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DOI: 10.4230/LIPIcs.FORC.2026.2

Learning Rate Scheduling with Matrix Factorization for Private Training

Authors: Nikita P. Kalinin and Joel Daniel Andersson

Published in: LIPIcs, Volume 368, 7th Symposium on Foundations of Responsible Computing (FORC 2026)

Abstract

We study differentially private model training with stochastic gradient descent under learning rate scheduling and correlated noise. Although correlated noise, in particular via matrix factorizations, has been shown to improve accuracy, prior theoretical work focused primarily on the prefix-sum workload. That workload assumes a constant learning rate, whereas in practice learning rate schedules are widely used to accelerate training and improve convergence. We close this gap by deriving general upper and lower bounds for a broad class of learning rate schedules in both single- and multi-epoch settings. Building on these results, we propose a learning-rate-aware factorization that achieves improvements over prefix-sum factorizations under both MaxSE and MeanSE error metrics. Our theoretical analysis yields memory-efficient constructions suitable for practical deployment, and experiments on CIFAR-10 and IMDB datasets confirm that schedule-aware factorizations improve accuracy in private training.

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Nikita P. Kalinin and Joel Daniel Andersson. Learning Rate Scheduling with Matrix Factorization for Private Training. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kalinin_et_al:LIPIcs.FORC.2026.2,
  author =	{Kalinin, Nikita P. and Andersson, Joel Daniel},
  title =	{{Learning Rate Scheduling with Matrix Factorization for Private Training}},
  booktitle =	{7th Symposium on Foundations of Responsible Computing (FORC 2026)},
  pages =	{2:1--2:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-419-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{368},
  editor =	{Lin, Huijia (Rachel)},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2026.2},
  URN =		{urn:nbn:de:0030-drops-259738},
  doi =		{10.4230/LIPIcs.FORC.2026.2},
  annote =	{Keywords: differential privacy, machine learning, matrix factorization}
}

Document

DOI: 10.4230/LIPIcs.FORC.2025.10

Count on Your Elders: Laplace vs Gaussian Noise

Authors: Joel Daniel Andersson, Rasmus Pagh, Teresa Anna Steiner, and Sahel Torkamani

Published in: LIPIcs, Volume 329, 6th Symposium on Foundations of Responsible Computing (FORC 2025)

Abstract

In recent years, Gaussian noise has become a popular tool in differentially private algorithms, often replacing Laplace noise which dominated the early literature on differential privacy. Gaussian noise is the standard approach to approximate differential privacy, often resulting in much higher utility than traditional (pure) differential privacy mechanisms. In this paper we argue that Laplace noise may in fact be preferable to Gaussian noise in many settings, in particular when we seek to achieve (ε,δ)-differential privacy for small values of δ. We consider two scenarios: First, we consider the problem of counting under continual observation and present a new generalization of the binary tree mechanism that uses a k-ary number system with negative digits to improve the privacy-accuracy trade-off. Our mechanism uses Laplace noise and whenever δ is sufficiently small it improves the mean squared error over the best possible (ε,δ)-differentially private factorization mechanisms based on Gaussian noise. Specifically, using k = 19 we get an asymptotic improvement over the bound given in the work by Henzinger, Upadhyay and Upadhyay (SODA 2023) when δ = O(T^{-0.92}). Second, we show that the noise added by the Gaussian mechanism can always be replaced by Laplace noise of comparable variance for the same (ε, δ)-differential privacy guarantee, and in fact for sufficiently small δ the variance of the Laplace noise becomes strictly better. This challenges the conventional wisdom that Gaussian noise should be used for high-dimensional noise. Finally, we study whether counting under continual observation may be easier in an average-case sense than in a worst-case sense. We show that, under pure differential privacy, the expected worst-case error for a random input must be Ω(log(T)/ε), matching the known lower bound for worst-case inputs.

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Joel Daniel Andersson, Rasmus Pagh, Teresa Anna Steiner, and Sahel Torkamani. Count on Your Elders: Laplace vs Gaussian Noise. In 6th Symposium on Foundations of Responsible Computing (FORC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 329, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{andersson_et_al:LIPIcs.FORC.2025.10,
  author =	{Andersson, Joel Daniel and Pagh, Rasmus and Steiner, Teresa Anna and Torkamani, Sahel},
  title =	{{Count on Your Elders: Laplace vs Gaussian Noise}},
  booktitle =	{6th Symposium on Foundations of Responsible Computing (FORC 2025)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-367-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{329},
  editor =	{Bun, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2025.10},
  URN =		{urn:nbn:de:0030-drops-231376},
  doi =		{10.4230/LIPIcs.FORC.2025.10},
  annote =	{Keywords: differential privacy, continual observation, streaming, prefix sums, trees}
}

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Documents authored by Andersson, Joel Daniel

Learning Rate Scheduling with Matrix Factorization for Private Training

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Count on Your Elders: Laplace vs Gaussian Noise

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