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A dynamic stochastic quantizer lets distributed online stochastic optimization converge almost surely to the optimum while guaranteeing (0,δ^i)-local differential privacy on directed graphs even after infinite iterations.

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2026-06-29 05:54 UTC pith:EXQNW3RE

load-bearing objection Claims a.s. convergence plus (0,δ)-LDP over infinite steps on directed graphs via dynamic quantization, but the per-step privacy decay needed for composition may conflict with the error decay needed for convergence. the 2 major comments →

arxiv 2605.29845 v1 pith:EXQNW3RE submitted 2026-05-28 math.OC

Local Differential Privacy via Dynamic Quantization in Distributed Online Stochastic Optimization

Zhiguo Zhang , Cheng Kui , Qian Ma , Dongrui Wu This is my paper
classification math.OC
keywords distributed optimizationlocal differential privacystochastic optimizationquantizationdirected graphsonline learningalmost sure convergence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper presents a fully distributed algorithm for online stochastic optimization over directed graphs that inserts a dynamic stochastic quantizer before each communication step. The quantizer masks the exchanged values to enforce local differential privacy while the underlying updates still drive the agents' iterates to the optimal solution almost surely. The privacy guarantee holds for each agent i with parameter δ^i no matter how large the iteration count becomes. A reader would care because streaming-data optimization in sensor networks or collaborative learning often runs indefinitely on asymmetric communication links where information leaks must be controlled without sacrificing exact convergence.

Core claim

By employing an elaborately designed dynamic stochastic quantizer to mask exchanged information, the proposed algorithm converges almost surely to the optimal solution and achieves (0,δ^i)-local differential privacy for each agent i even when the number of iterations tends to infinity. The algorithm is fully distributed and applicable to directed graphs. This is the first work on distributed online stochastic optimization that simultaneously achieves exact convergence and rigorous local differential privacy over a directed graph by exploiting quantization effects.

What carries the argument

The dynamic stochastic quantizer, which adjusts its stochastic levels over time to mask information sufficiently for privacy while preserving the almost-sure convergence of the distributed updates.

Load-bearing premise

An elaborately designed dynamic stochastic quantizer exists that can mask the exchanged values enough to meet the local differential privacy bound without preventing the iterates from converging almost surely.

What would settle it

A concrete counter-example or long-horizon simulation on a directed graph in which the proposed quantizer either leaks more than δ^i probability mass for some agent or the sequence of iterates fails to converge almost surely to the claimed optimum.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The iterates converge almost surely to the optimal solution.
  • Each agent satisfies (0,δ^i)-local differential privacy for any finite or infinite number of iterations.
  • The algorithm runs in a fully distributed manner on arbitrary directed graphs.
  • The same quantizer-based approach works for streaming data tasks such as online classification and recognition.

Where Pith is reading between the lines

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

  • The quantization schedule could be adapted to trade communication bits against privacy level in other directed-network problems without introducing separate privacy primitives.
  • Real-time applications such as distributed sensor fusion on asymmetric topologies become feasible when both convergence and ongoing privacy are required.
  • The method suggests examining whether similar quantization dynamics can protect privacy in non-stochastic or non-convex distributed settings.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a distributed online stochastic optimization algorithm that employs a dynamic stochastic quantizer to mask information exchanged over a directed communication graph. It claims that the algorithm converges almost surely to the optimal solution while simultaneously guaranteeing (0, δ^i)-local differential privacy for each agent i, even as the iteration count tends to infinity. The approach is asserted to be fully distributed and the first to achieve both exact convergence and rigorous LDP on directed graphs by exploiting quantization effects; numerical experiments on three classification datasets are provided to illustrate performance.

Significance. If the central claims hold, the result would be significant for privacy-preserving distributed learning: it would establish the first rigorous combination of almost-sure convergence and non-vanishing (0,δ)-LDP over infinite horizons on directed graphs without requiring a central coordinator or sacrificing exact optimality.

major comments (2)
  1. [Abstract and theoretical analysis] Abstract and §3–§4 (theoretical analysis): the claim of (0,δ^i)-LDP holding for the infinite sequence of quantized messages requires that the per-step privacy losses satisfy ∑δ_t < ∞. Standard almost-sure convergence arguments for online stochastic gradient methods on directed graphs, however, require the quantization noise variance to obey ∑α_t²σ_t² < ∞. No explicit parameter regime or design rule for the dynamic quantizer is shown to reconcile these two summability conditions simultaneously under the stated assumptions on the objective, noise, and graph.
  2. [§4 (privacy analysis)] §4 (privacy analysis): the (0,δ^i) guarantee is stated to hold “even when the number of iterations tends to infinity.” Under standard composition, this requires the dynamic quantizer to drive δ_t → 0 sufficiently fast; the manuscript does not exhibit the explicit decay schedule of the quantizer parameters that achieves both the privacy sum and the convergence sum on a directed graph.
minor comments (2)
  1. [§2–§3] Notation for the dynamic quantizer (e.g., the dependence of the quantization levels or noise variance on iteration index t) should be introduced earlier and kept consistent between the algorithm statement and the convergence/privacy proofs.
  2. [Numerical experiments] The numerical experiments section would benefit from an explicit statement of the directed-graph topology used and how the step-size and quantizer schedules were chosen to satisfy the theoretical conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to explicitly reconcile the summability conditions for privacy and convergence. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and theoretical analysis] Abstract and §3–§4 (theoretical analysis): the claim of (0,δ^i)-LDP holding for the infinite sequence of quantized messages requires that the per-step privacy losses satisfy ∑δ_t < ∞. Standard almost-sure convergence arguments for online stochastic gradient methods on directed graphs, however, require the quantization noise variance to obey ∑α_t²σ_t² < ∞. No explicit parameter regime or design rule for the dynamic quantizer is shown to reconcile these two summability conditions simultaneously under the stated assumptions on the objective, noise, and graph.

    Authors: The manuscript parameterizes the dynamic quantizer so that its noise variance can be chosen to satisfy both requirements simultaneously: the convergence condition depends on the product α_t²σ_t² while the per-step privacy loss δ_t depends on the sensitivity divided by the effective noise scale. Because the quantization levels and dynamic range are free design parameters, one can select σ_t² = O(α_t / t^ε) for ε>0 (or an analogous schedule) to make ∑α_t²σ_t² < ∞ while ensuring ∑δ_t < ∞. We agree that this joint regime was not stated explicitly and will add a dedicated remark (or short subsection) in the revised §4 that lists the admissible decay rates under the paper’s standing assumptions on the objective, noise, and graph. revision: yes

  2. Referee: [§4 (privacy analysis)] §4 (privacy analysis): the (0,δ^i) guarantee is stated to hold “even when the number of iterations tends to infinity.” Under standard composition, this requires the dynamic quantizer to drive δ_t → 0 sufficiently fast; the manuscript does not exhibit the explicit decay schedule of the quantizer parameters that achieves both the privacy sum and the convergence sum on a directed graph.

    Authors: The privacy analysis already invokes the infinite-horizon composition theorem and relies on the quantizer becoming progressively finer. The explicit schedule is obtained by letting the quantization step size decrease at a polynomial rate compatible with the step-size sequence α_t used for convergence on the directed graph. We will insert the concrete decay rule (e.g., δ_t = O(1/t^{1+ε})) together with the corresponding choice of quantization parameters into the revised version of §4 so that both summability conditions are visibly satisfied. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on analysis of a novel dynamic quantizer construction

full rationale

The paper proposes a new distributed online stochastic optimization algorithm that incorporates an elaborately designed dynamic stochastic quantizer. It then states that theoretical analysis establishes both almost-sure convergence to the optimum and (0,δ^i)-LDP for each agent even as the iteration count tends to infinity, on directed graphs. No quoted step, equation, or self-citation in the provided abstract reduces the privacy or convergence result to a fitted parameter, a renaming of a known pattern, or a prior result whose only justification is the authors' own earlier work. The derivation is presented as a direct proof for the constructed algorithm rather than a tautology or composition of self-referential definitions. This is the normal case of an independent construction whose correctness is left to the reader to verify externally.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate specific free parameters or axioms; the dynamic stochastic quantizer is described as elaborately designed, suggesting possible design choices not specified here.

pith-pipeline@v0.9.1-grok · 5718 in / 1120 out tokens · 32337 ms · 2026-06-29T05:54:32.651157+00:00 · methodology

0 comments
read the original abstract

Distributed online stochastic optimization has received extensive attention in large-scale distributed learning and other related fields due to its unique advantage in processing streaming data. However, information exchange through the communication network during the optimization process may lead to privacy leakage. To address this issue, this paper proposes a locally differentially private distributed online stochastic optimization algorithm that employs an elaborately designed dynamic stochastic quantizer to mask the exchanged information prior to communication. Theoretical analysis shows that the proposed algorithm not only converges almost surely to the optimal solution but also achieves $(0,\delta^i)$-local differential privacy for each agent $i$ even when the number of iterations tends to infinity. Furthermore, the algorithm is fully distributed and applicable to scenarios where the interaction network among agents is a directed graph. To the best of our knowledge, this is the first work on distributed online stochastic optimization that simultaneously achieves exact convergence and rigorous local differential privacy over a directed graph by exploiting quantization effects. Numerical experiments of distributed online training on the mushroom classification dataset, handwritten digits recognition dataset, and brain-computer interface dataset verify the effectiveness of the proposed method.

Figures

Figures reproduced from arXiv: 2605.29845 by Cheng Kui, Dongrui Wu, Qian Ma, Zhiguo Zhang.

Figure 1
Figure 1. Figure 1: Topology structure of the directed graph [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the prediction accuracy of Algorithm [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the maximum δ i under different quan￾tization stepsizes In each iteration, each agent randomly collects two labeled samples for training. The dynamic quantization stepsize and the iteration stepsize are configuerd as d i t = d i 0 (t+1)ς i with d i 0 = 2 + 0.01i, ς i = 0.6 + 0.01i and λt = 0.5 (t+1)0.71 for i = 1, 2, · · · , 5, respectively. Prior to formal training, each agent sequentially r… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the training and testing accuracy of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of iDLG attacker’s image inference re [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the cross-subject classification accu [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of iDLG attacker’s EEG inference results [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

discussion (0)

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