the concrete machinery of Combinatorial Hodge Theory Lim 2020 Jiang et al. 2011). This approach analyzes flows on a graph, providing a rigorous and accessible way to quantify structural inconsistencies (cycles) the data preferences.
4.1 The Graph of Hypotheses and Preference Flows
We focus on finite subset of relevant hypotheses Hy = (h1.... J (e.g., models in an ensemble or candidates in model selection). We construct complete graph K where the vertices are the hypotheses Hy.
We analyze the preferences induced by the data over these hypotheses. As introduced earlier, we view the collection of datasets D as "voters." In practical contexts, such as analyzing an existing ensemble on a test set (as in Experiment 2), individual data points also serve the voters, providing empirical estimate of the preferences.
Definition 4.1 (Preference Profile). A preference profile P collection of preference relations {>s}SED over He-
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We interested in how these preferences manifest as flows the edges of the graph K. Definition 4.2 (Cochains/Flows).
O-cochain is a function assigning values to vertices (e.g., a loss or utility function).
1-cochain (or flow) is a function assigning values to edges (e.g., pairwise comparisons).
The fundamental operator connecting these is the coboundary operator d. transforms a 0-cochain (po- lential) ʃ into 1-cochain (fow):: (d')) f)),,, h2) - [(hg) [(h))
Definition 4.3 (Gradient Flow). A flow C is gradient flow (or coboundary) it can be derived from a global potential function, i.., C' = d f. Gradient flows represent perfectly consistent preferences corresponding to a global ranking.
4.2 Aggregating Preferences: The Importance of Ordinal Data
To analyze stability, we must aggregate the local preferences in the profile P into a single global flow Cp on the graph K. The mothod aggregation is critical.
Cardinal Aggregation. If we aggregate the losses (cardinal values) directly, e.g., by averaging the loss This construction always results in gradient flow, as it is the gradient of the aggregate loss function Lugg (h) = Es [s, Therefore, Cpard
differences across datasets: (hard h2) Lis-pL(h,, Lh,,, S)).
cannot detect Condorcet cycles. To see this explicitly, define Logg (h) == Es( S)}. Then:
Cgard h2) Es|L(hi, 5) L(h;, S)]
= Laga (h1) - Logg (1x) = (dl Lngg) hz).
Thus Cyard d Lagg is gradient flow by construction.
Ordinal Aggregation (Pairwise Majority Vote). To detect structural inconsistencies analogous to those in social choice thcory, we must aggregate the rankings (ordinal preferences). We define the Ordinal Preference Flow Czrd using Pairwise Majority Vote (PMV): (h,, h,) = PswD(h, >s h,) - Ps-pphh > h.). This represents the net preference for h; over hy across the dataset distribution. Crucially, Op is generally not a gradient flow.
Aggregation Across Distributions. In the general distributional setting where datasets in D may arise from different source distributions P E P, the ordinal aggregation via PMV naturally captures conflicts between distributional preferences.
Consider a scenario where different subregions of the data space are governed by different distributions (e.g., different demographic groups fairness applications, different domains in transfer learning, or different wilderness areas in ccological classification). Each distribution P, induces local preferences over hypotheses. The aggregated flow Card synthesizes these potentially conflicting local preferences into a global structure.
When datasets from distribution P: consistently prefer ha over hss while datasets from distribution P3 consistently prefer h2 over ha, the ordinal aggregation reveals this structural conflict. If such conflicts form cycles across multiple distributions, the resulting Cyrd exhibits a non-zero cyclical component, indicating that no single hypothesis can simultaneously satisfy the preferences induced by all distributions in P.
Formally, we can decompose the preference flow by distribution:
Card (hi,
wp [
Σwr
Chr
(1)
PEP
where up is the weight (prevalence) of distribution P in the mixture, and Cord is the preference flow induced by datasets sampled from P alone. Cyclical obstructions arise when the local flows (Cord), } are mutually inconsistent in their global aggregation.
Empirical estimation: In practice, we estimate the preference profile using a test set. Individual test points reveal the preference structure induced by training different data samples: Cord hT)
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1 Exex ((0 Lethy ((hha))) where 2(h,2) is the pointwise loss and 1 denotes the indicator
function. This empirically estimates the aggregate preference relationship between hypotheses.
4.3 Hodge Decomposition and Cyclical Obstructions
We use the Hodge decomposition theorem analyze the structure of the ordinal preference flow Oprd, The divergence operator 5 : C + C is the adjoint of do with respect to the standard inner product. A flow C' is diveryence-free (or harmonic) if 8²C = 0, meaning it has zero net flow at each vertex.
Theorem 4.4 (Hodge Decomposition on Graphs). Any flow C on the gruph K can be uniquely and orthog- onally decomposed as: C == Cgradkent + Ceycle. Where Cgradient is a gradient flow (the consistent part), and Ceyele divergence-free flow (the cyclical part).
(See Appendix A.4 for mathematical details and the definition of the divergence operator).
This theorem provides the central tool for our framework. It allows us to quantity the degree to which preference profile deviates from global consensus.
Definition 4.5 (Cyclical Obstruction). The component Cayele in the Hodge decomposition of the ordinal preference flow Cprd is the cyclical obstruction. It mathematically characterizes the structural inconsistency (Condorcet cycles) inherent in the preference profile P.
The magnitude ||Ceycte|| || quantifies the severity of the obstruction. When ||Ceyctel| = 0, the preferences are perfectly consistent. When ||Ceyatel| > 0, a structural inconsistency exists. In the language of algebraic topology, Ceyole corresponds non-trivial element in the first cohomology group (H+)). We the standard 2 norm: for 1-cochain C, ||C|| = / C(h;, h2)2.
Example 4.6 (The Condoreet Cocycle). The profile (A > B,B> C,C'> A) from Example 1.1 if dominant across the data, rosults in Cprd where the flow is purcly cyclical (C (Cgradient 0). This is the canonical example of structural obstruction.
4.4 Mapping Obstructions to Instability
We connect this mathematical structure back to the Stable Selection Problem. While the existence of cyclical obstruction (||Cayctell > 0) indicates inconsistency, instability arises when this inconsistency is sufficiently strong and localized.
Definition 4.7 (Strong Structural Inconsistency). A preference profile P exhibits a strong structural in- consistency with margin 7 > if there exists a cycle of hypotheses ( h& and disjoint regions of the dataset space, D, S D, such that for S E Dr, h4 is strongly preferred over h4+1 (indices modulo k): I(h,, < [)
Remark 4.8. Strong structural inconsistency (Definition 4.7) is a sufficient condition for non-zero cyclical obstruction when the cycle is sufficiently prevalent in the dataset distribution. Specifically, when disjoint regions D, with the cycle structure induce net positive flow Chrd(h, h++1) > 0 for each edge (h, h++1) in the cycle, the ordinal aggregation via PMV necessarily yields ||Ceyetel| > 0.
Justification: If Chi 1141) > 0 for all edges in the cycle, then summing around the cycle gives Eim (hi, hi41)= 0. However, any gradient flow f satisfies
Σ ()hs h441) 11(hhm =
because each value (hyg appears exactly once with a positive sign (when j = + 1) and once with a negative sign (when j = i), causing all terms to cancel. This contradiction implies Oprd is not a gradient flow, forcing ||Coyctell > by the Hodge decomposition.
This condition implies that different regions of the data space have strongly conflicting preferences, forcing accurate algorithm to choose different. hypotheses in different regions.

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Proposition 4.9 (Instability Induced by Strong Obstructions). If the preference profile induced by the data contains strong structural ònconsistency with margin 7, and for sufficiently small 5 the accuracy construint implies metric localization (i.e., there exists (6) + such that dx(A(S), h4) < 8' when E Di), then for < 4/2 and sufficiently small e, the Stable Selection Problem is unsolvable.
Proof. The condition of strong structural inconsistency ensures a separation margin 7, We choose the accuracy tolerance 7/2. If S E D,, assuming h2 is near-optimal, L(h,. L(hi+1, Since y 26, hi41 cannot be in the near-optimal set: h4+1 & Os(S). This confirms that the optimal sets in different regions are disjoint with respect to the hypotheses in the cycle.
To satisfy the accuracy constraint, an algorithm A must select A(S) E Os(S). The structural inconsistency forces the algorithm to choose between distinct hypotheses in different regions. We demonstrate that stability prevents this.
Intuitive Case: Discrete Metric. Consider the simplified case where has the discrete metric (d(ha; hn) if o b) and 1. Stability implies A(S) A(S") for adjacent S, Since D is connected (Assumption 3.5), this forces single global choice h" However, the strong structural inconsistency (a cycle) implies that for any h", there exists region D. where ht is strongly dispreferred and thus h" Φ Og(S) for S D, Thus, the problem is unsolvable.
General Case: General Metric Space. We generalize the argument using the triangle inequality. This requires that the structural inconsistency manifests metrically (standard regularity conditions assumed in Scction 3.2).
Let h; and hy be two distinet hypotheses in the cycle, separated by distance D d (Hi, h2) 0. Consider the corresponding regions D; and Dj. We assume that for sufficiently small 8, the accuracy constraint implies metric localization. Formally, we assume there exists 8 (which decreases as 8 decreases) such that for 5 E D, (A((A < 8, and for S' E DD, (x((A hy) < 8'. We choose 5 small enough such that 28' < D/2.
Since D connected, there exists a finite path of adjacent datasets S where So D; and SM D,.
By the stability constraint, (((A(S)): S.. By the triangle inequality along the path:
M-1
(x(((S))AA(MM) S
Σ (((A(S)) s -.c.
k=0
Now we relate this to the separation using the triangle inequality again:
D = du h1) < dx A(S6),,((6)) + (x)((()) + (A(SM)),
8' + (M (M) +
This gives the requirement D < 28' + +M
The distance D and path length M are fixed by the problem structure. We have chosen such that 28' < D/2. We can now choose sufficiently small such that M e S D/2. Then 28' + M e < D. This violates the derived inequality, creating a contradiction. Thus, the Stable Selection Problem unsolvable.
When strong cyclical obstruction exists, no single-valued algorithm can simultaneously satisfy the incon- sistent preferences and the metric constraints of stability.
Metric localization follows from standard contimuity assumptions, such 9-Lischhizz continuity the loss function in the hepothesis space: (L(h,S) (()) B du (h).

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4.5 Sheaf- Theoretic Interpretation: -ocal-to-Global Obstructions
The distributional framework admits elegant interpretation through the lens of sheaf theory, revealing the preference structure as a sheaf over distribution space and cyclical obstructions as cohomological obstructions to gluing local sections.
The Preference Sheaf. Let (P. be the space of distributions over X, equipped with a suitable topology (e.g., the weak topology or topology induced by the Wasserstein metric). We define the preference sheaf f over as follows:
For each open set U S P, the space of sections FUU) consists of hypothesis selection rules UreU Dp + H that are locally optimal and stable within U.
For each point P E P, the stalk Pp represents the space of stable selections for datasets drawn from distribution P alone, characterized by the local preference structure Op
Restriction maps pvv (U) - (V) for V U restrict the selection rule to the smaller region.
Local Consistency, Global Obstruction. A global section a E # ((P) would be hypothosis selection rulc that simullaneously accurate and stable across all distributions P. The exislence such global section is precisely the solvability of the Stable Selection Problem (Definition 3.4) in the distributional setting.
The cyclical obstruction Ceyele + 0 is the manifestation of a cohomological obstruction: the local sections (stable selections within each distribution P) cannol be glued into global section due to incompatible tran- sition functions on overlaps. In classical sheaf cohomology, such obstructions are measured by cohomology groups H² ().
The Standard Case. When p = {Po} a single point, the sheaf theory trivializes: there only one stalk, and the cohomological obstruction reduces to the local structure within that single distribution. This recovers the standard algorithmic stability setting. The power of the sheaf perspective emerges when is non-trivial, capturing scenarios with gemuine distribution shift or heterogeneity.
Remark 4.10. This sheaf-theoretic interpretation provides rigorous mathematical foundation for un- derstanding why certain learning problems admit stable solutions (global sections exist) while others do not (cohomological obstructions). It also suggests natural generalizations: higher-order cohomology groups (P(P, F) for q 2 2 may characterize more complex multi-way conflicts beyond pairwise cycles, direction for future investigation.
5 A Mathematical Explanation for Stability Methods
Our framework, centered the Hodge decomposition, rigorously distinguishes between two sources of instability and unifies the methods used to address them based the presence absence of structural obstructions (Ceyate).
5.1
Addressing Structural Obstructions (Ceyate + 0)
When the data preferences contain fundamental inconsistencies, the instability is structural. This requires methods that either resolve the ambiguity enforce consistency.
5.1.1 Obstruction Resolution: Target Space Enlargement
When obstructions are inherent (Caycle + 0), a stable single-valued solution may not exist (Proposition 4.9 The mathematica solution is to change the larget space.
In social choice thcory, when a single consensus winner does not exist due to cycles, the solution to identify consensus set, such as the Top Trading Cycle (TTC) set-thet the minimal set hypotheses H that dominate those outside H* Inflated operators, rigorously analyzed by Adrian et. al. (2024) and Liang et. al. (2025)
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implement this principle by enlarging the target space from H to P(H). For set-valued outputs, use the Hausdorff distance on P(H):
(A, max sup inf dx(a, b), sup inf du(a,b) aaEA bEB
beB aEA
Alternatively, stability can be defined via set intersection: two set-valued outputs are stable if Os(S) n Os(S") + 0. Definition 5.1 (Inflated Argmax (Adrian et al. 2024 ) argmaz,(w) = {ilwy 2 mazyws - c}.
Interpretation 5.2. Cohomological Significance of Inflation Inflated operators resolve cyclical ob- structions by systematic target space enlargement. The set-valued output of the inflated operator approzimates the Top Trading Cycle set associated with the underlying preference cycle.
Argument. Consider a profile P with a significant cyclical obstruction, represented by a cycle h4 >..>> hy > h1. As shown in Proposition 4.9 if this obstruction is strong, it leads to the failure of the single-valued Stable Selection Problem.
The Top Trading Cycle (TTC) set associated with this cycle is H fhi,..., h+)- Now consider the appli- cation of the inflated operator (e-g., returning the near-optimal set Os(S)) II for each pair h2 by H* the losses satisfy (h,, S) - SSI S é' uniformly across datasets $ in the relevant region (eycle balanced), then the losses of the hypotheses in the cycle will be close across the relevant region of the datasct space: L(h4, S) S)] < e' for h4, hy E H*
If we choose the tolerance parameter 2 é', the near-optimal set Os(S) will contain Hr hypotheses outside the cycle have significantly higher loss, then 0s(5) H*
The inflated operator thus identifies the set of hypotheses involved in the structural ambiguity characterized by the obstruction. Under bounded loss variations (a standard regularity assumption), for adjacent datasets S, and balanced cycle, if h E H has L(h,, S) S ming S(h,S) + 6, then by continuity L(h,, < ming L(h, S') + 8' for 8' = 8. Thus all hypotheses in the cycle remain near-optimal on both datasets, guarantecing H C (0(SS)nO((S)
5.1.2 Obstruction Prevention: Enforcing Consistency
Al Alternatively, we can attempt to prevent obstructions by enforcing consistency during the learning process. Regularization based ensemble stability adds penalty terms that discourage disagreement (e.g., Eytue = Σ. E., [1/x(x) (x)]1,, where (zx denotes the prediction of model h; on input æ, and the sum is taken over some discrete set of inputs on which agreement is required and all pairs of predictors). This explicitly enforces consensus among the ensemble members. By minimizing Lytues the optimization procedure forces the local preferences to align. When models are forced to produce similar outputs, their rankings across the data become more consistent, directly minimizing the inconsistencies that contribute to the cyclical component Ceyele. Our proposed method in Section 6.1 is a higher-order generalization of this approach.
5.2 Addressing Statistical Variance (Ceycle ~ 0)
When the cyclical component is near zero, the aggregaled preferences are globally consistent. However, instability can still occur due to statistical variance the sensitivity of the learning algorithm or the selection procedure (like argmax) to noise, ncar-tics, or specific data samples.
5.2.1 Bagging: Variance Reduction
Bagging (Breiman 1996) utilizes bootstrap samples {S.} from S and aggregates the results. Its effectiveness is statistically grounded in variance reduction (Soloff al. 2024
Interpretation 5.3. Role of Bagging in the Hodge Framework Bagging is a mechanism for reducing statistical instability when the underlying preference structure is largely consistent (Ceycle 0). is not designed to resolve strong structural obstructions (Geyete # 0).
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Argument. Let VSy(S) = S) 8) be the cardinal preference score. When Caycle 2 the ordinal preferences are largely aligned with global ranking. Instability occurs when Vi)(S) has high variance, causing the sign of (S) (the ordinal preference) to flip frequently across the dataset distribution, especially in near-tics.
The bagged learner Atag smooths the cardinal preferences, resulting ( (S) with significantly reduced variance. This stabilization makes the ordinal preferences more robust and less likely fluctuate, thereby improving stability.
However, if a strong structural obstruction exists (Cagete # 0), the data inherently contains conflicting prefer- ences. Variance reduction alone camnot create a stable consensus, as the underlying problem is fundamentally ambiguous. Bagging averages over the conflicting preferences but does not resolve the cycle itself. As we demonstrate in Experiment 1, bagging fails to stabilize the selection process in the presence of a strong Condorect cycle.
6
Obstruction-Aware Ensembling: A Framework Inspired by Cohomological Insights
Our cohomological perspective suggests principled approaches to designing stable algorithms by systemati- cally targeting these mathematical structures.
6.1 CohomologicallyyInspiee Regularization
Motivated by the mathematical structure of disagreement cycles, we propose regularization terms that target higher-order inconsistencies, moving beyond pairwise agreement. Consider an ensemble with models {ha} and learnable alignment maps {øjs} between their representation spaces. Global consistency requires the alignment maps to satisfy the cocycle condition: Φ&i = Φkj o Oji-
Definition 6.1 (Obstruction-Awarc Ensemble Loss). A cohomolggically-impprrrd loss includes a term en- forcing higher-order consistency: Leoeyale = Eigz 1l - Φk @,ill? The total loss is Ltotal = Ltosk + Ag + A2Cooeydee
Minimizing Leocyele encourages global consistency in the geometric relationships between models, directly targeting the structures identified by the framework in the representation space.
Computational Complexity and Stochastic Approximation. The proposed Leoeyete involves a sum- mation over all triplets (i.j, k), resulting in a computational complexity of O(N3) for an ensemble of size N. This can be computationally prohibitive for very large ensembles. However, this complexity can be readily mitigated using stochastic approximation. Instead of calculating the loss over all triplets at every step, we randomly sample fixed number of triplets K and calculate the average loss:
Lstochastio-cocycle
Σ (Aap.))
| Øky Q
This reduces the complexity to O(K) per training step and provides an unbiased estimator of the aver- age cocycle defoct. This stochastic approach effectively enforces the cocycle constraint while maintaining computational tractability.
Connection to Cycle Consistency. The mathematical form of Looeyale is analogous to the 'Cycle- Consistency Loss' widely used in unsupervised learning domains like image translation and unsupervised machine translation Grover el al. (2020). In those domains, cycle consistency serves as an engineering heuristic to constrain ill-posed, unsupervised problems. Our work re-contextualizes this mechanism: we derive the cocycle constraint from thcoretical framework, arguing that violations of this consistency are manifestations of mathematical obstruction with direct consequences for stability. This elevates the mech- anism from heuristic to principled, theoretically-grounde tool for improving robustness.
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6.2 Adaptive Target Space Enlargement
Inspired by the success of inflated operators, propose adaplive mechanisms that automatically transition between single-valued outputs when consensus exists (suggesting Coyole R 0) and set-valued outputs when obstructions may be present (Ceyate # 0), based on mcasures of ensemble disngreement.
7 Computational Experiments
We validate our thcoretical framework through three complementary experiments. First, we demonstrate the core mechanism in a controlled setting with engineered structural obstructions, confirming that inflated oper- ators succeed where bagging fails (Experiment 1). Second, we validate the fundamental empirical prediction that standard single-istribution supervised learning cxhibits negligible structural obstructions, explaining why variance-reduuciinn methods dominate practice (Experiment 2). Third, we demonstrate that significant structural obstructions emerge naturally in multi-objective fairnes-constrainned model selection, confirming the framework's diagnostic power for identifying boundary conditions (Experiment 3).
7.1 Experiment 1: Structural Instability (Engineered Condorcet Cycle)
This experiment validates the core distinction of our framework: structural instability requires structural solutions (inflation), while statistical methods (bagging) fail. We Lest this in a scenario with pure structural obstruction.
We engincered a scenario with three hypotheses (A, B, C) where the underlying data distribution is mixture of three regions (Z1, Z2, Za) inducing a balanced Condorcet cycle (A > B > C> A). We simulate the process of drawing dataset S and an adjacent dataset S' (by perturbing the sampling weights by small amount). We ran 5000 trials.
Sctup:
Loss function: dataset S is represented by weights Ws = (wi, W2, wa) indicating the sampling proportion from each region. Each hypothesis h; has a utility vector Uj = (U:1, U12 Uz3) where uzj its utility in region j. The aggregate loss is:
L(h;, 5) =- (U,, Ws)
Ex=1 Wi
W1 + U2 + W3
Σ} Wi
(For this experiment: UA = -3, 1,2], UB = [2, 3,1, 1), Uc = (1)
Methods: We compare the stability of:
Standard 'argmax:
Inflated 'argmax' (e = 0.01) (Structural solution).
3. Bagged 'argmax' (50 bootstraps) (Statistical solution).
Metrics: Agreement Rate (probability the winner is the same for S and S') for single-valued methods; Intersection Consistency (probability the output sets intersect) for the inflated method.
Table 1: Experiment 1: Stability under a pure structural obstruction (Condorect cycle).
Method
Type
Stability (Agreement/Interrsctiin Avg. Output Size
Standard Argmax
Bascline
0.4094
1.00
Bagged Argmax
Statistical
0.3432
1.00
Inflated Argmax
Structural
1.0000
3.00
Results: As shown in Table 1 the standard 'argmax' exhibits low stability (0.4094). Crucially, the Bagged 'argmax' also fails stabilize the selection, performing slightly worse (0.3432). This confirms Interpreta- tion 5.3 statistical variance reduction ineffective against structural instability. In contrast, the Inflated
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'argmax' achieves perfect stability (1.0000) by identifying the ambiguity inherent in the cycle and returning the consensus set {A, B, Cj.
7.2 Experiment 2: Validating the Absence of Structural Obstructions in Standard Classification
This experiment validates the central empirical prediction of our framework: standard single-distribution supervised learning with i.i.d. data, exhibits negligible structural obstructions (l|Ceystel = 0), explaining the empirical success of variance-reduction methods like bagging.
Setup: We use the scikit-learn handwritten digits dataset (8x8 pixel images of digits 0-9, N-1797) as a representative standard supervised learning task. We train ensembles of K 20 classifiers on bootstrap samples of the training data to induce diversity, mirroring standard bagging practice. We then compute the Hodge decomposition of the ordinal preference flow on the test set.
Model Architectures: We test two canonical model families; multi-layer perceptrons (single hidden layer with 64 units, trained for 20 epochs with early stopping) and decision trees (maximum depth 10, unpruned otherwise).
Bootstrap sampling ensures the ensemble contains diverse but related hypotheses trained on the same un- derlying distribution.
Methodology:
For multiple trials with different random seeds:
1. Split data into train (70%) and test (30%) with stratification
2. Train K = 20 models on bootstrap samples from training set
3. Compute ordinal preference cochain via PMV on test set (using 0-1 loss)
Apply Hodge decomposition to measure ||Ceyetel|
Table 2: Experiment 2: Structural obstructions in standard classification (Digits dataset). Values at machine precision confirm the absence of cyclical inconsistencies.
Model Type
Trials
|[Ceysie|| || (mean)
|[Coydel || (std)
Interpretation
MLP
10
2.33 x 10-16
1.29 x 10-16
Machine precision
Decision Tree
5
2.05 x 10-16
6.96 10-17
Machine precision
Results and Interpretation: As shown in Table 2 the measured cyclical norms are consistently at the level of machine precision (l[Cayctell ze 10-11), effectively indistinguishable from zero across both model families and all trials. This empirically validates the theoretical prediction that standard supervised learning with i..d. data produces consistent preference structures.
The absence of structural obstructions provides mathematical explanation for fundamental empirical pattern: the dominance of variance-reductioon methods (bagging, ensembling) in the machine learning liter- ature (Breiman 1996 Soloff et al. 2024). When ||Caystel ~ 0, instability is purely statistical arising from sensitivity to noise, near-ties, or specific samples and methods that smooth cardinal preferences without requiring structural resolution are both sufficient and effective.
This contrasts sharply with the multi-objective fairness setting (Exporiment 3), where structural obstructions emerge (l/Ceyete| || 0.857, approximately 10 times larger) and require fundamentally different approaches.
Theoretical Grounding: This empirical observation aligns with theoretical framework (Section 4). In the standard single-distribution setting (/P) = 1) with convex or near-convex loss landscapes, the aggre- gated preferences naturally form a consistent total order. Each hypothesis has a well-defined expected loss Es( and the preference profile induced by the data distribution corresponds to the gradient this expected loss, resulting in Ceyele by construction (as shown in Scction 4.2 for cardinal aggregation). While we aggregate ordinal preferences, the underlying consistency of the expected loss landscape ensures the ordinal flow also exhibits negligible cyclical structure.
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7.3 Experiment 3: Structural Obstructions in Chirness-Consrriing Model Selection
Motivation and Distributional Interpretation. This experiment demonstrates that structural obstruc- tious-absent in standard single-distribution classification tasks emerge naturally in multi- cualiiddsttibutioaall settings with conflicting objectives. Fairnos--aware machine learning provides a canonical testbod: different fairness criteria correspond to different distributional priorities (e.g., optimizing performance on reweighted demographic subpopulations), which are mathematically incompatible (Kleinberg et al. 2017
We leverage the distributional generalization of our framework (Section 3.1, Section 4.5 Each stakeholder perspective can be interpreted as representing different distribution P. P over the data space, where P. reweights or resamples the base distribution to prioritize specific fairness objectives. For instance:
Demographic Parity: Distribution PDP reweights minority groups equalize selection rates.
Equalized Odds: Distribution PEo emphasizes equal error rates across groups.
Predictive Parity: Distribution Ppp focuses on calibration within groups.
Models trained or selected under these different distributional objectives induce different local preferences. The aggregation of these distributional preferences via PMV (Section 4.2) reveals whether a globally stable selection exists or whether the objectives are fundamentally incompatible (inducing cycles).
Setup. We the UCT Adult dataset. (Becker & Kohavi 1996) to simulate fairmess-constraineed selection. We train five Logistic Regression models using different training heuristics (reweighting, regularization ad- justments) to generate a diverse set of models exhibiting varied fairness-accuracy tradcoffs. These heuristics are not intended to optimally achieve specific metrics, but to simulate the pool of candidate models that would be available when optimizing under different distributional constraints.
We define five distributional perspectives (Business: accuracy-focused;; Civil Rights: demographic parity- focused; Equal Opportunity: equalized odds-focused; Calibration: predictive parity-focused; Regulator: balanced), cach representing different implicit data distribution or prioritization scheme. Each perspoc- tive evaluates models according to utility functions that weight accuracy and fairness metrics differently, effectively representing preferences under different distriibttinnl constraints.
Crucially, we aggregate preferences across these distributional perspectives using PMV (not across datasets with a single loss function, as in Experiments 1-2). This models the real-world scenario where decision- makers must reconcile models optimized under different distributional objectives- a natural instance of the multi-distrbuutional framework.
(See Appendix B.5 for detailed methodology.)
Methodology: For each of 20 trials with different train-test splits, we:
Trained the five models with different fairness-accuracy tradeoffs.
2. Computed stakeholder preferences based on model performance.
3. Constructed the ordinal preference flow Card via Pairwise Majority Vote (PMV).
4. Applied Hodge decomposition to measure |[Cayetel| and the Cycle Ratio ([[Crytell/|] Ctotaal)))))
Unlike Section 4.2 where we aggregate over random datasets with iden- tical loss functions, here we aggregate over stakcholders with different utility functions. Each stakcholder's utility encodes a normative priority over fairness-aceuracy tradeoffs. This models real-world scenarios where multiple maresssn with conflicting objectives must. select from a common model pool. The PMV aggregation captures the democratic aggregation of these preferences.
Note on Aggregation Method:
Results and Interpretation:
The results, summarized Table 3 reveal a striking contrast with standard classification tasks. The mean cyclical norm is ||Gaydel|| 0.857, approximately 10 times larger than the
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Table 3: Experiment 3: Structural obstructions fairness-constraineed model selection (UCI Adult). Multi- objective problems exhibit cyclical norms approximately 10t5 times larger than standard singlo-distribution classification (Experiment 2).
Task Type
||Cryeeel (mean)
Cycle Ratio (mean)
vs. Standard Tasks
Standard Classification (Digits, Exp. 2)
2.3 x 10-16
~ 0
Baseline
Fairness Selection (Adult, Exp. 3) 0.857 + 0.173
0.363 0.075
100 x larger
machine- machine--reccsion values observed in Experiment 2. The mean cycle ratio is 0.363, indicating that over 36% of the preference disagreement is fundamentally structural rather than statistical.
Thcorctical Significance: This validates the core distinction of our framework on real data. Standard learning problems produce negligible structural obstructions (Ceyele R 0), explaining the effectiveness of selection involves genuinely conflicting objectives that create
bagging. In contrast, strong cyclical obstructions (Coyole + 0), making single-model selection fundamentally unstable.
Topological Characterization of Fairness Impossibility: This provides a topological characterization of known fairness impossibility theorems (Kleinberg et al., 2017]. The incompatibility between fairness definitions manifests as non-trivial cohomology (H² * 0) in the preference space. Our contribution is not discovering that fairness metrics conflict-the is well-established but providing the first rigorous quantifi- cation of this conflict via topological invariants (I]Ceyete ll); unifying these impossibilities with the broader theory of algorithmic stability and social choice theory under a common mathematical framework.
Practical Implications: When Ceyote # O, Proposition 4.9 establishes that stable, single-valued selection is impossible. The appropriate structural solution (Interpretation 5.2) is target space enlargement via inflated operators, returning a portfolio of models (the Pareto frontier). Crucially, this does not "solve" the fairness conflict; it merely acknowledges the impossibility of a single solution characterized by the cycle norm, and externalizes the decision to policymakers. The framework clarifies that the challenge is structural and cannot be resolved by purely statistical means.
8 Limitations and Future Directions
While our framework provides a unifying perspective on algorithmic stability, it has several limitations that suggest promising directions for future research.
8.1 Limitations
The framework of topological social choice theory and Hodge decom- position is primarily developed for finite set of alternatives H4. Extension to continuous hypothesis spaces requires additional technical machinery, such analysis function spaces or discretization schemes that preserve the relevant topological structure.
Continuous Hypothesis Spaces:
Our framework characterizes two distinct mathematical sources of instability: structural obstructions (cyclical inconsistencies) and statistical variance. The empir- ical findings of Experiment that ||Ceystel| Rs in standard classification tasks consststtt a significant theoretical result rather than a limitation. They explain mathematically why the field has empirically con- verged on variance reduction methods: the structural obstructions that would defeat such methods simply do not arise in well-posed supervised learning problems with convex near-convex loss surfaces and i.i.d. data. The cases where Cayde # 0 represent important boundary conditions where fundamentally different mathematical structures emerge. We provide concrete evidence for this in Experiment 2, demonstrating that significant structural obstructions arise fairness scenarios due to incompatible metrics. Investigating the precise conditions under which structural obstructions manifest. other domains (c.g., adversarial settings, multi-agent systems), and their prevalence in these specialized domains, remains an important direction for future theoretical and empirical work.
Theoretical Scope and Empirical Findings:
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Computational Scalability of Analysis: Calculating the Hodge decomposition for theoretical analysis (as in Experiment 2) requires operations such as computing the pseudoinverse of the graph Laplacian, which scale as ((K)) where K is the number of hypotheses. This limits the direct application of the decomposition as an online diagnostic tool for very large hypothesis Thic This stochastic approach effectively enforees the cocycle constraint while maintaining computational tractability.
While we rigorously distinguish between ordinal aggregation (which detects cycles) and cardinal aggregation (which does not), the precise relationship between high variance in cardinal preferences and the emergence of near-cycles in ordinal preferences remains to be fully characterized. A deeper understanding of this connection would strengthen the thcoretical link between variance reduction (bagging) and cycle suppression.
Connection Between Ordinal and Cardinal Structures:
8.2 On the Rarity of Structural Obstructions and Boundary Conditions
Our framework distinguishes structural from statistical instability based on the presence or absence of cyclical obstructions ([[Ceyctell) II). A critical empirical question is: when do structural obstructions occur in practice?
Standard Supervised Learning: The Null Caso. Experiment 2 provides definitive empirical validation that standard single-distribution supervised learning exhibits negligible structural obstructions. Across mul- tiple trials with two different model families (MLPs and decision trees) on standard classification task, we consistently measured ||Caydeel ~ 2 x 10-16 effectively zero at the level of machine precision.
This observation has deep theoretical grounding. In the standard selting where datasets are sampled i.i.d. from a single distribution (IP) = 1), the loss landscape convex or near-convex, and hypotheses are evaluated under fixed objective- theheee-eeeeeeeeee aggregated preferences naturally form a consistent total order. Each hypothesis has well-defined expected loss Es [h, S)], and the preference profile Card aligns with the gradient of this expected loss, resulting in Ceyele 0.
This mathematical characterization explains fundamental empirical pattern in machine learning: the dom- inance of variance-reduccion methods like bagging (Breiman 1996 Soloff et al. 2024). When ||Cayetel| |l z 0, instability is purely statistical (sensitivity to noise, near-ties, specific samples), ; and methods that smooth cardinal preferences without requiring structural resolution are both sufficient and effective. The field has empirically converged on these methods precisely because the structural obstructions that would defeat them are absent in standard well-posed learning problems.
Boundary Conditions:
When Obstructions Emerge. Our framework characterizes the boundary conditions under which structural obstructions arise, transforming the nature of the stability problem. Ex- periment& 3 demonstrates one such boundary condition: multi-objective learning with incompatible criteria. The mcasured eyclical norm in the fairness setting (l[Ceyaell = 0.857) is approximately 1045 times larger than in standard classification, with 36% of the preference structure being fundamentally cyclical rather than statistical.
Structural obstructions (l[Cogetell * 0) emerge under three characteristic conditions. First, when differ- ent distributions P. P induce conflicting preferences- as in fairness-awars learning where optimizing for different demographic groups or incompatible fairness metrics creates genuine Condorcet cycles. Experi- ment 3 demonstrates this quantitatively: aggregating preferences across stakeholders prioritizing accuracy, demographic parity, equalized odds, and predictive parity yields ||Ceystel| = 0.857, indicating fundamental preference conflicts that cannot. be reconciled.
Second, highly multimodal loss surfaces with multiple local minima of comparable quality can create local preference cycles, particularly in adversarial or tame-theoretic settings where different regions of the data space favor fundamentally different solutions.
Third, multi-task learning, Pareto optimization, or settings with mathematically incompatible objectives (ac- curacy vs. interpretability vs. computational efficiency) naturally generate preference cycles when different criteria rank hypotheses in mutually inconsistent orders.
Diagnostic Framework and Contribution. Our contribution not discovering that these boundary conditions exist. fairness impossibilities (Kleinberg et al. 2017), multi-objective conflicts, and adversarial
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instabilities are well-documented. Rather, provide the first rigorous quantitative framework for diagnosing the source of instability via |[Cayeell as precise, computable diagnostic; for characterizing when structural vs. statistical methods are appropriate (Experiment 2 shows bagging suffices for standard ML while Exper- iment 3 shows would fail for fairness-constraaned selection); and for unifying seemingly disparate stability phenomena under a common topological language.
Standard supervised learning is not merely special case of our framework, but the null case where structural obstructions vanish (|Coyotel = 0). The boundary conditions where [|Cryetell & 0 represent fundamental limits of learning that cannot be overcome by purely statistical means- they require structural solutions such as target space enlargement (inflated operators) or acknowledgment that no single stable solution exists (as in fairness trade-offs). The framework provides the mathematical tools to diagnose which regime a learning problem occupies and select appropriate stabilization strategies accordingly.
8.3 Future Directions
Further development of obstruction-awar algorithms. This includes design- ing efficient methods to estimate when Coyele * 0 during training and adaptively deploying regularization (structural) or bagging (statistical) strategics based on the diagnosed source of instability.
Algorithmic Applications:
Theoretical Extensions: Investigating the precise mathematical relationship between the smoothing effect of bagging (variance reduction) and the reduction of the cyclical component in the Hodge decomposition. Furthermore, exploring the use of higher-order cohomology groups (H9 for q 2 2) to identify more complex structural inconsistencies beyond pairwise cycles.
Connections to Fairness and Explainability: Preference cycles often underlie paradoxes in fairness and explainability, where different. metrics or perspectives lead to conflicting conclusions. Extending this cohomological framework to analyze the stability and consistency of fairness metrics a promising direction.
9 Conclusion
We have presented a unified framework for algorithmic stability based on Combinatorial Hodge Theory and the geometry of preferences. This framework rigorously distinguishes between structural instability, caused by cyclical obstructions (Condorcet cycles, Cayete > 0), and statistical instability, caused by variance (Coyete 0). We demonstrated that structural instability requires targeted solutions like inflated operators novel obstruction-awwn regularization, while statistical instability is addressed by methods like bagging, Our experiments validate this distinction, showing that bagging fails in the presence of strong structural obstructions but succeeds when instability is statistical. This framcwork provides a mathematical foundation for understanding and improving the robustness of machine learning systems.
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A Appendix A: Mathematical Foundations: Cohomology of Preferences
This appendix provides the rigorous mathematical details underlying the connection between social choice theory, algebraic topology, and the cohomological obstructions discussed in the main text. We adapt the framework established by Chichilnisky (980)) and Baryshnikov (1993). and incorporate tools from Hodge theory for ranking.
A.1 The Geometry of Rankings and Preferences
Let H4 (hi,...,hh, h&3 be finite set of hypotheses.
Definition A.1 (Space of Preferences). A utility function U : H₂ - R induces preferences. The space of utility functions is 2 = R A preference profile is a collection of utility functions {Us) Alternatively, we can consider the space of strict linear orders P.
A.2 The Source of Obstructions: Topological Social Choice
The motivation for analyzing preference cycles comes from Topological Social Choice Theory (Chichilnisky 1980 (Baryshniovv 1993). This theory analyzes the topology of the space of all possible preferences (P) to determine if fair aggregation rules F PN + P exist.
A.2.1 Topological Interpretation (Chichilnisky's Approach)
Chichilnisky 1980)) showed that the impossibility of fair aggregation (satisfying certain axioms like Arrow's conditions) stems from the topological properties of P. To define a continuous aggregation rule, P must be equipped with a topology. The standard approach identifies the space of preferences P with a subset of the space of utility functions 4 R (c.g., normalized utility voctors). The topology on P then induced by the standard Euclidean metric on U. This means preferences are considered "close" if their utility values are close.
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Theorem A.2 (Chichilnisky, 1980). If continuous aggregation rule F PN + P erists satisfying certain desirable arioms (e.g, anonymity, unanimily), then P must be contractible.
A space is contractible it has no topologiaal "holes" (i.c., its homology and cohomology groups are trivial).
A.2.2 Cohomology of Preference Spaces
We analyze the topology of the space of strict linear orders Sg. This space is known to be non-contractible for & 2 3.
The Condorcet cycle (A B, C,C A) represents a loop in the space of preferences. This loop cannot point while maintaining the structure of the preferences.
be contracted
Lemma A.3. The existence of a non-contructbble loop in the space of preferences corresponds to non-trivial element in the first cohomology group (Sp).
This abstract theory shows that the possibility of Condorcel cyeles arises because the space of preferences itself has non-trivial topology (it is not contractible). This motivates the search for these structures in specific preference profiles.
A.3 Connecting Topology to Stability
While the abstract theory analyzes the space of all preferences, in machine learning analyze specific profile P induced by the data. Combinatorial Hodge Theory provides the tools to detect the manifestation of these topological obstructions within specific profile by analyzing flows the graph of hypotheses K.
A.4 Algebraic Tools: Cochains and Hodge Decomposition
This section provides the rigorous definitions for the tools used in Section 4.
A.4.1 Cochains and the Coboundary Operator
Let K be the 1-skeleton (the complete graph) on the verlices Hu-
C (K; O-cochains (functions assigning values vertices, ie., utility functions).
Ca R); 1-cochains (functions assigning values edges, ie., pairwise comparisons or flows).
The coboundary operator d² C C maps a potential function f E C to flow: (hi,)
(h; - (hh). This represents perfectly consistent ranking.
A.4.2 Preference Cochain Construction: Cardinal vs. Ordinal
We define the preference cochain Cp E (K, R). R). 4.2.
The construction method is critical, detailed in Section
Cardinal Aggregation: If we aggregate the losses dircetly (c.g., average loss difference): Cpard Chard(h; hy) So (L(,,,))) S) - L(h,, (S)..(()) This construction always results in a coboundary, as it is the gradient of the aggregate loss function Tagg (h) = L(h, ()),((). Therefore, C'gard cannot detect Condorcet cycles.
Ordinal Aggregation (Pairwise Majority Vote): To detect. cohomological obstructions relevant to social choice theory, we must aggregate ordinal preferences. Let Rs be the ranking induced by L(,S)). (hr, h,) = Jp sign(Rs(h,) Rs(h:))d((S)) In practice (Experiment 2), this calculated as: (Proportion of voters preferring h; over h₂) (Proportion preferring h, over h.). This construction allows for Cp to be non-coboundary.
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A.4.3 Hodge Decomposition
We use the Hodge decomposition theorem to analyze flows (cochains) on the graph K.
Theorem A.4 (Hodge Decomposition on Graphs). The space of 1-cochains C con be orthogonally decom- posed as: = im((d) @D ker(6') where 8 the adjoint of d (the divergence operotor).
Here, im is the space of coboundaries (gradient flows, Coradient). ker(ō') is the space of harmonic cochains (cyclical flows, Ceyete). This harmonic component corresponds to the cohomological obstruction identified in the lopological framework.
This theorem allows us rigorously decompose a preference profile Cp into its consistent part (Caradient) and its cyclical obstruction (Ceyate); providing the diagnostic tool used in Section 7.
B Appendix B: Experimental Details and Reproducibility
This appendix provides the detailed configurations and methodologies for the experiments presented in Section 7, ensuring reproducibility. We utilized Python with the following key libraries: NumPy, SciPy, scikit-learn (including fetch_ fetch_covtype), and PyTorch.
B.1 Experiment 1: Structural Instability
Objective: Validate that structural instability (Condorcet cycles) is resolved by inflation but not by bagging.
Setup Details:
Hypotheses: K = (A, B, C).
Data Regions (defining the cycle):
Z1 (prefers >> B >> C): Utility vector U1 [3.0, 2.0, 1.0].
Zz (prefers B > C > A): Utility vector U2 = [1.0,3.0 2.0]. Z3 (prefers C > >> B): Utility vector U3 [2.0, 1.0,
Simulation: 5000 trials.
Dataset Generation: dataset S is simulated by sampling weights Wg for the three regions around base weight of 100 (with uniform noise [-0.5,0.5]
Perturbation: An adjacent dataset S' is generated by perturbing one weight in Ws by =1.
Argmax: Standard argmax with random tie-breaking.
Inflated Argmax: 0.01.
Bagged Argmax: Simulated by drawing B = 50 bootstrap samples (using a multinomial distribution basod on Ws) and aggregating the winners by majority vote (with random tic-breaking).
B.2 Experiment 2: Absence of Obstructions in Standard Classification
Objective: Validate that ||Ceyete|| || Rs O in standard single-distribution supervised learning.
Setup Details:
Dataset: Scikit-learn digits (8x8 handwritten digits, 1797 samples)
Models: MLPs (hidden layer 64 units) and Decision Trees (max depth 10)
Ensemble size: K 20 models per trial
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Bootstrapping: Models trained on bootstrap samples to induce diversity
Test split: 30% of data
Trials: 10 for MLPs, 5 for Decision Trees
Hodge Decomposition: Applied ordinal preference cochain constructed via PMV on test using 0-1
loss.
Key Results:
MLPs: Mean [|Coycell 2.33 10-16 (std 1.29 10 16)16)
Trees: Mean ||eyctel|| = 2.05 x 10-16 16 (std 6.96 x 10-17)
All values at machine precision level, confirming negligible structural obstructions
B.3 Experiment 3: Structural Obstructions in Fairness
Objective: Demonstrate the emergence of structural obstructions fairness-const model selection the UCI Adult dataset.
Setup Details:
Dataset: UCI Adult Income (Becker & Kohavi 1996 (fetched via OpenML). N-48842.
Features: 12 features (5 numeric, 7 calegorical) processed via standardization and one-hot encoding, Missing values handled by mode imputation.
Sensitive Attribute: Sex (Male/Female)
Target: Income (>50K <=50K).
Models (K=5): Logistic Regression models trained with different heuristics. The objective is to induce diversily in the fairness-accuracy trade-ofls, not necessarily to optimally achieve the target metrics:
Accuracy: Standard optimization (C=1.0).
Demographie Parity: Reweighting minority group (weight-2.5).
Equalized Odds: Reduced regularization (C=0.1).
Predictive Parity: Using class haaas__wegghtt"aalaneeo
Balanced: Modorate reweighting (weight=1.5) and regularization (C=0.5).
Stakeholders (Voters=5): Simulated utility functions prioritizing different metrics: Business (Aceu- racy only), Civil Rights (DP/EO), Equal Opportunity (BO/Acc), Calibration (PP/Acc), Regulator (Balanced).
Aggregation: Pairwise Majority Vote (PMV) across the stakeholders.
Trials: 20 trials with different randomized train-test splits (70/30).
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Reviews:
[{"review_id": "UE9xPWdbbR", "reviewer": "Reviewer_gu4X", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "__Summary:__\nThis paper presents the robustness of algorithms from the viewpoint of algebraic topology. In particular, the authors characterize the lack of algorithmic stability in terms of the existence of a non-trivial first cohomology group of the 1D cochain complex derived from the utility function. The authors demonstrate that standard techniques to increase robustness, such as bagging and regularization, can be understood from this perspective -- as reducing this topological obstruction term.\n\n__Strengths:__\n- The perspective presented in this paper is unique and interesting.\n\n__Weaknesses:__\n- I find some of the arguments in the paper to be unclear and could be supported with more details (see below in __Requested Changes__).\n- The proposed obstruction-aware ensembling loss in Definition 6.1 has cubic complexity (due to the triple sum). This doesn't seem to be very practical in real-world settings.\n- I find the experiments to be quite weak overall and unconvincing. All of them are on simple synthetic datasets that are quite far from realistic ML tasks. Explanations are ambiguous and there's not enough details to undetstand what it is doing. More concretely,\n    - Experiment 1 is confusing to me; what is the purpose if the proposed solution to the instability is a method that return _all_ hypotheses?\n    - In Experiment 2, how reliable is the result if the magnitudes observed of the obstruction term are of the order of magnitude of machine precision? Couldn't the observed \"improvement\" be just noise?\n    - I don't understand what Experiment 3 is showing. What is the hypothesis space and the model that is being trained? When discussing \"improvements\", it is improvements with respect to what?\n- Methods such as bagging and regularization are typically used to increase generalizability (practically, this is robustness on a separate held-out validation set). I don't see the immediate link between this and the training data stability that is considered in this work.", "claims_and_evidence_yn": "No", "claims_explanation": "I find the presentation in the paper to be vague overall and do not provide enough convincing evidence about its claims. Theoretically, some of the steps in the proof are brushed over, relying more often than not on plain language arguments than concrete mathematical claims. Additionally, I find the experiments to be too simplistic, detatched from real-world ML problems, and lacking in concrete details.", "audience_yn": "Yes", "audience_explanation": "If presented more convincingly, I believe there will be some interest in the direction of the work; of trying to understand model instability from a topological viewpoint.", "requested_changes": "Overall, I find the paper to be very mixed in terms of mathematical rigour. Some concepts are introduced precisely, yet others are vague (or perhaps, at least to those who are not familiar with social choice theory). I would encourage the authors to be more precise throughout. For example:\n- Could you please state the construction of the 0-cochains, 1-cochains, and the coboundary operator more concretely in Section 4.2? For example, what is \"pairwise comparison\" and what does it mean to \"transform utility into induced comparisons\".\n- How is $\\succ$ defined without reference to a dataset? In Definition 3.2 $\\succ\\_S$ is defined as a preference ordering with respect to a dataset $S$, but in Section 4, a \"global\" consensus ranking $\\succ\\_G$ is introduced without a precise definition, and thereafter, $\\succ$ without reference to a specific dataset is used, with no mention of what it means.\n\nSome of the arguments in the proofs are not clear, and the mathematical rigor is questionable. For example:\n- In the proof of Proposition 4.4, can you be more mathematically precise about the statement \"If the stability constraint $\\epsilon$ is too tight relative to the separation of the hypothesis and the geometry of $\\mathcal{D}$, the algorithm cannot transition between the preferred hypotheses $h\\_i$, while remaining stable\".\n- In the proof of Proposition 5.1, I don't understand why \"Instability increases the probability that the local preferences form a cycle\".\n- The statement about the bagged learner decreasing the variance of $V_{ij}$ in Proposition 5.1 is not very rigorous.\n- As before, I don't understand why \"the occurence of localized inconsistencies (local cycles) decreases\" as \"the preferences become more stable\".\n- The statement of Proposition 5.1 talks only about decreasing the topological obstruction term, but algorithmic interpretation is provided only in the cases when the term is either trivial (a global preference exists) or non-trivial (a structural inconsistency exists). What does are the algorithmic implications in the non-trivial case, when the obstruction term is large or small?", "broader_impact_concerns": "The work is mostly theoretical. To the best of my knowledge, there are no negative ethical implications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "B4STmlDSPL", "reviewer": "Reviewer_tpE2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper draws interesting connections between topological social choice theory and algorithmic stability. It considers a finite collection of datasets, $\\mathcal{D}$, viewed as a graph where two datasets are adjacent if they differ by only one data point. A learning algorithm is a map $\\mathcal{D} \\to \\mathcal{H}$. From the point of view of social choice theory, we can view each dataset in $\\mathcal{D}$ as a voter, each having some preferences over the models in $\\mathcal{H}$ induced by a loss function. The learning algorithm selects the winning model given each dataset based on those preferences. \n\nTopological social choice theory asserts that attempts to derive a global ranking of $\\mathcal{H}$ based on local preferences usually lead to Condorcet cycles. The paper observes that these Condorcet cycles are a significant source of algorithmic instability: the learning algorithm does not choose \"similar\" hypotheses under small perturbations in the data (when datasets differ by only one data point). \n\nThe paper then analyzes how methods such as bagging and regularization prevent Condorcet cycles by smoothing the preference landscape, while inflated operators resolve Condorcet cycles by enlarging the target space. It also proposes other methods for reducing these obstructions.\n\n**Strengths**\n* The connection made between two seemingly very different fields is highly intriguing. It strengthens our existing understanding of algorithmic robustness by showing that inconsistent global rankings, which are somewhat inevitable, directly lead to algorithmic instability. This is a very novel perspective to me, and I believe it will be of interest to the ML community.\n\n**Weaknesses**\n* My primary concern with this paper lies in the writing. I believe additional effort is required to make the presentation clearer. \n\t* Since the mathematical framework heavily builds on topological social choice theory, I found the current Appendix 1 too brief and unclear to provide the necessary mathematical background. Furthermore, some section titles do not accurately reflect their content, and the overall logical flow of the paper feels disjointed.  (See \"Requested Changes\" for more specific feedback.)\n* The \"Propositions\" and \"Proofs\" in this paper feel more like informal mathematical explanations rather than rigorous statements and proofs that clarify the key assumptions being made. While this level of rigor might be acceptable to some extent, it would still be better if the authors could formalize these more rigorously. If the authors instead have reasons against this, they might wish to refrain from labeling them as \"Propositions/Proofs,\" or they could explicitly add comments in the paper clarifying this lack of formality.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes. The proofs are given for the propositions, and experiment results support the arguments of the paper.", "audience_yn": "Yes", "audience_explanation": "Yes, I believe the findings of the paper make a valuable contribution to enhancing our understanding of algorithmic robustness and would be of interest to TMLR's audience.", "requested_changes": "These comments suggest optional changes that I believe could make the paper more readable, though the authors may disagree. The comments also highlight a few confusions I had while reading the paper --- please consider making changes where my confusions seem reasonable or might be shared by other readers.\n\n**Introduction**\n*(Optional)* I believe it would improve clarity to add a short, intuitive overview in the introduction explaining the connection between social choice theory and algorithmic stability, given that they initially seem like very different areas. For example, you could include a paragraph explaining how \"datasets\" are viewed as voters, and how social choice theory reveals the inevitability of the \"Condorcet Paradox,\" which directly leads to algorithmic instability.\n\n**Related Work**\nIs there a specific reason why topological social choice theory is not addressed as its own subsection in the \"Related Work\" section? It seems like a good opportunity to introduce the main ideas and key conclusions of this field—especially those most relevant to the paper.\n\n**Section 3**\n* The titles of the sections and subsections in Section 3 do not align well with the content. It takes reading much of the later sections to understand what these terms mean, making the organization feel confusing.\n    * For example, Section 3 is titled \"Stability as Constrained Preference Aggregation.\" This is confusing since the term \"preference aggregation\" isn't fully explained until the end of Section 4.1 and Appendix A.2. Moreover, it is not clear why or how \"stability\" corresponds to \"constrained preference aggregation,\" as the title suggests.\n    * Similarly, Section 3.2 is titled \"Accuracy-Stability Trade-Off,\" but the section content does not actually discuss any trade-offs. Instead, it defines \"The Stable Selection Problem\" as one that has both accuracy and stability. The \"trade-off\" seems to emerge later in Proposition 4.4, but the current organization makes this connection unclear.\n\n**Section 4**\n* **(Confusion)** Line 2 in Section 4: What does \"stable aggregation\" mean? Is \"algorithmic stability\" considered a type of \"aggregation\"? If so, how?\n* **Terminologies**: To clarify, are the following interpretations correct? \n    - \"Preference\" refers to an ordering on the hypotheses.\n    - \"Preference relation\" is essentially the same concept as \"preference.\"\n    - \"Space of preferences\" is the space consisting of all possible orderings.\n    - \"Preference profile\" is a collection of \"preferences,\" one per dataset.\n    \n    If this is correct, I suggest rephrasing Definition A.1 for greater clarity and moving at least part of it to the main text.\n* Once again, the section/subsection titles do not accurately reflect the content:\n    * Section 4.1 introduces \"preference profiles\" but does not explore the \"geometry\" of preferences, despite the section's title.\n    * The title of Section 4.3, \"Mapping Stability to Cohomological Obstruction,\" is also vague and hard to follow. What exactly do you mean by \"mapping\" in this context?\n* **Cohomology Construction in Section 4.2**: I struggled to follow the necessary background for understanding the material in Section 4.2. Unfortunately, Appendix A does not provide enough clarity either. I had to conduct additional reading to fully grasp the concepts. It would be helpful if the authors provided a more intuitive explanation, emphasizing:\n    - The motivation behind the cohomology construction.\n    - Why coboundaries correspond to consistent preferences.\n    - Why the cohomology group captures information about Condorcet cycles.\n* **Proof of Proposition 4.4**: The sentence of the proof at the end of page 4 is rather vague and unclear. Could you please make it more precise?\n\n**Section 5**\n* **Section 5.1.1**:\n    * **(Confusion)** Proposition 5.1 indicates that bagging reduces cohomological obstruction in the aggregate ordinal preferences, but the aggregated ordinal preference cochain $C_P$ is only one of the cochains. Does this mean the result does not provide a general reduction in $H^1$? Clarity on this point would be helpful.\n\n**Section 6**\n* **(Confusion)** Could you elaborate on why minimizing $\\mathcal{L}_{cocycle}$ reduces $H^1$? The one-line explanation provided is not sufficiently clear.\n\n**Section 7**\n* Could you provide more experimental details in the Appendix? As it stands, the setup of the experiments is not detailed enough to permit reproducibility.\n\n**Appendix A**\n* Generally, I think more background information is needed to make Appendix A self-contained and accessible to readers without prior familiarity with topological social choice theory.\n    * An intuitive explanation of Theorem A.2 would be particularly helpful given its importance to this work. Specifically:\n        - What is the topology on the space of preferences $\\mathcal{P}^N$ that defines continuity in Theorem A.2?\n        - How should we interpret \"continuous aggregation rule\" in this context? \n    * Additionally, the ordering of the subsections in Appendix A could be improved. For example, the cohomology group (Lemma A.3) is currently introduced before the definition of cochains and the coboundary operator (Section A.4.1), which makes the presentation harder to follow.", "broader_impact_concerns": "This work has no ethical implications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "l8Zj0FeaA1", "reviewer": "Reviewer_EcaC", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors revisit algorithmic stability from a topological perspective. A classification algorithm (mapping datasets S to hypotheses \nh) is stable if similar datasets are classified the same way. \n\n- They analyze this property in terms of the absence of cycles in a graph over hypotheses induced by pairwise (ordinal) preferences. \n- They then argue that well-known methods (e.g., bagging or output-space enlargement) can be viewed as reducing or eliminating such cycles. \n- Finally, they propose a “cohomologically inspired” regularization that discourages these cycles when minimizing the training loss.", "claims_and_evidence_yn": "No", "claims_explanation": "My main issue is that it’s hard to make sense of the mathematical results as stated: I couldn’t connect the definitions to the proofs. Since the paper proposes a new framework, that framework needs to be well defined and clearly presented.\n\n\n\n___________________________\n\n\n**Example 1**\n\nTheorem 4.2: \"*A preference profile P contains a Condorcet cycle if and only if the configuration of local preferences leads to a structural inconsistency. This inconsistency is mathematically characterized by the existence of a non-trivial class in the first cohomology group H1 of the underlying topological space of preferences.*\"\n\n- It is never explained what it means for a \"preference profile P to contain a Condorcet cycle\". An example of Condorcet cycle is given somewhere: \"A>B, B>C,C>A\", but what is the preference profile there? I have no idea what this means.\n- What does it means that there is \"a non-trivial class in the first cohomology group H1 of the underlying topological space of preferences\"? For this to make sense, there should be topology on the space of preferences. Where is this topology defined? \n\nThe proof points to Appendix A, but that didn’t resolve these gaps for me.\n\n__________\n\n**Example 2:**\n\nProposition 4.4. *If the preference profile induced by the data contains a structural inconsistency (a Condorcet cycle characterized by a non-trivial H1 class), then for sufficiently small accuracy tolerance δ and stability parameter ε, the Stable Selection Problem is unsolvable.*\n\nI still don’t know what the assumption precisely means. Reading the proof, it seems to rely on a stronger condition: there are disjoint regions of datasets where different hypotheses are preferred with a uniform margin (which is not what the statement says I believe). Maybe that stronger condition is true, maybe not. As written, the definition isn’t clear, and there’s a statement–proof mismatch.\n\n\n__________\n\n**Example 3:**\n\nProposition 5.1. *Bagging transforms the original preference profile P induced by an unstable base learner A into a smoother profile Pbag induced by the bagged learner Abag. This operation systematically reduces the magnitude of the cohomological obstruction in the aggregated ordinal preferences.*\n\nIn the proof, the paper shifts to a different (and much clearer) object: a preference cochain C_P^ord (= function on edges), which admits a standard Hodge decomposition into a gradient part plus a cyclical part (C_P^ord)_{cycle}.  The proof is saying that on expectation, after bagging, \n\n$$ \n|C^{P_{bag}}_{ord,cycle}| \\leq |C^{P_{bag}}_{ord}|,  (*)\n$$\n\nie \"bagging reduces the number of cycles\". That’s a precise mathematical target! But then:\n- Should earlier claims (Theorem 4.2, Prop. 4.4) also be framed directly in terms of C^P_{ord} and its Hodge split? If so, say so up front. This seems central but is only detailed in the appendix.\n- The argument for (∗) reads as a heuristic, not a proof. Heuristics are fine (this is not a math journal), but it should be clearly marked as such.\n______\n\nSummary: The paper’s big picture is interesting, but the claims aren’t backed by clear, precise definitions and proofs. At minimum, the paper should (i) define what is the topology of the preference profile space or switch consistently to the graph/Hodge setting with  C^P_ord, (ii) align statements with the assumptions actually used in proofs, and (iii) clearly distinguish formal results from heuristics.\n\t​", "audience_yn": "Yes", "audience_explanation": "The overall idea is fun and appealing. I especially like the attempt to view several methods that seem unrelated at first (e.g., bagging and output-space enlargement) as actually trying to achieve the goal.\n\nOne criticism: at times the paper leans heavily on cohomology jargon, but the concrete machinery used is the Hodge decomposition of a flow on a graph. Therefore, I’d minimize the topological vocabulary (you likely only need graph theory here) to make the paper more accessible to the average ML researcher (the primary audience for TMLR).", "requested_changes": "Requested changes were given in the two previous sections.\n\nAdditional comments:\n- What is the dataset space D? It is said that \"D is the set of finite training datasets\", which is seen \"as a graph where edges connect adjacent datasets (differing by one point)\", and then later that we assume that \"the dataset space D is connected.\" But the set of all finite subsets of a given set is clearly connected right? I am confused on what D should be.", "broader_impact_concerns": "No particular concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)