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REVIEW 3 major objections 4 minor 38 references

Privacy-aware Berrut Approximated Coded Computing applied to general distributed learning

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper generalizes Private Berrut Approximate Coded Computing to three distributed-learning configurations—centralized training, secure aggregation, and fully decentralized training—and reports that all three keep per-participant…

desk verdict A broader PBACC with two working modes and one unproved mode: secure training over decentralized data relies on an invalid interpolation assumption, and the privacy bound is circular. read the letter →

arxiv 2505.06759 v1 pith:7I7HK4NM submitted 2025-05-10 cs.LG cs.CRcs.DCcs.ITmath.IT

Xavier Martínez-Luaña , Manuel Fernández-Veiga , Rebeca P. Díaz-Redondo , Ana Fernández-Vilas This is my paper · ORCID
classification cs.LGcs.CRcs.DCcs.ITmath.IT MSC 68P2765D0594A1768T05
keywords privatecodedcomputingBerrutrationalinterpolationfederatedlearningprivacysecureaggregationdistributedtrainingdecentralizedmutualinformationleakageapproximate
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 claims that Private Berrut Approximate Coded Computing, a scheme that encodes data as Gaussian-noised rational interpolation shares before sending it to workers, can be exported from federated aggregation to the full range of distributed learning layouts. Three adapted protocols are given: secure training when one owner distributes a dataset, secure aggregation when many owners train locally and a master combines their models, and secure training when both the model and the data stay encoded end to end. The engineering value is that the scheme works for arbitrary nonlinear learning functions on real-valued tensors, needs no input quantization, tolerates stragglers, and carries a per-data-element privacy guarantee. In experiments with convolutional networks, variational autoencoders, and Cox regression, the encoded protocols keep mutual-information leakage at or below one bit per participant while model quality is largely unchanged, except for the fully decentralized training variant.

What carries the argument

The load-bearing object is the PBACC interpolating rational function $$u_X(z)=\frac{\sum_{j=0}^{K-1}\frac{(-1)^j}{z-\alpha_j}X_j+\sum_{j=0}^{T-1}\frac{(-1)^{j+K}}{z-\alpha_{K+j}}R_j}{\sum_{k=0}^{K+T-1}\frac{(-1)^k}{z-\alpha_k}},$$ with $X_j$ the data slices, $R_j$ Gaussian noise tensors, $\alpha$ the Chebyshev interpolation nodes of the first kind, and evaluations at Chebyshev nodes of the second kind $\beta_j$ forming the shares sent to workers. The decoder is the Berrut rational interpolant built from the fastest workers' outputs, which recovers $f(X_j)$ approximately at the $\alpha_j$ nodes. The privacy metric $i_L$ is the normalized worst-case mutual information between the private inputs and the shares observed by up to $c$ colluding nodes, bounded through a MIMO channel-capacity formula; the randomness $T,\sigma_n$ sets the leakage level. This same mechanism carries all three distributed-learning protocols, with the only differences being which object (dataset, local model, or global model) is encoded and which entity decodes.

What would settle it

Run the secure-training-over-decentralized-data protocol with two or more workers whose local datasets are strongly non-IID, for example disjoint MNIST digit classes, then compare the decoded global model against the plain average of the same workers' local models. If the decoded model's accuracy is much worse than the average, or if permuting which worker's share sits at which interpolation point changes the decoded model materially, the delta-stability-as-average justification fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the PBACC encoding, built from a barycentric rational function whose numerator places data points and Gaussian random tensors at distinct interpolation nodes, is not tied to a single master-owner setup. The scheme is generalized to tensor-valued inputs and to multiple data owners, yielding three workable modes: distributed training over centralized data, secure aggregation over decentralized data, and secure training over decentralized data. In the third mode, each worker trains on an encoded copy of the global model with its own private data, and the master decodes a new model by Berrut interpolation; the paper argues this is meaningful because Berrut rational interpolation is delta-stable, so the decoded result approximates an average of the trained models. The measured consequence is that the three protocols attain less than one bit of leakage per data element for groups of honest-but-curious colluding workers, with model accuracy essentially preserved for CNN and Cox experiments and a clear accuracy cost only in secure training over decentralized data.

Load-bearing premise

The load-bearing assumption is that decoding the complete training outputs produced by different workers on their own private data yields a meaningful global model: the paper invokes delta-stability of Berrut interpolation to say the decoded result behaves like an average, but it does not prove that interpolating different data-dependent training functions produces a valid aggregate.

Editorial extensions

If this is right

  • The three PBACC protocols give a single privacy mechanism for distributed learning that does not require exact computations, finite fields, or quantized inputs.
  • At the tested security levels (leakage at most one bit per element under twenty percent colluding nodes for the CNN and Cox cases), CNN and Cox model quality is essentially the same as uncoded training, so privacy can be added without hurting those models.
  • The fully decentralized training variant is the most private but costs accuracy and convergence speed; for the Cox model this cost nearly vanishes, so simpler models tolerate the strongest setting.
  • Because the approximation error does not accumulate with the number of workers, the scheme behaves differently from differential privacy as the network grows.
  • Encoding and decoding cost scales with the split parameter $K$ and with the degree of decentralization, giving a tunable communication-versus-computation trade-off, such as $N(N-1)$ share exchanges for secure aggregation versus unchanged per-round communication for secure training.

Reading between the lines

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

  • A testable extension is to treat PBACC secure training as a privacy-preserving approximation of FedAvg and measure the gap between its decoded model and the explicit average of local models as a function of data heterogeneity; the paper's delta-stability argument predicts the gap stays small, but does not test it.
  • Because the leakage bound is derived from a MIMO channel model under Gaussian noise, the same machinery could yield explicit privacy-versus-utility curves for other interpolation point sets beyond Chebyshev nodes, which might improve the accuracy of the fully decentralized mode without lowering privacy.
  • The numerical overflow that forced the VAE experiments to use a Gumbel-Softmax reparameterization suggests the encoded-training mode will need numerical safeguards when applied to models with exponentials; this is a practical boundary of the method rather than a tested claim.
  • A system designer could exploit the $K$ trade-off differently from the paper, for instance by encoding only selected layers of a model or by using larger $K$ in secure aggregation to improve precision at a predictable communication cost.
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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

3 major / 4 minor

Summary. The manuscript extends Privacy-aware Berrut Approximated Coded Computing (PBACC) to three distributed learning configurations: secure training over centralized data (DLCD), secure aggregation over decentralized data (DLDD), and secure training over decentralized data (DLDD). It proposes tensor-based encoding/decoding, a MIMO-capacity privacy bound (Eq. (12)) with leakage per data element iL, and reports experiments on CNN/MNIST, VAE/Fashion-MNIST, and Cox/METABRIC. The central claim is that PBACC is a general solution providing bounded leakage below one bit with minimal quality loss in all three configurations.

Significance. The paper addresses a real gap: extending approximate coded computing with privacy to non-linear ML training and decentralized data. Its strengths are the breadth of the experimental campaign (three model families, three configurations, convergence and timing measurements) and a clear threat model of colluding honest-but-curious nodes. The efficiency comparison in Table II is useful. However, the theoretical support for the main generalization is not sound: the secure decentralized training mode lacks a valid decoding justification, and the privacy bound in Eq. (12) is dimensionally inconsistent. The reported leakage values are determined by the same equation used to select parameters, so they are not independent verifications. If the DLDD secure-training claim were removed and the privacy analysis corrected, the remaining contributions (secure aggregation and DLCD secure training) would be more modest incremental extensions of [25].

major comments (3)
  1. [Section V-B2] The decoding operation in secure training over decentralized data is not justified. PBACC decoding via Eq. (17) is valid when the interpolated points are evaluations of one fixed function at the encoded points; here each worker j evaluates a different function full_train(·; X_j) on its private data and from a different encoded starting point u_theta(beta_j). The δ-stability of Berrut interpolation bounds the interpolation operator and does not imply that the decoded value approximates the desired global model (e.g., the FedAvg aggregate of full_train(theta; X_j)). No proof or convergence argument is supplied. The experiments in Tables IV and VII show a substantial quality loss for this configuration (CNN accuracy 0.86 vs 0.98; VAE loss 461.1 vs 289.8), and Tables V and VIII show severe degradation as sigma_n increases, contradicting the abstract's claim of minimally altered quality. Since this configuration is presented as one of the three modes of the general solution, the central claim is not established.
  2. [Section III-C, Eqs. (12)-(13)] The privacy bound is formally invalid as written. The matrix \tildeΣ_c defined in Eq. (13) is c×T and Σ_c is c×K, yet Eq. (12) uses \tildeΣ_c^{-1}, which only exists if c=T and the matrix is square. Furthermore, the matrices in Eq. (13) contain interpolation basis evaluations q_i(beta_j), not covariance matrices of encoded input and noise; the MIMO channel capacity formula in Eq. (9) applies to covariance matrices of the transmitted signal and additive noise, and the paper does not derive how these basis-evaluation matrices arise as covariances. Because the parameters T and sigma_n in Tables III, VI, and IX are chosen using Eq. (12), the reported leakage bounds are design constraints imposed by the equation rather than independent measurements of the protocol's privacy. The privacy guarantee stated in the abstract is therefore unsupported.
  3. [Section VII-A, Tables IV-V] The paper's own numerical results contradict the 'minimally altered' claim for the DLDD secure-training configuration even at the parameter settings chosen by the authors. In Table IV, the DLDD secure-training accuracy is 0.86 versus 0.98 for the uncoded baseline, and in Table V the accuracy degrades to 0.45 at σ_n=100 and 0.10 at σ_n=400. The text acknowledges this outlier but frames it as a privacy benefit; however, without a proof that the decoded model is a meaningful aggregate, the degradation is evidence that the interpolation-of-different-functions argument does not hold. This is a load-bearing issue for the claimed generality of PBACC.
minor comments (4)
  1. [Equation (16)] In the second sum of Eq. (16), the denominator has (−1)^j/(z−α_k) but the summation index is k; this appears to be a typo for (−1)^k. Please fix the index and ensure the denominator is consistent with Eq. (5).
  2. [Abstract and Tables III, VI, IX] The abstract states leakage is bounded 'strictly to less than a fraction of one bit per participant,' but Table VI reports a leakage bound of ≤ 1.0 bit for the VAE experiments. The wording should be aligned with the actual numeric values.
  3. [Section VII] All experiments appear to be single runs with no indication of seeds, variance across runs, or statistical significance. Given the large timing differences and the noise-based privacy mechanism, reporting means and standard deviations over multiple runs would strengthen the empirical claims.
  4. [Section V] There is a typo in the category name 'Dentralized Data' in the paragraph introducing DLCD and DLDD; it should read 'Decentralized Data.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the privacy thresholds are chosen design parameters, and the main generalization content is not reduced to its inputs.

full rationale

The only apparent circularity candidate is the leakage column: the entries in Tables III/VI/IX are bounds computed from Eq. (12), and Section V-A explicitly says that T and sigma_n were chosen to make that bound fall below 1 bit. That is a design constraint rather than a fitted prediction or an independently measured leakage value, so it does not qualify as a circular reduction under the stated rules. The PBACC encoding/decoding and the iL metric are presented with their defining equations and an information-theoretic derivation in Section III, with the authors' prior work [25] used as a pointer rather than as the sole load-bearing proof. The paper's genuine weakness is Section V-B2: the assertion that delta-stability of Berrut interpolation makes the decoded output of different per-worker training functions 'theoretically similar to an average' is an unsupported correctness claim about interpolating data-dependent functions, but that is a proof gap, not a circularity. No equation or numerical claim in the paper reduces by construction to its own input.

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

The central method depends on a set of design parameters (K, T, sigma_n, gamma) that are tuned to satisfy the leakage target and to improve convergence. The privacy proof rests on a Gaussian noise assumption and a MIMO capacity bound from the literature. The most fragile addition is the unproved averaging assumption for decentralized training.

free parameters (5)
  • K (number of data interpolation points) = 1 or 10
    Chosen per scenario to balance communication cost, precision, and leakage; directly affects the privacy bound and model accuracy.
  • T (number of random coefficients) = 18, 30, or 42
    Chosen so that the leakage formula returns below about one bit; appears in Eq. (12) as the number of noise terms.
  • sigma_n (noise standard deviation) = 10 or 30
    Chosen to meet the leakage target; larger values reduce leakage but degrade model quality in several experiments.
  • Learning rate gamma = 0.001 or 0.025
    Tuned per scheme to minimize convergence round, so accuracy results depend on this choice.
  • Noise interpolation shift b = unspecified
    Introduced in Eq. (7) for shifted Chebyshev points but never given a value or studied; it affects the noise basis.
assumptions (4)
  • standard math Berrut rational interpolation is delta-stable and approximates any target function with bounded error that decreases with the number of received results.
    This is the core BACC property taken from prior work [4] and used throughout the decoding steps in Section III-A and Eq. (17).
  • domain assumption The encoded noise from random coefficients is Gaussian and uncorrelated, so the MIMO capacity formula bounds the mutual information.
    Assumed in Section III-C before Eq. (12); the paper states 'assuming the noise is uncorrelated' but does not justify the covariance structure of the rational-function-weighted noise.
  • ad hoc to paper Decoding full_train outputs from different workers approximates an average of trained models because Berrut interpolation is stable.
    Stated in Section V-B2 without proof. Each worker computes a different data-dependent function, so interpolation of their outputs is not a standard function interpolation.
  • domain assumption The target function can be composed with the rational encoding and recovered via Berrut interpolation with tolerable error.
    Used in all three schemes; the paper notes conditions are discussed in Section V but never formalizes which function classes are admissible.

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Pith. "Pith review of Privacy-aware Berrut Approximated Coded Computing applied to general distributed learning." pith.science (2026). https://pith.science/paper/7I7HK4NM

@misc{pith2026250506759,
  author       = {Pith},
  title        = {Pith review of: Privacy-aware Berrut Approximated Coded Computing applied to general distributed learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7I7HK4NM}},
  note         = {Machine review of arXiv:2505.06759}
}
read the original abstract

Coded computing is one of the techniques that can be used for privacy protection in Federated Learning. However, most of the constructions used for coded computing work only under the assumption that the computations involved are exact, generally restricted to special classes of functions, and require quantized inputs. This paper considers the use of Private Berrut Approximate Coded Computing (PBACC) as a general solution to add strong but non-perfect privacy to federated learning. We derive new adapted PBACC algorithms for centralized aggregation, secure distributed training with centralized data, and secure decentralized training with decentralized data, thus enlarging significantly the applications of the method and the existing privacy protection tools available for these paradigms. Particularly, PBACC can be used robustly to attain privacy guarantees in decentralized federated learning for a variety of models. Our numerical results show that the achievable quality of different learning models (convolutional neural networks, variational autoencoders, and Cox regression) is minimally altered by using these new computing schemes, and that the privacy leakage can be bounded strictly to less than a fraction of one bit per participant. Additionally, the computational cost of the encoding and decoding processes depends only of the degree of decentralization of the data.

Figures

Figures reproduced from arXiv: 2505.06759 by the authors.

Figure 1
Figure 1. Distributed training over centralized data [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Secure aggregation over decentralized data [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Secure Training over decentralized data VI. COMPARISON OF PBACC DECENTRALIZED LEARNING MODELS Once the different operation models have been described, we provide a comparison according to privacy and efficiency from a theoretical perspective. The privacy analysis considers input privacy, taking into account who owns the data and if it reveals it to another entity. The efficiency analysis has been focused on each rou… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Comparison of the Accuracy evolution of all scenarios for the CNN experiments [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Comparison of the Accuracy evolution of all scenarios for the VAE experiments [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Comparison of the Accuracy evolution of all scenarios for the COX experiments [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.