Private Learning with Public Feature Conditioning

Nicolas Mayoraz; Shuli Jiang; Walid Krichene

arxiv: 2606.18773 · v1 · pith:KVRBHZVCnew · submitted 2026-06-17 · 💻 cs.LG · cs.AI

Private Learning with Public Feature Conditioning

Shuli Jiang , Walid Krichene , Nicolas Mayoraz This is my paper

Pith reviewed 2026-06-26 21:17 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords differentially private learningpublic featuresconditioning matrixDPSGDprivate regressionlabel DPoptimization landscape

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The pith

Cond-DP constructs a conditioning matrix from public features to accelerate convergence of differentially private SGD in regression without extra privacy cost.

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

The paper introduces Cond-DP as a variant of DPSGD that incorporates a conditioning matrix derived directly from public, non-sensitive features. This matrix reshapes the optimization landscape when those features have rapidly decaying spectra, yielding provable faster convergence rates than standard DPSGD for convex, strongly convex, and non-convex objectives. The approach recovers DPSGD when the matrix is the identity and incurs no additional privacy loss because the conditioning uses only public data. Empirical results show consistent outperformance over baselines in label-DP regression across datasets and architectures.

Core claim

Cond-DP incorporates a data-driven conditioning matrix derived from public feature matrices to reshape the optimization landscape in differentially private stochastic gradient descent, enabling faster convergence in private regression tasks while recovering standard DPSGD when the matrix is the identity.

What carries the argument

The data-driven conditioning matrix constructed from public feature matrices with rapidly decaying spectra.

If this is right

Private linear regression achieves provably faster convergence rates than DPSGD with no added privacy cost.
Convergence guarantees apply across convex, strongly convex, and non-convex problem settings.
Standard DPSGD is recovered exactly when the conditioning matrix is set to the identity.
Empirical performance improves over prior label-DP baselines on a range of datasets and model architectures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could extend to other DP optimization tasks where public covariates are available and exhibit similar spectral decay.
If public features in recommendation systems routinely satisfy the spectral condition, the privacy-utility gap for regression models could narrow in production settings.
Testing whether the same conditioning improves performance when labels are fully private rather than label-DP would clarify the scope of the gains.

Load-bearing premise

Public feature matrices exhibit rapidly decaying spectra that allow a data-driven conditioning matrix to meaningfully reshape the optimization landscape.

What would settle it

A dataset in which the singular values of the public feature matrix decay slowly, such that Cond-DP shows no faster convergence or worse performance than plain DPSGD under the same privacy budget.

Figures

Figures reproduced from arXiv: 2606.18773 by Nicolas Mayoraz, Shuli Jiang, Walid Krichene.

**Figure 2.** Figure 2: Example feature matrix spectra. Building on the above ideas, we make a related observation: in many applications, the public feature matrix may not be strictly low-rank; but may exhibit a rapidly decaying spectrum. See [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: Model with conditioning matrix C trained by Cond-DP. arg minΘpub ,Θpriv,ω L(Θpub , Θpriv, ω; D) be optimal parameters and L ∗ be the optimal loss. Finally, Nm×n(0, σ2 ) denotes the distribution of real m×n matrices with each entry sampled from the Gaussian distribution N (0, σ2 ). 4. Cond-DP: Conditioning Model Parameters with Public Features We propose Cond-DP (Algorithm 1), a variant of DPSGD (Abadi et… view at source ↗

**Figure 5.** Figure 5: Spectra of singular values in synthetic. and train all models for 1024 epochs using full-batch optimization [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Validation MSEs of different algorithms for private linear models on real-world datasets across varying privacy budgets ϵ. Owing to the larger dataset sizes, training is performed for 1000 epochs using batch size 100. The spectra of the datasets can be found in Appendix G.3. The results are presented in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: Validation MSEs of different algorithms for private linear models with linear input transformations and a two-layer MLP prediction layer (16, 8 hidden units) on real-world datasets across varying privacy budgets ϵ. 1,732,721 examples, and clip the conversion value at 400 Euro, which corresponds to the 95th percentile of the value distribution. We randomly split the data into 80% for training and 20% for t… view at source ↗

**Figure 8.** Figure 8: Validation MSEs of different algorithms on Criteo Sponsored Search Conversion across varying privacy budgets ϵ. (left) linear models; (right) models with linear input transformations and a two-layer MLP prediction layer (128, 64 hidden units). is well known that nonlinear models, such as models with MLP-based prediction layers, substantially outperform linear models on this dataset (Rendle, 2010), and are… view at source ↗

**Figure 9.** Figure 9: Singular value spectra of datasets used in the experiments. 6https://aws.amazon.com/ec2/instance-types/g6e/ 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

read the original abstract

We study differentially private (DP) regression in settings where each data sample includes public, non-sensitive features -- common in applications such as recommendation and advertising systems. While such label-DP or semi-sensitive-feature settings have been primarily explored in the context of classification, effective approaches for regression remain underexplored. We introduce Cond-DP, a conditioned variant of DPSGD that leverages the structure of public feature matrices to improve optimization under privacy constraints. Motivated by the observation that these public features often exhibit rapidly decaying spectra, Cond-DP incorporates a data-driven conditioning matrix to reshape the optimization landscape and accelerate convergence. We provide convergence guarantees for convex, strongly convex, and non-convex settings, and recover standard DPSGD as a special case when the conditioning matrix is the identity. We show how to construct an effective conditioning matrix for Cond-DP directly from public features, enabling provably faster convergence than DPSGD in private linear regression without incurring additional privacy cost. Empirically, Cond-DP with this conditioning matrix consistently outperforms state-of-the-art baselines across a wide range of datasets and model architectures under label DP, demonstrating strong and robust performance in practice.

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.

Desk Editor's Note private letter to a colleague

Cond-DP adds public-feature conditioning to DPSGD for label-private regression and recovers the base method as a special case, but the speed-up claim rests on spectral decay that the abstract treats as given rather than tested.

read the letter

The main takeaway is that this paper defines Cond-DP, a variant of DPSGD that inserts a conditioning matrix built from public features. The matrix is meant to improve the optimization landscape for private linear regression and other models, with convergence bounds given for convex, strongly convex, and non-convex cases. Standard DPSGD appears when the matrix is the identity, and the construction uses only public data so no extra privacy budget is spent.

The contribution sits in the explicit construction of the conditioning matrix from the public feature structure and the adapted analysis that shows when the method can beat plain DPSGD. The practical setting (label-DP regression with public covariates) matches real recommendation and advertising workloads, and the abstract reports consistent gains over baselines across datasets and architectures.

The central assumption is that public feature matrices exhibit rapidly decaying spectra; the abstract presents this as an observed property that justifies the approach rather than a condition the authors prove holds or test for robustness. If the spectra decay slowly, the conditioning matrix supplies no advantage and the method reduces to DPSGD with no gain. The paper would be stronger with either a robustness result or experiments on data where the spectral property is weak. Details on how the matrix is formed exactly, how error bars are computed, and whether any post-selection occurs also need verification in the full text.

The work is aimed at researchers who already study DP-SGD variants and need to handle mixed public-private features. Someone looking for a targeted improvement in private regression optimization will find the construction and guarantees worth reading. The idea is concrete enough, the privacy accounting stays clean, and the empirical claims are specific, so it deserves a serious referee rather than a desk reject.

Referee Report

1 major / 1 minor

Summary. The paper introduces Cond-DP, a conditioned variant of DPSGD for differentially private regression that constructs a data-driven conditioning matrix from public (non-sensitive) features. It claims this yields provably faster convergence than standard DPSGD for private linear regression with no additional privacy cost, provides convergence guarantees across convex/strongly convex/non-convex regimes, recovers DPSGD when the matrix is the identity, and shows consistent empirical outperformance over baselines under label DP.

Significance. If the central claims hold, the work provides a practical mechanism to exploit public feature structure for improved private optimization in regression without extra privacy expenditure, which is relevant for applications like recommendation systems. The explicit recovery of DPSGD as a special case when the conditioning matrix is the identity is a clear strength that anchors the contribution.

major comments (1)

[Abstract] Abstract (motivation paragraph): The claim of 'provably faster convergence than DPSGD' is motivated by the observation that public feature matrices 'often exhibit rapidly decaying spectra,' but this spectral property is presented only as motivation rather than a derived condition or robustness result. Since the analysis recovers standard DPSGD for the identity matrix, any strict improvement is load-bearing on this unproven spectral assumption; without it the guarantee reduces to the baseline.

minor comments (1)

The abstract states empirical outperformance 'across a wide range of datasets and model architectures' but provides no detail on error-bar handling, number of runs, or statistical testing; this should be clarified in the experimental section for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive comment on the abstract. We address the point below and indicate the planned revision.

read point-by-point responses

Referee: [Abstract] Abstract (motivation paragraph): The claim of 'provably faster convergence than DPSGD' is motivated by the observation that public feature matrices 'often exhibit rapidly decaying spectra,' but this spectral property is presented only as motivation rather than a derived condition or robustness result. Since the analysis recovers standard DPSGD for the identity matrix, any strict improvement is load-bearing on this unproven spectral assumption; without it the guarantee reduces to the baseline.

Authors: We agree that the provable improvement over DPSGD is not unconditional and depends on the spectrum of the constructed conditioning matrix. The convergence analysis in Sections 3–4 derives explicit rates that improve upon standard DPSGD precisely when the conditioning matrix (built from public features) yields a more favorable effective condition number; the identity-matrix case recovers the baseline rates as a special case. The construction procedure in Section 5 is designed to produce such a matrix whenever the public feature covariance exhibits the decaying spectrum that is typical in the applications considered. We acknowledge that the abstract presents this spectral property only as motivation. In the revision we will update the abstract to state explicitly that the faster convergence guarantee holds under the condition that the public features admit a conditioning matrix with improved spectrum (as derived in the analysis), thereby framing the spectral decay as a derived condition rather than mere motivation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs the conditioning matrix explicitly from public features (external to the private optimization) and provides general convergence guarantees that recover DPSGD when the matrix is the identity. The claimed improvement is conditional on the matrix being effective due to public feature spectra, which is stated as motivation rather than a derived or fitted quantity. No equations reduce a prediction to a fitted input by construction, no self-citation is load-bearing for the central claim, and no ansatz or uniqueness result is smuggled in. The analysis is independent of the target faster-convergence result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the domain assumption that public features have rapidly decaying spectra and on standard convex/non-convex optimization assumptions; no free parameters are explicitly fitted in the abstract description, and no new entities are postulated.

axioms (2)

domain assumption Public feature matrices exhibit rapidly decaying spectra
Motivation paragraph states this observation drives the conditioning matrix design.
standard math Standard assumptions for convex, strongly convex, and non-convex convergence analysis hold
Guarantees are stated for these regimes.

pith-pipeline@v0.9.1-grok · 5728 in / 1338 out tokens · 18753 ms · 2026-06-26T21:17:58.565197+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 3 canonical work pages

[1]

The algorithmic foundations of differential privacy.Found

Dwork, Cynthia and Roth, Aaron , title =. 2014 , issue_date =. doi:10.1561/0400000042 , journal =

work page doi:10.1561/0400000042 2014
[2]

Proceedings of Thirty Fourth Conference on Learning Theory , pages =

(Nearly) Dimension Independent Private ERM with AdaGrad Rates\\ via Publicly Estimated Subspaces , author =. Proceedings of Thirty Fourth Conference on Learning Theory , pages =. 2021 , editor =

2021
[3]

2024 , eprint=

Training Differentially Private Ad Prediction Models with Semi-Sensitive Features , author=. 2024 , eprint=

2024
[4]

2024 , URL=

Training Differentially Private Ad Prediction Models with Semi-Sensitive Features , author=. 2024 , URL=

2024
[5]

2023 , eprint=

Private Learning with Public Features , author=. 2023 , eprint=

2023
[6]

The Eleventh International Conference on Learning Representations , year=

Regression with Label Differential Privacy , author=. The Eleventh International Conference on Learning Representations , year=
[7]

Advances in Neural Information Processing Systems , editor=

Deep Learning with Label Differential Privacy , author=. Advances in Neural Information Processing Systems , editor=. 2021 , url=

2021
[8]

Badih Ghazi and Yangsibo Huang and Pritish Kamath and Ravi Kumar and Pasin Manurangsi and Chiyuan Zhang , booktitle=. Label. 2024 , url=

2024
[9]

Label differential privacy and private training data release , year =

Busa-Fekete, R\'. Label differential privacy and private training data release , year =. Proceedings of the 40th International Conference on Machine Learning , articleno =
[10]

2021 , eprint=

Label differential privacy via clustering , author=. 2021 , eprint=

2021
[11]

Antipodes of label differential privacy: PATE and ALIBI , year =

Malek, Mani and Mironov, Ilya and Prasad, Karthik and Shilov, Igor and Tram\`. Antipodes of label differential privacy: PATE and ALIBI , year =. Proceedings of the 35th International Conference on Neural Information Processing Systems , articleno =
[12]

2023 , eprint=

Label Differential Privacy via Aggregation , author=. 2023 , eprint=

2023
[13]

Factorization Machines , year=

Rendle, Steffen , booktitle=. Factorization Machines , year=
[14]

and Perescu-Popescu, Liliana and Mastorakis, Nikos , title =

Popescu, Marius-Constantin and Balas, Valentina E. and Perescu-Popescu, Liliana and Mastorakis, Nikos , title =. WSEAS Trans. Cir. and Sys. , month = jul, pages =. 2009 , issue_date =

2009
[15]

Brendan and Mironov, Ilya and Talwar, Kunal and Zhang, Li , title =

Abadi, Martin and Chu, Andy and Goodfellow, Ian and McMahan, H. Brendan and Mironov, Ilya and Talwar, Kunal and Zhang, Li , year=. Deep Learning with Differential Privacy , url=. doi:10.1145/2976749.2978318 , booktitle=

work page doi:10.1145/2976749.2978318
[16]

Proceedings of The 24th International Conference on Artificial Intelligence and Statistics , pages =

Evading the Curse of Dimensionality in Unconstrained Private GLMs , author =. Proceedings of The 24th International Conference on Artificial Intelligence and Statistics , pages =. 2021 , editor =

2021
[17]

The influence of a matrix condition number on iterative methods' convergence , year=

Pyzara, Anna and Bylina, Beata and Bylina, Jarosław , booktitle=. The influence of a matrix condition number on iterative methods' convergence , year=
[18]

Proceedings of the 24th Annual Conference on Learning Theory , pages =

Sample Complexity Bounds for Differentially Private Learning , author =. Proceedings of the 24th Annual Conference on Learning Theory , pages =. 2011 , editor =

2011
[19]

2017 , eprint=

Semi-supervised Knowledge Transfer for Deep Learning from Private Training Data , author=. 2017 , eprint=

2017
[20]

2018 , eprint=

Scalable Private Learning with PATE , author=. 2018 , eprint=

2018
[21]

Proceedings of the 36th International Conference on Machine Learning , pages =

On Sparse Linear Regression in the Local Differential Privacy Model , author =. Proceedings of the 36th International Conference on Machine Learning , pages =. 2019 , editor =

2019
[22]

Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds , year=

Bassily, Raef and Smith, Adam and Thakurta, Abhradeep , booktitle=. Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds , year=
[23]

3rd International Conference for Learning Representations, San Diego, 2015 , year=

Adam: A method for stochastic optimization , author=. 3rd International Conference for Learning Representations, San Diego, 2015 , year=

2015
[24]

Proceedings of the 29th International Coference on International Conference on Machine Learning , pages =

Rakhlin, Alexander and Shamir, Ohad and Sridharan, Karthik , title =. Proceedings of the 29th International Coference on International Conference on Machine Learning , pages =. 2012 , isbn =

2012
[25]

Ashkan Yousefpour and Igor Shilov and Alexandre Sablayrolles and Davide Testuggine and Karthik Prasad and Mani Malek and John Nguyen and Sayan Ghosh and Akash Bharadwaj and Jessica Zhao and Graham Cormode and Ilya Mironov , journal=. Opacus:
[26]

Trafficformer: An efficient pre-trained model for traffic data

Mahloujifar, Saeed and Guo, Chuan and Suh, G. Edward and Chaudhuri, Kamalika , booktitle =. 2025 , volume =. doi:10.1109/SP61157.2025.00179 , url =

work page doi:10.1109/sp61157.2025.00179 2025

[1] [1]

The algorithmic foundations of differential privacy.Found

Dwork, Cynthia and Roth, Aaron , title =. 2014 , issue_date =. doi:10.1561/0400000042 , journal =

work page doi:10.1561/0400000042 2014

[2] [2]

Proceedings of Thirty Fourth Conference on Learning Theory , pages =

(Nearly) Dimension Independent Private ERM with AdaGrad Rates\\ via Publicly Estimated Subspaces , author =. Proceedings of Thirty Fourth Conference on Learning Theory , pages =. 2021 , editor =

2021

[3] [3]

2024 , eprint=

Training Differentially Private Ad Prediction Models with Semi-Sensitive Features , author=. 2024 , eprint=

2024

[4] [4]

2024 , URL=

Training Differentially Private Ad Prediction Models with Semi-Sensitive Features , author=. 2024 , URL=

2024

[5] [5]

2023 , eprint=

Private Learning with Public Features , author=. 2023 , eprint=

2023

[6] [6]

The Eleventh International Conference on Learning Representations , year=

Regression with Label Differential Privacy , author=. The Eleventh International Conference on Learning Representations , year=

[7] [7]

Advances in Neural Information Processing Systems , editor=

Deep Learning with Label Differential Privacy , author=. Advances in Neural Information Processing Systems , editor=. 2021 , url=

2021

[8] [8]

Badih Ghazi and Yangsibo Huang and Pritish Kamath and Ravi Kumar and Pasin Manurangsi and Chiyuan Zhang , booktitle=. Label. 2024 , url=

2024

[9] [9]

Label differential privacy and private training data release , year =

Busa-Fekete, R\'. Label differential privacy and private training data release , year =. Proceedings of the 40th International Conference on Machine Learning , articleno =

[10] [10]

2021 , eprint=

Label differential privacy via clustering , author=. 2021 , eprint=

2021

[11] [11]

Antipodes of label differential privacy: PATE and ALIBI , year =

Malek, Mani and Mironov, Ilya and Prasad, Karthik and Shilov, Igor and Tram\`. Antipodes of label differential privacy: PATE and ALIBI , year =. Proceedings of the 35th International Conference on Neural Information Processing Systems , articleno =

[12] [12]

2023 , eprint=

Label Differential Privacy via Aggregation , author=. 2023 , eprint=

2023

[13] [13]

Factorization Machines , year=

Rendle, Steffen , booktitle=. Factorization Machines , year=

[14] [14]

and Perescu-Popescu, Liliana and Mastorakis, Nikos , title =

Popescu, Marius-Constantin and Balas, Valentina E. and Perescu-Popescu, Liliana and Mastorakis, Nikos , title =. WSEAS Trans. Cir. and Sys. , month = jul, pages =. 2009 , issue_date =

2009

[15] [15]

Brendan and Mironov, Ilya and Talwar, Kunal and Zhang, Li , title =

Abadi, Martin and Chu, Andy and Goodfellow, Ian and McMahan, H. Brendan and Mironov, Ilya and Talwar, Kunal and Zhang, Li , year=. Deep Learning with Differential Privacy , url=. doi:10.1145/2976749.2978318 , booktitle=

work page doi:10.1145/2976749.2978318

[16] [16]

Proceedings of The 24th International Conference on Artificial Intelligence and Statistics , pages =

Evading the Curse of Dimensionality in Unconstrained Private GLMs , author =. Proceedings of The 24th International Conference on Artificial Intelligence and Statistics , pages =. 2021 , editor =

2021

[17] [17]

The influence of a matrix condition number on iterative methods' convergence , year=

Pyzara, Anna and Bylina, Beata and Bylina, Jarosław , booktitle=. The influence of a matrix condition number on iterative methods' convergence , year=

[18] [18]

Proceedings of the 24th Annual Conference on Learning Theory , pages =

Sample Complexity Bounds for Differentially Private Learning , author =. Proceedings of the 24th Annual Conference on Learning Theory , pages =. 2011 , editor =

2011

[19] [19]

2017 , eprint=

Semi-supervised Knowledge Transfer for Deep Learning from Private Training Data , author=. 2017 , eprint=

2017

[20] [20]

2018 , eprint=

Scalable Private Learning with PATE , author=. 2018 , eprint=

2018

[21] [21]

Proceedings of the 36th International Conference on Machine Learning , pages =

On Sparse Linear Regression in the Local Differential Privacy Model , author =. Proceedings of the 36th International Conference on Machine Learning , pages =. 2019 , editor =

2019

[22] [22]

Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds , year=

Bassily, Raef and Smith, Adam and Thakurta, Abhradeep , booktitle=. Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds , year=

[23] [23]

3rd International Conference for Learning Representations, San Diego, 2015 , year=

Adam: A method for stochastic optimization , author=. 3rd International Conference for Learning Representations, San Diego, 2015 , year=

2015

[24] [24]

Proceedings of the 29th International Coference on International Conference on Machine Learning , pages =

Rakhlin, Alexander and Shamir, Ohad and Sridharan, Karthik , title =. Proceedings of the 29th International Coference on International Conference on Machine Learning , pages =. 2012 , isbn =

2012

[25] [25]

Ashkan Yousefpour and Igor Shilov and Alexandre Sablayrolles and Davide Testuggine and Karthik Prasad and Mani Malek and John Nguyen and Sayan Ghosh and Akash Bharadwaj and Jessica Zhao and Graham Cormode and Ilya Mironov , journal=. Opacus:

[26] [26]

Trafficformer: An efficient pre-trained model for traffic data

Mahloujifar, Saeed and Guo, Chuan and Suh, G. Edward and Chaudhuri, Kamalika , booktitle =. 2025 , volume =. doi:10.1109/SP61157.2025.00179 , url =

work page doi:10.1109/sp61157.2025.00179 2025