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REVIEW 4 major objections 5 minor 93 references

Concurrent vertical and horizontal federated learning with fuzzy cognitive maps

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A new federated learning framework built on fuzzy cognitive maps lets participants train together when neither their samples nor their features are the same, and the paper's experiments show that the participants' mean accuracy improves…

desk verdict Square FL with FCMs is a genuinely novel idea, but the weighting descriptions contradict the equations and the evaluation avoids the hard feature-alignment case. read the letter →

arxiv 2412.12844 v1 pith:5YQBEFQX submitted 2024-12-17 cs.LG

Jose L Salmeron , Irina Arévalo This is my paper · ORCID
classification cs.LG
keywords federatedlearningfuzzycognitivemapssquareverticalhorizontalnon-IIDdataprivacy-preservingmachineaggregationstrategies
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a federated learning framework that handles both horizontal and vertical data partitions at once, using fuzzy cognitive maps (FCMs) as the shared model. It introduces "square federated learning," in which each participant trains a local FCM on its own samples and features, sends only the learned adjacency matrix (not raw data) to a server, and the server merges the matrices into one augmented FCM that covers the union of all participants' feature nodes. The authors test four aggregation strategies—constant, accuracy-based, AUC-based, and precision-based weights—on twenty-two binary classification datasets, with five participants that each lose three randomly chosen features. They report that the mean accuracy of the participants' local models improves after federation under all four strategies, while the F1-score improvement is more sensitive to the aggregation choice. A sympathetic reader cares because this is a step toward privacy-preserving collaborative training in settings where participants' data schemas only partially overlap, as is common in healthcare and finance.

What carries the argument

The central object is the augmented adjacency matrix of a fuzzy cognitive map (FCM). Each participant trains a local FCM with particle swarm optimisation, producing an $n\times n$ weight matrix $W_i$; the server aggregates these with Eq. 2, computing for each cell the mean over all participants of that participant's weight if the edge exists in its model and zero otherwise. The result is a matrix with rows and columns for every node appearing in any participant's map, so models with different feature sets can be married into one global FCM; this is what lets horizontal and vertical federation proceed simultaneously. The four federation strategies differ only in the weights of the averaging: constant $1/n$, or normalised accuracy, normalised AUC, or normalised precision of each participant's local model on its test set.

What would settle it

Run the square federated protocol with two participants whose feature sets are completely disjoint, then compare each participant's test accuracy before and after federation; because the augmented matrix has zero blocks connecting the feature sets, no cross-participant information exists, so any accuracy gain would be an averaging artifact rather than evidence that the method exploits differing feature spaces.

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Extended reading notes

Core claim

The central claim is that fuzzy cognitive maps make it possible to federate models across participants whose data differs in both the sample dimension (horizontal federation) and the feature dimension (vertical federation), a combination the paper calls square federated learning and presents as the first such framework for FCMs. Because an FCM is an adjacency matrix whose entries are learned causal weights between concept nodes, the server can merge participants' local matrices into a single augmented adjacency matrix, averaging the weights on nodes that appear in several participants and placing participant-specific nodes in disjoint blocks. The experiments use five participants drawn from the same datasets, with each participant missing three features, and all four aggregation strategies raise the mean accuracy of the local models after federation; the constant-weight strategy also improves mean F1 score on 20 of 22 datasets, accuracy-based weights on 19, precision-based weights on 18, and AUC-based weights on fewer than 70% of datasets. The authors do not claim to beat centralized training, and note that the value of the proposal lies in enabling training in distributed settings where centralised training is not possible.

Load-bearing premise

The method assumes that a feature with the same name means the same concept in every participant and that averaging the independently learned weights of such features yields a global model at least as good as the local ones; if participants differ in how they encode or interpret a shared column, the aggregation loses its meaning.

Editorial extensions

If this is right

  • Participants with partially overlapping data schemas can train a shared FCM without any raw data leaving their premises, since only learned adjacency matrices are exchanged.
  • The federation step does not require a central server: the paper notes the aggregation can be performed by one participant or by all participants in a peer-to-peer arrangement.
  • A single augmented FCM replaces the usual separation of horizontal and vertical federated pipelines, so consortia whose members' data differs along both axes can be handled by one protocol.
  • If F1 score is the target metric, the constant-weight aggregation is the safest of the four tested strategies, while AUC-based weighting improves mean F1 on fewer than 70% of the experimental datasets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Eq. 2 divides by the total number of participants even when only some participants contain a given edge, participants with heavily pruned feature sets are effectively diluted in the shared nodes; averaging only over the participants that possess the edge might change results on skewed overlaps.
  • A stress test the paper does not run is to vary the number of features removed per participant; the experiments fix three removals, so behaviour at extreme vertical splits with only one shared feature remains unknown.
  • The aggregated FCM could double as a diagnostic artifact: edges with large post-federation weights may reveal which concepts consistently matter across participants, a privacy-preserving byproduct the paper does not discuss.
  • If the square idea transfers to other model families whose parameters can be aligned and averaged (such as sparse linear models with a shared index space), the approach might generalize beyond FCMs; the paper explicitly leaves this generalization open.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes "square federated learning," a framework in which multiple participants with partially overlapping feature spaces and different sample spaces train local fuzzy cognitive maps (FCMs) via particle swarm optimization and send their adjacency matrices to a server. The server aggregates these matrices using one of four weighting schemes—constant, accuracy-based, AUC-based, and precision-based—via the augmented adjacency-matrix construction of Eq. (2). The authors report experiments on 21 PMLB classification datasets (though the text says 22) with five participants per dataset, and they claim that mean post-federation accuracy improves over pre-federation accuracy in all four strategies while F1 improvements depend on the strategy. The paper frames this as the first approach to concurrently address horizontal and vertical federated learning with FCMs.

Significance. The conceptual idea is attractive and potentially useful: FCMs have a natural aggregation operation through augmented adjacency matrices, so they are a plausible model class for federated settings with heterogeneous feature spaces. A working square-FL method for FCMs would be a genuine contribution to the federated-learning literature, and the paper explicitly and honestly states that its goal is not to beat centralized baselines but to enable training under distributed, non-IID feature sets. However, the paper's central empirical claim is currently supported mainly by point estimates without variance, significance tests, or a disclosed evaluation protocol that prevents the aggregation weights from being derived from the same test labels used for evaluation. The manuscript also contains a direct internal contradiction between the stated intent of the weighting strategies and the equations that define them. If these issues are fixed, the contribution could be valuable; in its present form, the quantitative evidence is not yet convincing.

major comments (4)
  1. [§4.2 and §5, Eqs. (7)-(9), Tables 3-6] The aggregation weights in Eqs. (7)-(9) are computed from each participant's test set, and the same test set is then used to report the post-FL accuracy and F1 values in Tables 3-6. This makes the reported improvements an in-sample evaluation: the federated model is partially selected to perform well on the exact labels used for the headline numbers. The authors should compute the weighting metrics on a separate validation set and reserve a disjoint test set for final evaluation, or at minimum demonstrate that the improvement persists under such a protocol.
  2. [§4.2, Eqs. (7)-(9)] The prose after Eqs. (7), (8), and (9) states that the aggregation gives more weight to less accurate or less precise models, and that individual models contribute "inversely proportional to their performance metric." The equations, however, divide by the sum of the metric values, so a participant with higher accuracy, AUC, or precision receives a larger weight. This is the opposite of the stated intent and directly affects how the experimental comparisons among strategies are interpreted. The text or the equations must be corrected.
  3. [§5, Tables 3-6] Every reported result is a single mean over five participants, with no standard deviation, no number of PSO runs, no seeds, no train/test split details, and no code. Since PSO is stochastic and the random removal of features changes the participants' feature spaces, the claim that accuracy improves "in all cases" is not supported without evidence that the differences are stable across runs. The authors should report distributions over repeated runs, provide significance tests or confidence intervals, and release code or at least exact seeds and split procedures for reproducibility.
  4. [§4.1 and Eq. (2)] The augmented-matrix aggregation in Eq. (2) is only semantically meaningful when shared feature nodes denote identical concepts across participants and when the locally learned weights estimate the same causal relations. In the experiments this alignment is guaranteed by construction: all participants are derived from a single PMLB dataset and shared columns are identical by definition. The paper identifies partial overlapping attribute skew as the target scenario in §4.1 but provides no experiment or argument for cases where feature names align while meanings differ, where the same concept is labeled differently, or where independently optimized PSO solutions are not aligned by the objective. The authors should either add experiments with deliberately misaligned or renamed shared features or clearly scope the claim to feature-compatible settings.
minor comments (5)
  1. [§5.2, Tables 2-6] The text says the experiments use "twenty-two open classification datasets," but Tables 2 through 6 each contain only 21 dataset rows. Please correct the count or add the missing dataset.
  2. [§3.1.1, Eq. (2)] Eq. (2) is difficult to parse: the symbol M is used without being defined, and the dimensions of the submatrices are not stated precisely. Please define M and clarify the indexing so that the averaging over existing entries is unambiguous.
  3. [§5.1, Table 2] The centralized neural-network baseline is described only as having "6 hidden layers and dropout" with no optimizer, learning rate, epochs, or architecture details. This is not load-bearing for the main claim, but the baseline would be more interpretable if those details were provided.
  4. [§3.1.1] The notation for the direct-sum operator and the augmented aggregation operator is inconsistent with the later use of Eq. (2); please unify the notation across the section.
  5. [General] The paper repeatedly emphasizes the semantic significance and interpretability of FCM weights. Given that emphasis, a short discussion of whether averaging independently learned weights preserves semantic meaning under non-IID data would strengthen the presentation.

Circularity Check

1 steps flagged · score 4.0 of 10

Evaluation protocol for metric-weighted federation uses the same per-participant test set both to set the aggregation weights and to report post-FL accuracy/F1, making the measured gains partly self-confirming; the framework itself is not circular.

  1. fitted input called prediction [Section 4.2, Eqs. 7-9; Section 5 experimental setup, Tables 4-6]
    "Accuracy-based weights: Weighted average using the normalised accuracy of the model computed in a test set of each participants ... Φ′j = Σ_{i=1}^{n} (acc_{ji}/Σ_{k=1}^{n} acc_{jk}) · Φ_{ji} ... The results are presented in tables with the average accuracy and F1 metrics for all participants ... These metrics will be computed using a test dataset in each participant."

    The per-participant test set is used twice: Eqs. 7-9 compute accuracy/AUC/precision weights from a test set, and the post-FL accuracy and F1 scores in Tables 4-6 are measured on the same type of per-participant test set. Thus the metric used to select the aggregation weights is the same metric reported as the outcome; improvements in the accuracy-, AUC-, and precision-weighted experiments are partly self-confirming because the federation is optimised on the evaluation data. The constant-weight experiment avoids this reuse, so the circularity is confined to the metric-weighted evaluation.

full rationale

The central methodological derivation is not circular: Eq. 2 is a stated aggregation rule for augmented FCMs, Eqs. 6-9 are standard weighted averages, and participant FCMs are learned by PSO from local data without fitting the target result. The framework's novelty claim for square FL with FCMs is not derived from a prior result by the authors; self-citations appear as background and are not load-bearing. The main circularity is in the empirical protocol for the metric-based aggregations. Section 4.2 defines accuracy/AUC/precision weights using a test set for each participant, and Section 5 says the reported pre/post accuracy and F1 are computed using a test dataset in each participant. The same test-set metrics are therefore inputs to the aggregation in Eqs. 7-9 and outputs in Tables 4-6. Improvements in the metric-weighted experiments are partly self-confirming: the federation is tuned on the data used to measure success. The constant-weight experiment is unaffected, so the framework itself is not circular and the score is moderate rather than high. A separate limitation, not circularity, is that the experiments guarantee shared-feature semantic alignment by removing features from a single PMLB dataset, so the claim for arbitrary non-IID feature spaces is not independently established. There is also a consistency issue: the prose says weights are inversely proportional to performance while Eqs. 7-9 give larger weight to higher metric values; that is a correctness concern, not a circularity concern.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The framework rests on known FCM aggregation and PSO learning, plus several unstated assumptions about node alignment, weight averaging, privacy, and evaluation. The only truly new part is the combination of these pieces, which is why the ledger lists no invented entities but does list hand-chosen experimental parameters.

free parameters (5)
  • PSO iterations = 50
    Fixed for all experiments with no sensitivity analysis.
  • PSO swarm size = 10
    Fixed for all experiments with no sensitivity analysis.
  • State convergence tolerance = 1e-5
    Used as the stopping condition for FCM state iteration.
  • Number of participants = 5
    All experiments use five participants with no variation.
  • Features randomly removed per participant = 3
    Creates the vertical split, but no protocol is given for datasets with fewer than three features.
assumptions (4)
  • domain assumption Augmented FCM aggregation (Eq. 2) produces a valid global FCM from local adjacency matrices when participants share some nodes.
    Taken from prior work (ref. 26), and the paper relies on it for the square FL mechanism; no proof or validation that averaging shared-node weights preserves classification behavior.
  • domain assumption PSO-learned FCM weights preserve semantic meaning and classify unseen data.
    The paper states weights must not be random for interpretability but does not analyze whether PSO produces stable or comparable weights across participants.
  • domain assumption Sharing only adjacency matrices is privacy-preserving.
    No cryptographic, differential privacy, or model-inversion analysis; the claim that confidentiality standards are met is asserted.
  • domain assumption The same test set used to compute aggregation weights is a valid set for reporting final performance.
    The protocol in Section 4.1 step 6 and Section 5 uses test-set metrics to weight participants and then reports those same metrics, which conflates model selection with evaluation.

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Cite this review

Pith. "Pith review of Concurrent vertical and horizontal federated learning with fuzzy cognitive maps." pith.science (2026). https://pith.science/paper/5YQBEFQX

@misc{pith2026241212844,
  author       = {Pith},
  title        = {Pith review of: Concurrent vertical and horizontal federated learning with fuzzy cognitive maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YQBEFQX}},
  note         = {Machine review of arXiv:2412.12844}
}
read the original abstract

Data privacy is a major concern in industries such as healthcare or finance. The requirement to safeguard privacy is essential to prevent data breaches and misuse, which can have severe consequences for individuals and organisations. Federated learning is a distributed machine learning approach where multiple participants collaboratively train a model without compromising the privacy of their data. However, a significant challenge arises from the differences in feature spaces among participants, known as non-IID data. This research introduces a novel federated learning framework employing fuzzy cognitive maps, designed to comprehensively address the challenges posed by diverse data distributions and non-identically distributed features in federated settings. The proposal is tested through several experiments using four distinct federation strategies: constant-based, accuracy-based, AUC-based, and precision-based weights. The results demonstrate the effectiveness of the approach in achieving the desired learning outcomes while maintaining privacy and confidentiality standards.

Figures

Figures reproduced from arXiv: 2412.12844 by the authors.

Figure 1
Figure 1. A federated learning process. Adapted from [67]. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Federated Learning categories the parameters of the local models at iteration j, denoted as Φj = [Φj1, Φj2, · · · , Φjn], where n represents the number of participants, Φji represents the parameters of the local model for participant i at iteration j, and Φ′ j represents the parameters (adjacency matrices in FCM models) of the federated model, the functions of the parameters that will be discussed are as fol￾lows: •… view at source ↗

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