Temporally Discounted Differential Privacy for Evolving Datasets on an Infinite Horizon

Farhad Farokhi

REVIEW 3 major objections 4 minor 42 references

Temporally Discounted Differential Privacy for Evolving Datasets on an Infinite Horizon

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Discounted differential privacy relaxes the definition enough that an infinite stream of noisy reports can be released with bounded noise, whereas standard differential privacy forces noise to grow without bound.

desk verdict The bounded-noise result is mathematically correct under the paper's own definition, but the definition's normative discounting premise is what does the work, and the paper should be clearer about what it does and does not guarantee. read the letter →

arxiv 1908.03995 v2 pith:3JNBSNLR submitted 2019-08-12 cs.CR cs.SYecon.THeess.SPeess.SY

Farhad Farokhi This is my paper · ORCID

classification cs.CRcs.SYecon.THeess.SPeess.SY

keywords differentialprivacyevolvingdatasetstemporaldiscountinglossLaplacenoiseinfinitehorizonexponentialhyperbolic

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

Standard differential privacy treats a privacy loss from long ago as exactly as serious as one from today, and for evolving datasets that forces the added noise to grow without bound over an infinite horizon. This paper proposes discounted differential privacy, in which a privacy loss at time $k$ is weighted by $\alpha^{t-k}$ (exponential) or $1/(1+\beta(t-k))$ (hyperbolic) when judged at time $t$. The main result is that under exponential discounting a constant Laplace noise scale $b = \Delta f/(\epsilon(1-\alpha))$ satisfies the discounted guarantee for every report time, so report quality never degrades. Under hyperbolic discounting the noise still must grow, but only as $\sqrt{k}$. The paper demonstrates the practical difference on household smart-meter data, where the discounted mechanisms keep expected relative error low over a year while standard differential privacy degrades quickly.

What carries the argument

The load-bearing object is the discounted privacy loss, which replaces the uniform sum of per-report privacy losses with a weighted sum in which older losses count less. The privacy-loss decomposition $p(Y(t)|X(t))=\prod_{k=1}^t p(y(k)|X(k))$, inherited from standard differential privacy, turns the definition into the inequality $\sum_{k=1}^t \alpha^{t-k}\rho(k)\le \epsilon$. Because the geometric series $\sum_{j=0}^{\infty}\alpha^j = 1/(1-\alpha)$ is finite, a constant per-report privacy loss of $\epsilon(1-\alpha)$ fits inside the budget for every $t$, which is exactly why the Laplace scale $b=\Delta f/(\epsilon(1-\alpha))$ works. Hyperbolic discounting gives a divergent series, which is why its noise must still grow, but the slower divergence yields only $\sqrt{k}$ growth.

What would settle it

Run a controlled study in which people state how much compensation they would demand for releasing a sensitive reading from 1, 10, or 100 days ago; if the demanded compensation does not decline roughly like $\alpha^{t-k}$ or $1/(1+\beta(t-k))$, the normative premise that justifies bounded noise does not describe actual privacy preferences.

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Extended reading notes

Core claim

The paper's central claim is that a time-dependent weighting of privacy losses converts an impossibility into a feasibility result: an infinite sequence of reports can be protected with noise of constant magnitude if old losses are allowed to count for less. Formally, for reports $y(t)=f_t(X(t))+w(t)$ with i.i.d. Laplace noise, standard $\epsilon$-differential privacy requires $\sum_{k=1}^t \Delta f_k/b_k \le \epsilon$, so with constant query sensitivity $\Delta f$ the noise scale $b_k$ must diverge. The paper defines discounted privacy loss, $\rho(k,t)=\alpha^{t-k}\rho(k)$, and proves that $(\epsilon,\alpha)$-exponentially discounted differential privacy holds for all $t$ with the constant choice $b_k=\Delta f/(\epsilon(1-\alpha))$. Hyperbolic discounting, with weight $1/(1+\beta(t-k))$, gives a slower-growing scale proportional to $\sqrt{k}$. These are relaxations: setting $\alpha=1$ or $\beta=0$ recovers standard differential privacy, so the bounded-noise conclusion is bought precisely by the decision to forget.

Load-bearing premise

The load-bearing premise is that an individual's concern about a privacy leak decays with the time since the leak, so old losses can be down-weighted; if every past leak matters as much as today's, the total privacy loss of an infinite release remains unbounded despite the relaxation.

Editorial extensions

If this is right

Under exponentially discounted differential privacy, a curator can publish an endless stream of noisy answers without ever increasing the noise, so reports do not become meaningless over time.
Standard $\epsilon$-differential privacy on the same problem forces the noise scale to grow without bound; the paper gives the explicit choice $b_k = \Delta f \pi^2 k^2 / (6\epsilon)$ as one valid schedule.
Hyperbolic discounting still requires growing noise, but only as $\sqrt{k}$, which is slow enough that on a one-year smart-meter horizon the degradation is barely visible.
The results apply to any sequence of queries with bounded sensitivity when reports are generated by adding i.i.d. Laplace noise, not only to averaging queries.

Reading between the lines

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

One could calibrate the discount factor to a measured 'privacy half-life' from behavioral studies and check whether the resulting noise level matches what people accept in real data-sharing decisions.
The same reweighting of per-time privacy losses could be applied to other definitions that compose over time by summing a privacy-loss measure, potentially yielding bounded-noise analogues for privacy-loss random variables with finite moments.
If a legal retention period is known in advance, a curator could set $\alpha$ so that the discounted weight of any record falls below a chosen threshold by the end of the period; discounted differential privacy would then formalize forgetting without deleting data.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The paper does what it says: it defines discounted differential privacy, proves sufficient Laplace-noise conditions, and shows that exponential discounting lets the noise scale stay constant over an infinite horizon. Theorems 6, 11, 15, and Corollary 12 are straightforward and, as far as I checked, correct. The genuinely new piece is Corollary 16, where hyperbolic discounting leads to noise that grows like sqrt(k), much slower than the standard unbounded-growth result. The paper also credits Bolot et al. for the exponential case, so the novelty claim is honest. The empirical section is simple but decent; the Ausgrid data is public and the explicit formulas make reproduction easy.

The soft spot is not the math, it is the interpretation. Definition 10 bounds the weighted sum alpha^{t-k} rho(k), not the undiscounted privacy loss of the full transcript. The stress-test example is the right one: if every report is the same query about the same individual, the mechanism satisfies the definition with constant noise, yet an adversary who averages the reports learns that fixed secret with confidence growing linearly in time. The paper's own conclusion admits that capturing this adversary is future work. So the bounded-noise claim is real, but it is meaningful only if you accept the normative premise that old leaks matter less. For an adversary or a user who cares equally about all leaks of one secret, the relaxation does not control disclosure.

A smaller issue: Theorem 5 is stated without the conditional-independence assumption that its proof relies on. For the additive-noise mechanisms later in the paper the assumption holds because the noise is i.i.d., but the theorem as written overstates its generality. That should be fixed, along with the abstract's phrasing, which does not distinguish the exponential result from the hyperbolic one.

The hyperbolic proof in Corollary 16 uses integral approximations and a monotonicity argument. I did not machine-check it, but the steps look plausible and nothing in the result is implausible. The citation pattern is fine, and the overlap with Bolot et al. is disclosed.

Who benefits: readers working on continual release, stream privacy, or privacy-utility trade-offs under non-standard loss models will find a clean formal starting point. It is not a replacement for differential privacy, and the authors do not claim it is. If you accept the discounting premise, the exponential result gives a genuinely bounded-noise mechanism; if you do not, the definition is just a weighted sum with no adversary-based justification.

I would send this to serious peer review, with the repeated-query example and the scope of the guarantee made explicit before publication. The derivations deserve to be in the literature; the interpretation needs to be more honest about what the definition does not control.

Best

Referee Report

3 major / 4 minor

Summary. This paper defines discounted differential privacy for evolving datasets: at evaluation time t, the privacy loss incurred at time k is weighted by α^(t−k) (exponential discounting) or by 1/(1+β(t−k)) (hyperbolic discounting). For additive i.i.d. Laplace noise, it shows that exponential discounting allows a constant noise scale b=Δ_f/(ε(1−α)) over an infinite horizon, whereas ordinary ε-differential privacy forces the noise scale to grow without bound (e.g., quadratically, Corollary 8). A hyperbolic variant yields b_k=O(√k). The mechanisms are evaluated on real smart-meter data from Ausgrid, comparing expected relative error under the three definitions.

Significance. If the discounted privacy definition is accepted, the paper is a useful contribution: it makes the implicit temporal weighting in privacy loss explicit, gives elementary sufficient conditions for additive Laplace mechanisms, and demonstrates a clear utility advantage on a real dataset. The core calculations in Theorems 6 and 11 are standard and correct, and the paper is self-contained. The main caveat, which the paper does not address, is that the relaxation is normative: it assumes that old leaks are less sensitive, and it does not bound the undiscounted cumulative privacy loss of the transcript. The practical significance therefore hinges on whether the discounted-loss premise is appropriate for the application.

major comments (3)

[§III-A, Definition 10 and Corollary 12] The central claim that bounded noise gives a meaningful relaxation of differential privacy over an infinite horizon is not supported as stated. Definition 10 bounds only the discounted sum ∑_{k=1}^t α^{t−k}ρ(k); it does not bound the undiscounted privacy loss of the full transcript. To see the gap, take f_t(X(t))=x_1(1) for all t (each report is the same first measurement of individual 1) with sensitivity Δ_f=Δ and b=Δ/(ε(1−α)). Corollary 12 applies and the mechanism is (ε,α)-exponentially discounted DP for every t, since ∑_{k=1}^t α^{t−k}Δ/b ≤ ε. However, the worst-case log-likelihood ratio between neighboring datasets differing in x_1(1) is tΔ/b = tε(1−α), which grows linearly in t, and an adversary who averages the t independent Laplace reports obtains a standard error of order b/√t, so confidence about the fixed secret improves over time. The bounded noise scale is thus purchased by discounting repeated leaks of the same secret. I recommend that the paper either prove an additional guarantee on undiscounted transcript leakage under reasonable conditions (e.g., each individual's data changes over time, or queries are distinct), or explicitly state this limitation in the abstract and conclusions and temper the claim that this is a meaningful relaxation of differential privacy.
[§II-B, Theorem 5] Theorem 5 is stated without the conditional independence assumption that its proof requires. The decomposition p(Y(t)|X(t)) = ∏_{k=1}^t p(y(k)|X(k)) is valid only if the reports are conditionally independent given X(t), together with causality; without such an assumption the theorem is false. Please add this assumption explicitly to the statement of Theorem 5 (and clarify whether Definitions 9–14 inherit it). Note that the specific mechanism (5) with i.i.d. Laplace noise does satisfy the assumption, so the later results are not affected.
[§III-B, Corollary 16] The proof of Corollary 16 is not fully justified. The integral approximations used to bound the two parts of the sum are not accompanied by error analysis: the first inequality integrates over [0,t0−1] instead of [0,t0], and the second integrates over [t0+1,t]. Because Corollary 16 is a central contribution (bounded-noise variant for hyperbolic discounting), please supply a complete proof or state the approximation error explicitly.

minor comments (4)

[§IV-A] The sentence 'smart meter measurements of of households' contains a duplicated preposition; also, 'The individuals in both these datasets' refers to only one dataset.
[§IV-B] The phrase 'avrage expected relative error' should be 'average expected relative error'.
[§II-A, Corollary 7] The statement 'lim_{k→} b_k = +∞' is missing the target of the limit; it should read lim_{k→∞} b_k = +∞.
[§III-B, Eq. (14)] The argument of atanh in the displayed formula should be √(β/(1+β)) rather than √β/(1+β), to match the derivation in the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

Definitional discounting is explicit, and all privacy-guarantee derivations are self-contained sufficient conditions; no circularity found.

full rationale

The paper's contribution is a new privacy definition (Definitions 9, 10, 13, and 14) plus sufficient conditions for bounded-noise Laplace reports (Theorem 11, Corollary 12, Theorem 15, and Corollary 16). The derivation chain is self-contained: Theorem 5 factors the likelihood ratio under the mechanism's conditional independence; Theorem 6 computes the per-step privacy loss as rho(k) = Delta_f / b_k; Definition 10 substitutes rho(k) into the discounted sum; Corollary 12 then solves for the constant noise scale b = Delta_f / (epsilon * (1 - alpha)). The only free inputs are the user-chosen discount factor alpha and the data-independent sensitivity Delta_f; nothing is fitted to the Ausgrid data to obtain the privacy guarantee. The numerical section uses the real data only to report expected relative error, not to set b. There is no load-bearing self-citation: the paper cites external sources such as Dwork and Roth, Bolot et al., and the economics discounting literature, and no prior work by the same author is invoked to establish the central theorem. The hyperbolic case follows from the same substitution plus an integral bound, again without fitting. The normative premise that privacy loss in the distant past matters less is an explicit definitional choice, not a hidden equivalence: the paper never claims to derive discounting from differential privacy; it defines discounted differential privacy and proves sufficient conditions that satisfy that definition. The skeptical concern that undiscounted privacy loss can still grow without bound for a query that repeats the same secret is a limitation of the definition's semantics, not circular reasoning within the paper's derivation chain.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests primarily on the normative choice to discount privacy losses and on standard DP composition machinery. No new physical or mathematical entities are invented. Free parameters are user choices or data-dependent inputs, not fitted to make the theorems work.

free parameters (3)

alpha (discount factor)
User-chosen in (0,1); controls how quickly past privacy losses are forgotten. Not fitted to data.
beta (hyperbolic discounting coefficient)
User-chosen >= 0; controls hyperbolic decay. Not fitted.
Delta_f (query sensitivity) = 200/300 kWh
In the numerical section, set as the maximum change in a household's daily consumption read from the histogram; data-dependent and not accompanied by uncertainty bounds.

assumptions (4)

domain assumption Reports Y(k) are conditionally independent given X(t).
Needed for the product factorization p(Y(t)|X(t)) = product_k p(y(k)|X(k)) in Theorem 5. Holds for the Laplace mechanism with independent noise, but not for general mechanisms.
ad hoc to paper Privacy losses in the distant past can be discounted via alpha^(t-k) or 1/(1 + beta (t - k)).
This is the defining normative premise of discounted DP, motivated by behavioral economics but not derived from any adversary model.
domain assumption The evolving dataset has a fixed set of rows per individual (with placeholder symbol for absent times).
Assumed in the dataset model (1) and Remark 1; enables the neighboring-dataset definition.
domain assumption Query sensitivity Delta_f_k is known and bounded.
Required to set the Laplace noise scale; in practice must be estimated from data.

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Pith. "Pith review of Temporally Discounted Differential Privacy for Evolving Datasets on an Infinite Horizon." pith.science (2026). https://pith.science/paper/3JNBSNLR

@misc{pith2026190803995,
  author       = {Pith},
  title        = {Pith review of: Temporally Discounted Differential Privacy for Evolving Datasets on an Infinite Horizon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3JNBSNLR}},
  note         = {Machine review of arXiv:1908.03995}
}

read the original abstract

We define discounted differential privacy, as an alternative to (conventional) differential privacy, to investigate privacy of evolving datasets, containing time series over an unbounded horizon. We use privacy loss as a measure of the amount of information leaked by the reports at a certain fixed time. We observe that privacy losses are weighted equally across time in the definition of differential privacy, and therefore the magnitude of privacy-preserving additive noise must grow without bound to ensure differential privacy over an infinite horizon. Motivated by the discounted utility theory within the economics literature, we use exponential and hyperbolic discounting of privacy losses across time to relax the definition of differential privacy under continual observations. This implies that privacy losses in distant past are less important than the current ones to an individual. We use discounted differential privacy to investigate privacy of evolving datasets using additive Laplace noise and show that the magnitude of the additive noise can remain bounded under discounted differential privacy. We illustrate the quality of privacy-preserving mechanisms satisfying discounted differential privacy on smart-meter measurement time-series of real households, made publicly available by Ausgrid (an Australian electricity distribution company).

Figures

Figures reproduced from arXiv: 1908.03995 by the authors.

**Figure 2.** Quality of reports for the Ausgrid data under various [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Quality of reports for the Ausgrid data versus [left] [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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