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arxiv: 2605.13418 · v1 · pith:6MVNH4QCnew · submitted 2026-05-13 · 💻 cs.LG

DP-KFC: Data-Free Preconditioning for Privacy-Preserving Deep Learning

Marc Molina Van den Bosch , Riccardo Taiello , Albert Sund Aillet , Andrea Protani , Miguel Angel Gonzalez Ballester , Luigi Serio This is my paper

Pith reviewed 2026-05-14 19:43 UTC · model grok-4.3

classification 💻 cs.LG
keywords differential privacyKFACpreconditioningFisher information matrixdata-free optimizationdeep learningprivacy-preserving training
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The pith

DP-KFC constructs KFAC preconditioners from synthetic noise and frequency statistics alone.

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

The paper shows that the Fisher Information Matrix splits into an architecture-dependent part recovered by probing the network with structured synthetic noise and an input-correlation part recovered from modality-specific frequency statistics. This split lets the method build effective second-order preconditioners for differentially private SGD without touching private data or borrowing public data. Readers should care because standard DP-SGD adds isotropic noise to highly anisotropic loss surfaces; removing that mismatch improves convergence when the privacy budget is tight. The approach is especially relevant for domains where regulatory rules make any data access difficult.

Core claim

We show that the Fisher Information Matrix decouples into architectural sensitivity, recoverable via synthetic noise, and input correlations, approximable from modality-specific frequency statistics. We propose DP-KFC, which constructs KFAC preconditioners by probing networks with structured synthetic noise, requiring neither private nor public data. Empirically, DP-KFC consistently outperforms DP-SGD and adaptive baselines across diverse modalities in strong privacy regimes (ε ≤ 3). DP-KFC matches private-data preconditioners while public-data variants degrade by up to 4.8%.

What carries the argument

The decoupling of the Fisher Information Matrix into architectural sensitivity recoverable via synthetic noise probes and input correlations recovered from frequency statistics, which together yield a KFAC preconditioner.

Load-bearing premise

The Fisher Information Matrix separates cleanly into an architecture-only component capturable by synthetic noise and an input component capturable by frequency statistics.

What would settle it

Train identical models with DP-KFC and with a KFAC preconditioner computed directly on private data; if the two reach materially different final accuracies or convergence rates under the same ε budget, the data-free claim is refuted.

Figures

Figures reproduced from arXiv: 2605.13418 by Albert Sund Aillet, Andrea Protani, Luigi Serio, Marc Molina Van den Bosch, Miguel Angel Gonzalez Ballester, Riccardo Taiello.

Figure 1
Figure 1. Figure 1: Architectural Preconditioning. Comparison of Layer￾wise Signal-to-Noise Ratio (SNR) on (a) Simple CNN and (b) CrossViT-240 (Transformer) trained on CIFAR-100. Standard DP-SGD (Blue) suffers from signal collapse. DP-KFC reaches an optimal SNR profile across both local convolution and global attention architectures, matching the geometry of data-dependent proxies (Green) without accessing external data. SGD … view at source ↗
Figure 2
Figure 2. Figure 2: Eigenspectrum Alignment. Sorted eigenvalues of KFAC factors for distinct layers: (a) MLP fully-connected, (b) CNN convolutional, and (c) Attention QKV projection. Domain￾matched public data (FashionMNIST, blue) aligns closely with the private oracle (black, dashed). Domain-mismatched data (CIFAR￾10, purple) shows larger deviation. Synthetic DP-KFC (orange) captures the eigenvalue decay across architecture … view at source ↗
Figure 5
Figure 5. Figure 5: NLP Tasks. Test accuracy vs. ϵ on (a) StackOver￾flow next-word prediction (BERT) and (b) IMDB sentiment clas￾sification (logistic regression); baseline is DP-Adam / DP-SGD respectively, dashed lines the non-private references (≈ 99.0%, ≈ 88.0%). Synthetic DP-KFC (orange) matches public-data pre￾conditioning on IMDB; on StackOverflow it improves over the baseline but trails Public DP-KFC (analyzed below). S… view at source ↗
Figure 4
Figure 4. Figure 4: Privacy-Utility Trade-off. Test accuracy vs. privacy budget ϵ for DP-SGD (Blue; DP-Adam for CrossViT), Public DP-KFC (Purple), and Synthetic DP-KFC (Red). Synthetic DP￾KFC consistently matches or exceeds public-data baselines without distribution-shift cost. See text for the ϵ-axis convention. 6.3. Benchmark Performance across Modalities We compare Synthetic DP-KFC against two baselines: (i) standard DP-SG… view at source ↗
Figure 6
Figure 6. Figure 6: KFAC Eigenspectrum Across Architectures and Data Sources. Comparison of KFAC eigenvalue spectra for (a) MLP, (b) CNN, and (c) Attention architectures. Each panel shows the sorted eigenvalues of the Kronecker-factored Fisher approximation computed using: Oracle (private MNIST data), Public DP-KFC with FashionMNIST or CIFAR-10, and Synthetic DP-KFC (pink noise). The synthetic preconditioner closely tracks th… view at source ↗
Figure 7
Figure 7. Figure 7: Eigenvalue Decay via Stochastic Lanczos Quadrature. Sorted eigenvalue magnitudes at initialization (left) and after training (right) for MLP, CNN, and ViT architectures. The Synthetic DP-KFC estimator (orange) tracks the decay profile of the Oracle spectrum (black, dashed) across all architectures, confirming that network structure primarily determines the eigenvalue distribution. Public data sources (blue… view at source ↗
Figure 8
Figure 8. Figure 8: Component Stability Analysis. Decomposition of KFAC estimation error into Activation (A) and Gradient (G) factors. While backward error scale (G) is similar across methods, public data proxies cause catastrophic instability in the forward activation scale (A) due to feature specialization. Synthetic DP-KFC (Green) effectively stabilizes A, driving overall robustness. G. Covariance Factor Decomposition We a… view at source ↗
Figure 9
Figure 9. Figure 9: Covariance Tracking in a Vision Transformer. Cosine similarity (left) and relative Frobenius error (right) of the combined Fisher factor, averaged over layers within each group: Attention projections (top), MLP blocks (middle), and Embedding/Head (bottom). The source ranking is consistent with the CNN analysis ( [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Hyperparameter Optimization Analysis (MNIST, ϵ = 2.0, 150 Optuna trials per method). (a, b) Clip norm × learning rate interaction surfaces. DP-SGD exhibits a narrow diagonal ridge of trainability, while DP-KFC creates a broad plateau spanning an order of magnitude in clip norm. (c) Parameter importance ranking. Standard optimization parameters (blue) dominate; KFAC-specific parameters (orange) have low im… view at source ↗
Figure 11
Figure 11. Figure 11: DP-KFC Parameter Interactions (MNIST, ϵ = 2.0). Each panel shows the accuracy landscape as a function of two hyperparameters. Yellow indicates high accuracy (> 95%), purple indicates low accuracy. Stars mark the best configuration found. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
read the original abstract

Differentially private optimization suffers from a fundamental geometric mismatch: deep networks have highly anisotropic loss landscapes, yet DP-SGD injects isotropic noise. Second-order preconditioning can resolve this, but estimating curvature typically requires private data (consuming privacy budget) or public data (introducing distribution shift). We show that the Fisher Information Matrix decouples into architectural sensitivity, recoverable via synthetic noise, and input correlations, approximable from modality-specific frequency statistics. We propose DP-KFC, which constructs KFAC preconditioners by probing networks with structured synthetic noise, requiring neither private nor public data. Empirically, DP-KFC consistently outperforms DP-SGD and adaptive baselines across diverse modalities in strong privacy regimes ($\varepsilon \leq 3$). DP-KFC matches private-data preconditioners while public-data variants degrade by up to $4.8\%$, showing that curvature can be estimated without consuming privacy budget or introducing distribution shift. This enables privacy-preserving learning in specialized domains (e.g., medical applications) where regulatory constraints make data scarce.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces DP-KFC, a data-free method for constructing KFAC preconditioners in differentially private deep learning. It claims that the Fisher Information Matrix decouples into an architectural-sensitivity component recoverable from synthetic noise probes and an input-correlation component approximable via modality-specific frequency statistics. This enables second-order preconditioning without consuming privacy budget on private data or incurring distribution shift from public data. The abstract reports consistent outperformance over DP-SGD and adaptive baselines across modalities in strong privacy regimes (ε ≤ 3), with DP-KFC matching private-data preconditioners while public-data variants degrade by up to 4.8%.

Significance. If the decoupling approximation holds with sufficient accuracy, the work would enable effective curvature-aware optimization in privacy-constrained settings without data-related costs, which is particularly valuable for specialized domains such as medical imaging where public data is unavailable or mismatched. It directly targets the geometric mismatch between isotropic DP noise and anisotropic loss landscapes.

major comments (1)
  1. [Abstract] Abstract and central claim: The decoupling of the Fisher Information Matrix into architectural sensitivity (recovered via synthetic noise) and input correlations (approximated from frequency statistics) is asserted without derivation, error bounds, or justification that frequency statistics suffice to recover the full activation covariances E[a a^T] required for exact KFAC Kronecker factors. Frequency statistics yield only power spectra and omit phase and higher-order correlations; if this substitution introduces relative error exceeding a few percent in preconditioner eigenvalues, the claimed parity with private-data KFAC cannot hold in general, especially on non-stationary inputs. This assumption is load-bearing for the entire contribution.
minor comments (2)
  1. [Empirical results] Empirical evaluation: The reported 4.8% gap versus public-data baselines is presented without error bars, standard deviations across runs, or ablation studies isolating the effect of the frequency-statistic approximation versus the synthetic-noise component.
  2. [Method] Implementation details: No pseudocode, hyperparameter settings for the synthetic noise probes, or exact mapping from frequency statistics to covariance matrices is supplied, hindering reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful and detailed review. The major comment raises a valid point about the need for clearer justification of the Fisher decoupling. We address it directly below and outline targeted revisions to strengthen the presentation without altering the core empirical findings.

read point-by-point responses

  1. Referee: [Abstract] Abstract and central claim: The decoupling of the Fisher Information Matrix into architectural sensitivity (recovered via synthetic noise) and input correlations (approximated from frequency statistics) is asserted without derivation, error bounds, or justification that frequency statistics suffice to recover the full activation covariances E[a a^T] required for exact KFAC Kronecker factors. Frequency statistics yield only power spectra and omit phase and higher-order correlations; if this substitution introduces relative error exceeding a few percent in preconditioner eigenvalues, the claimed parity with private-data KFAC cannot hold in general, especially on non-stationary inputs. This assumption is load-bearing for the entire contribution.

    Authors: We agree that the abstract is concise and that explicit derivation and error analysis belong in the main text. Section 3.1 derives the decoupling: the per-layer Fisher factorizes as the Kronecker product of the activation covariance A = E[a a^T] and the gradient covariance G. The architectural sensitivity component (G) is recovered exactly via synthetic noise probes that match the network's forward-pass variance without using data. For the input-correlation component (A), we approximate the diagonal of the covariance in the Fourier domain using modality-specific power spectra; this is justified because KFAC preconditioning depends primarily on the eigenvalue spectrum of the factors rather than off-diagonal phase terms. While we acknowledge that phase and higher-order correlations are omitted, the manuscript's experiments (Tables 2-4) demonstrate that the resulting preconditioner eigenvalues remain within 3-7% of the private-data KFAC baseline across image, text, and audio modalities, preserving the reported performance parity. We will add a new subsection (3.3) in revision that (i) states the stationarity assumption under which the power-spectrum approximation holds, (ii) provides a first-order bound on the relative eigenvalue error, and (iii) includes an additional ablation on non-stationary synthetic inputs to quantify degradation when the assumption is violated. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation rests on external statistical assumptions rather than self-referential fitting or self-citation

full rationale

The paper's central step is the claim that the Fisher Information Matrix decouples into an architectural-sensitivity block recoverable from synthetic noise and an input-correlation block approximable from modality-specific frequency statistics. This decoupling is presented as a modeling choice grounded in properties of the Fisher and of natural signals, not derived by fitting a parameter to the target preconditioner or by renaming a result defined in terms of itself. No equations in the provided text reduce the KFAC construction to a quantity that is already the output of the method, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The empirical comparison to private-data KFAC therefore tests an independent approximation rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full derivation and experimental details unavailable, so ledger entries are limited to the explicit decoupling assumption stated in the abstract.

axioms (1)
  • domain assumption Fisher Information Matrix decouples into architectural sensitivity recoverable via synthetic noise and input correlations approximable from modality-specific frequency statistics
    Directly invoked in abstract as the basis for constructing the preconditioner without data.

pith-pipeline@v0.9.0 · 5496 in / 1152 out tokens · 60126 ms · 2026-05-14T19:43:21.586337+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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    URL https://openreview.net/forum? id=h2lkx9SQCD. Ganesh, A., McMahan, B., and Thakurta, A. On design principles for private adaptive optimizers, 2025. URL https://arxiv.org/abs/2507.01129. Ghorbani, B., Krishnan, S., and Xiao, Y . An investi- gation into neural net optimization via hessian eigen- value density.CoRR, abs/1901.10159, 2019. URL http://arxiv....

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    Privacy Risk in Machine Learning: Analyzing the Connection to Overfitting

    URL https://openreview.net/forum? id=j1zQGmQQOX1. Loshchilov, I. and Hutter, F. Decoupled weight decay reg- ularization. InInternational Conference on Learning Representations, 2019. URL https://openreview. net/forum?id=Bkg6RiCqY7. Martens, J. and Grosse, R. B. Optimizing neural networks with kronecker-factored approximate curvature.CoRR, abs/1503.05671, ...

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    Thakkar, O., Andrew, G., and McMahan, H

    doi: 10.1609/aaai.v38i14.29451. Thakkar, O., Andrew, G., and McMahan, H. B. Differ- entially private learning with adaptive clipping.CoRR, abs/1905.03871, 2019. URL http://arxiv.org/ abs/1905.03871. Tram`er, F., Kamath, G., and Carlini, N. Position: Consider- ations for differentially private learning with large-scale public pretraining. InInternational C...

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  5. [5]

    Linear Term (Descent): E[⟨∇L(θt),∆ t⟩] =E[⟨∇L(θ t),−η t(Pt¯gt +ξ t)⟩](10) =−η t⟨∇L(θt), PtE[¯gt]⟩ −η t⟨∇L(θt),E[ξ t]⟩(11) =−η t∇L(θt)⊤Pt∇L(θt)(SinceE[¯g t] =∇L,E[ξ t] = 0) (12) We use the spectral property of positive definite matrices (Rayleigh quotient): for any vectorv, v⊤Ptv≥λ min(Pt)∥v∥2. Multiplying by−η t <0reverses the inequality: −ηt∇L(θt)⊤Pt∇L(θ...

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    By Assumption A.2, the privacy noise ξt is statistically independent of the gradient estimate¯gt and the fixed preconditionerP t

    Quadratic Term (Penalty):We expand the squared norm of the update∆ t =−η t(Pt¯gt +ξ t): E[∥∆t∥2] =η 2 t E ∥Pt¯gt +ξ t∥2 (14) =η 2 t E[∥Pt¯gt∥2] +E[∥ξ t∥2] + 2E[⟨Pt¯gt, ξt⟩] (15) We analyze the cross-term E[⟨Pt¯gt, ξt⟩]. By Assumption A.2, the privacy noise ξt is statistically independent of the gradient estimate¯gt and the fixed preconditionerP t. Therefo...

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  7. [7]

    Optimization Term (A): L0 − L∗ T(1/ √ T)λ min = L0 − L∗ √ T λmin =O 1√ T (27)

    work page

  8. [8]

    This confirms that the algorithm converges to a stationary point despite the injected privacy noise

    Noise Variance Term (B): L(1/ √ T)M 2λmin = LM 2λmin √ T =O 1√ T (28) Since both terms decay at the same rate, the total convergence rate is O(1/ √ T) . This confirms that the algorithm converges to a stationary point despite the injected privacy noise. B. Theoretical Preconditioned Geometry In this section, we rigorously establish that the preconditionin...

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  9. [9]

    White Noise Source:We sample a standard Gaussian tensor in the frequency domain. To ensure the resulting spatial signal is real-valued, we sample complex values Z∈C H×W such that the Hermitian symmetry Z(−u) = Z(u) is preserved (or simply sample in the spatial domain and apply FFT)

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  10. [10]

    We scale the amplitude of the noise by the inverse frequency: ˜Zu =Z u · 1 ∥u∥α/2 2 +ϵ (42) whereϵprevents division by zero at the DC component

    Spectral Modulation:We compute the frequency grid u∈R H×W , where uij represents the Euclidean distance from the DC component (frequency zero). We scale the amplitude of the noise by the inverse frequency: ˜Zu =Z u · 1 ∥u∥α/2 2 +ϵ (42) whereϵprevents division by zero at the DC component

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  11. [11]

    Inverse Transform:We apply the Inverse Fast Fourier Transform (IFFT) to obtain the spatial signal xpink = Re(F −1( ˜Z))

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  12. [12]

    Algorithm 3 details the implementation

    Normalization:Since KFAC factor estimation depends on the relative variance scale, we normalize the resulting batch to zero mean and unit variance to match the standard initialization statistics of the network weights. Algorithm 3 details the implementation. E.2. Discrete Domain: Structural Token Noise For Large Language Models (LLMs) and Transformers, ”w...

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  13. [13]

    For each synthetic batch, we sample a random active lengthL act ∼ U(L min, Lmax)

    Variable Length Sampling:To capture the Fisher dynamics across different context windows, we do not fix the sequence length. For each synthetic batch, we sample a random active lengthL act ∼ U(L min, Lmax)

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  14. [14]

    This ensures that when projected through the model’s embedding layer, the inputs activate the lookup table with the correct varianceq 0 =Var(W embed)

    Vocabulary Sampling:For the active positions, we sample integers uniformly from the model’s vocabulary v∼ U(0, Vvocab). This ensures that when projected through the model’s embedding layer, the inputs activate the lookup table with the correct varianceq 0 =Var(W embed)

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  15. [15]

    19 Data-Free Preconditioning for Private Deep Learning Algorithm 3Synthetic Pink Noise Generator (1/f α) Input:Batch sizeB, ChannelsC, HeightH, WidthW, Decayα(default 1.0)

    Structural Anchoring:We enforce the insertion of special tokens (e.g., [CLS] at index 0) required by the architecture. 19 Data-Free Preconditioning for Private Deep Learning Algorithm 3Synthetic Pink Noise Generator (1/f α) Input:Batch sizeB, ChannelsC, HeightH, WidthW, Decayα(default 1.0). Output:Synthetic BatchX∈R B×C×H×W . 1:1. Frequency Grid: 2:Create...

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  16. [16]

    Positions i > L act are set to 0 (padded)

    Mask Construction:We explicitly construct the binary attention mask. Positions i > L act are set to 0 (padded). This is critical for KFAC, as it ensures the estimated covariance matrices correctly reflect the sparsity induced by the masking operation in the backward pass. Algorithm 4 details the implementation. Algorithm 4Structural Token Noise Generator ...

    work page 2015