Knowing When to Stop Matters: A Unified Algorithm for Online Conversion under Horizon Uncertainty

Bo Sun; Hasti Nourmohammadi Sigaroudi; Omid Ardakanian; Xiaoqi Tan; Yanzhao Wang

REVIEW 2 major objections 5 minor 6 references

Knowing When to Stop Matters: A Unified Algorithm for Online Conversion under Horizon Uncertainty

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A single pseudo-cost algorithm is optimal for online conversion under known, notified, or unknown trading horizons.

desk verdict Nice framework, but the new box-constraint upper bound rests on an equality in Lemma 8 that is false in general — worth reviewing, not yet proven. read the letter →

arxiv 2502.03817 v1 pith:BLLOQOML submitted 2025-02-06 cs.DS cs.LG

Yanzhao Wang , Hasti Nourmohammadi Sigaroudi , Bo Sun , Omid Ardakanian , Xiaoqi Tan This is my paper · ORCID

classification cs.DScs.LG MSC 68W2768W4090C05

keywords onlineconversioncompetitiveanalysishorizonuncertaintyboxconstraintspseudo-costfunctionslearning-augmentedalgorithmsprimal-dualone-waytrading

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

The paper claims that one online algorithm, PseudoMax, is optimal under competitive analysis for the online conversion problem under all three horizon models: known, notified partway through, and completely unknown. The same algorithm handles non-trivial per-step trading caps, a case the paper says no previous algorithm covered tightly for a known horizon. The known-horizon competitive ratio is the root of Eq. (7), the notified-horizon ratio is $1+W((\theta-1)/e)$, and the unknown-horizon ratio recovers $1+\ln\theta$. The learning-augmented version splits the resource between a known-horizon copy and an unknown-horizon copy according to a confidence parameter $\lambda$, yielding the consistency and robustness bounds of Theorem 2. A reader should care because one rule, with the horizon uncertainty dialed up or down, degrades smoothly from knowing everything to knowing nothing.

What carries the argument

The load-bearing object is the pseudo-cost function $\phi_t$ of Definition 1, an $\alpha$-parameterized estimate of the resource's marginal value that updates with every past allocation through the product $\prod_{i=1}^{t-1}(1-\alpha x_i/k)$. The algorithm uses it in the per-step subproblem $x^*_t=\arg\max_{x\in[0,b]} p_t x-\int_0^x \phi_t(\beta)\,d\beta$, which makes the online primal-dual analysis go through: the pseudo-cost directly defines dual prices, so once the balance parameter $\alpha$ satisfies the fixed-point equation Eq. (7), the dual solution stays feasible and the primal stays within the resource budget. The two-phase decomposition with the switching step $\tau=\min\{t: k_{t-1}-x_t^*>b(T-t)\}$ is what reduces the box-constrained problem to a smaller single-period forced-trading problem.

What would settle it

Solve the reduced linear program of Proposition 2 for the price sequence $\sigma^{(\tau+1)}=(p_1,\ldots,p_\tau,p_{\min})$ with the switching step set to $\tau$, and compare its optimal value with the value $k\phi_{\tau+1}+b\sum_{t=1}^{\tau}\mu_t$ used in Eq. (39); if the offline optimum is strictly larger for any $(\theta,T,b,k)$ in the non-trivial box-constraint regime, the claimed bound in Eq. (7) does not follow.

Watch

Extended reading notes

Core claim

The central discovery is that the threat-based optimal strategy for a known horizon, the threshold-style strategies for unknown horizons, and a new notified-horizon strategy are all instances of one mechanism: at each step the algorithm solves $\max_{x_t\in[0,b]} p_t x_t - \int_0^{x_t} \phi_t(\beta|F_t,\alpha)\,d\beta$ with the pseudo-cost function $\phi_t(\beta|F_t,\alpha)=p_{\min}+(\alpha-1)p_{\min}/\left((1-\alpha\beta/k)\prod_{i=1}^{t-1}(1-\alpha x_i/k)\right)$, then switches to forced maximum-rate trading whenever remaining slack becomes insufficient. Theorem 1 states that with $\alpha=CR^*_{\mathrm{known}}$, the root of $CR=(T-\lceil k/b\rceil+1)\left[1-\left((CR-1)/(\theta-1)\right)^{1/(T-\lceil k/b\rceil+1)}\right]$, Algorithm 1 is $CR^*_{\mathrm{known}}$-competitive for the known-horizon case with non-trivial box constraints; with $\alpha=1+W((\theta-1)/e)$ it is $CR^*_{\mathrm{notice}}$-competitive for the notified case; and with a different pseudo-cost it is $CR^*_{\mathrm{unknown}}=1+\ln\theta$-competitive for the unknown case. Theorem 2 adds the prediction setting with consistency $\eta(\lambda)\le CR^*_{\mathrm{known}}CR^*_{\mathrm{unknown}}/(CR^*_{\mathrm{unknown}}+\lambda(CR^*_{\mathrm{known}}-CR^*_{\mathrm{unknown}}))$ and robustness $\gamma(\lambda)\le CR^*_{\mathrm{unknown}}/\lambda$.

Load-bearing premise

The proof of the known-horizon bound assumes that the best possible offline profit equals the value of a particular helper solution built from the algorithm's own decisions; if that helper solution is not actually optimal, the claimed bound in Eq. (7) does not follow.

Editorial extensions

If this is right

For the known-horizon case with non-trivial box constraints, the paper gives the first tight competitive ratio; without box constraints it reduces to the classical threat-based ratio $CR^*_{\mathrm{known}}=T[1-((CR^*_{\mathrm{known}}-1)/(\theta-1))^{1/T}]$.
For the notified-horizon case the optimal ratio is $1+W((\theta-1)/e)$, independent of the box constraint, so receiving a last-minute notice is nearly as good as knowing the horizon upfront.
For the unknown-horizon case the algorithm subsumes the known optimal ratio $1+\ln\theta$, making PseudoMax a single unified rule across all three uncertainty levels.
In the prediction setting, increasing trust in an accurate horizon prediction drives the consistency ratio toward the known-horizon optimum, while decreasing trust returns the worst-case robustness of the unknown setting.
Underestimation of the horizon is less costly than overestimation in the paper's energy-trading case study, with the best average performance at intermediate confidence when the horizon is under-predicted.

Reading between the lines

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

Going beyond the paper, the pseudo-cost mechanism suggests a direct route to the two-way trading problem named as future work: maintain separate buying-side and selling-side pseudo-costs and run the same marginal-value update on both sides.
The paper's own footnote limits the box-constrained known-horizon result to asymptotic optimality in $T$ and $b$; a matching lower bound for all finite parameters would decide whether the guarantee is exactly tight at every point or only in the limit.
The asymmetry between under- and over-predicted horizons visible in the case study implies that an adversary should overestimate the horizon; a confidence schedule that shifts weight to the unknown-horizon component after the predicted horizon elapses could harden the algorithm against that attack.
Because the pseudo-cost update depends only on cumulative allocations and observed prices, the same function can be reinterpreted as a shadow price for other divisible-resource settings such as battery charging with degradation costs.

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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

Two things to know. The pseudo-cost framework is a genuinely nice way to view threat-based and threshold-based algorithms in one language, and the two-phase reduction for box constraints is clever. But the main new upper bound — the fixed-horizon box-constraint ratio in Eq. (7) — is not established as written. Lemma 8's Eq. (39) asserts that OPT(σ(τ+1)) equals the objective of the specific dual solution built from the algorithm. That's an equality, and it's used to propagate the bound from τ to τ+1. Strong duality only gives OPT ≤ that dual objective; equality requires dual optimality, and the authors don't supply it. The concern is concrete: with k=1, b=0.75, T=3, pmin=1, pmax=10, and prices (10,5,1), the constructed dual has objective 10, but OPT is 8.75. So Eq. (39) is just false in general. Replacing it with the correct inequality breaks the algebra in the proof. That's the load-bearing step, so the claimed tightness of Eq. (7) currently rests on an unproved assertion.

That said, the paper does more than this one claim. The unified pseudo-cost algorithm gives a fresh perspective on the known optimal ratios for OC-Known (no box), OC-Notice, and OC-Unknown, and the reduction to single-period forced trading is a useful technique. The learning-augmented extension with horizon predictions is new, and the robustness/consistency bounds in Theorem 2 are explicit and believable. The empirical case study on V2G energy trading is a nice sanity check. The paper also honestly footnotes that Eq. (7) is only asymptotically optimal, which tempers the word 'tight' in the abstract.

The soft spots are in proportion: the gap is in the new box-constraint upper bound, not in the unification or the prediction bounds. The OC-Unknown and OC-Notice ratios are from prior work, so the genuinely new content is the box-constraint formula and the prediction setting. There is no sign of fabrication; the writing is clear and the structure is sound.

Who should read it: anyone working on online conversion, threshold policies, or algorithms with predictions. The framework is worth knowing even if the proof needs repair. My recommendation: send it to peer review, but make clear to the authors that Lemma 8 must be fixed or replaced by a correct argument before the box-constraint claim can be accepted. If they can prove the upper bound with a correct inequality, the paper is a solid contribution.

Referee Report

2 major / 5 minor

Summary. The paper studies the online conversion (OC) problem, where a divisible resource is sold at sequentially revealed prices under box constraints, focusing on four horizon-information models: known, notified, unknown, and predicted horizons. It proposes a unified pseudo-cost-based algorithm, PseudoMax, and claims that with a suitably chosen balance parameter α it achieves the optimal competitive ratio for OC-Known with non-trivial box constraints (Eq. (7)), the known optimal ratio for OC-Notice (Eq. (8)), and matches the optimal OC-Unknown ratio (Eq. (9)). A learning-augmented variant, Algorithm 2, is then analyzed for the OC-Prediction setting, with claimed robustness and consistency bounds in Theorem 2. The proofs use an online primal-dual framework, a reduction from multi-period to single-period forced trading, and a case study on vehicle-to-grid energy trading.

Significance. If the central claims hold, the paper provides a genuinely useful unification: a single algorithmic template covers the known, noticed, and unknown horizon settings, and it gives the first competitive-ratio treatment of OC-Known with non-trivial box constraints. The pseudo-cost viewpoint is a nice conceptual contribution, and the extension to horizon predictions addresses a real gap in the learning-augmented online algorithms literature. The no-box upper bounds appear to follow standard primal-dual arguments and match known lower bounds, and the experimental section adds useful evidence about practical behavior. However, the headline box-constraint result currently rests on a load-bearing identity in Lemma 8 that is neither proved nor true as stated, so the exact upper bound in Eq. (7) is not established. The paper also overstates the word 'tight' for the box-constraint case, since Footnote 1 explicitly limits the claim to asymptotic optimality. These issues affect the core advertised contribution, not just the presentation.

major comments (2)

[Appendix F.2, Eq. (39)] The proof of Lemma 8 asserts OPT(σ(τ+1)) = kφ_τ + b∑_{t=1}^τ μ_t for the particular dual solution built from Algorithm 1, and then uses a similar identity for OPT(σ(τ)) in the subsequent algebra. For a minimization dual, weak duality gives only OPT ≤ kφ + b∑ μ_t for any feasible dual; equality requires a dual-optimality argument, and no such argument is supplied. The asserted identity is false in general. For example, take k=1, b=0.75, pmin=1, pmax=10, T=3, τ=2, and prices (10,5,1). With α=α_2, Algorithm 1 gives φ_2=10 and μ_1=μ_2=0, so the right-hand side of Eq. (39) equals 10, while OPT(σ(3)) = 0.75·10 + 0.25·5 = 8.75. Since inequality (40) and the bound for τ+1 are derived by substituting this equality into both the numerator and the denominator, replacing the false equality with the valid inequality breaks the algebra. Consequently, Lemma 8 and the upper bound CR*_known in Eq. (7) are not established.
[§3.3, Footnote 1, and §1 contributions] The paper advertises a 'tight upper bound' and 'the first algorithm with tight guarantees' for OC-Known with non-trivial box constraints, but Footnote 1 states that CR*_known in Eq. (7) is only asymptotically optimal in the regime of large T or large b. No matching lower bound for finite horizons and arbitrary b∈(k/T,k) is proved anywhere in the manuscript. If the intended claim is asymptotic optimality, then Theorem 1 and the contribution statements must be weakened; if exact optimality is intended, a lower-bound construction is required. This distinction is load-bearing because the box-constraint result is the paper's main new theoretical contribution.

minor comments (5)

[§5.1] The stated design goal that the consistency η(λ) 'approaches 1 as λ approaches 0' is contradicted by Theorem 2, which gives η(0)=CR*_known>1. Since even the known-horizon online problem cannot be 1-competitive, the text should say that η(λ) approaches CR*_known, the best possible known-horizon guarantee.
[§4.3] The notation 'τ∈[τmin,τT]' appears to be a typo; it should be 'τ∈[τmin,T]' to match the surrounding discussion and Lemma 8.
[Appendix F.1] The derivation of P_τ uses x*_t in the summation at t=τ, while the actual last-step allocation is ̅x_τ = k−∑_{t=1}^{τ−1} x*_t. The proof should state explicitly how the last step is treated in the incremental primal-dual argument.
[Appendix B.2, Eq. (22)] The correction term for the final step is written with p_T/b_T, but b_T is never defined and the formula appears to conflate the final allocation with the box constraint. This should be clarified or corrected.
[Table 1 caption] There is a typo: 'hrizon' should be 'horizon'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fixed-point choice of alpha and the imported prior results are standard design and external support, not reductions of the claimed guarantees to their own inputs.

full rationale

The derivation chain is not circular. Algorithm 1's balance parameter alpha is chosen as the root of Eq. (7) (or Eq. (15)/Eq. (37)); the proof then shows, via the online primal-dual argument in Lemma 7, that this choice yields a feasible solution and hence a competitive ratio bounded by alpha. This is a standard fixed-point design: the parameter appears on both sides of the inequality, but the theorem does not define the worst-case ratio as the root; it uses the root as a sufficient condition and then bounds the ratio by it. The OC-Unknown optimal ratio 1 + ln theta and the OC-Notice optimality are imported from [Sun+21a] and [Lec+24]; although the author lists overlap with [Sun+21a], those are published external results with stated assumptions and are not restatements of this paper's own conclusion, so under the stated rules they count as independent support rather than circularity. The learning-augmented bounds in Theorem 2 compose those established OC-Unknown guarantees with Theorem 1's OC-Known bound; no fitted quantity is renamed as a prediction. The main mathematical weakness is not circularity but a correctness gap: Appendix F.2's Eq. (39) asserts the equality OPT(sigma(tau+1)) = k phi_tau + b sum mu_t for a dual-feasible solution without proving dual optimality; strong duality alone gives only an inequality. If that equality fails, Lemma 8's recursive step is unsupported. That is an unproved step in the derivation, not a reduction of the claim to its own inputs, so it does not raise the circularity score. Similarly, the paper's claim that Eq. (7) is 'optimal' for OC-Known with box constraints is not supported by a matching lower bound in the text, but the absence of a lower bound is a completeness/correctness issue rather than circularity. Overall, the paper's central algorithm is evaluated against external worst-case benchmarks, and its fixed-point alpha is an optimization parameter, not a fitted prediction of its own competitive ratio.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No constants are fitted to data; alpha is fixed by the fixed-point equations and lambda is a user-supplied confidence parameter. The load-bearing external inputs are the cited optimal ratios for OC-Unknown and OC-Notice, plus two internal structural reductions (Proposition 1 and the Eq. 39 equality) that are not fully proved. No new physical entities are introduced.

assumptions (7)

domain assumption Prices in every step lie in a known interval [pmin, pmax].
Assumption 1 in Section 2.2; all competitive ratios are expressed through theta = pmax/pmin.
domain assumption The resource is divisible, decisions are irrevocable, and the objective is linear with no transaction costs.
Problem statement in Section 2.1; standard online conversion model.
domain assumption The optimal competitive ratio for OC-Unknown is 1 + ln theta.
Appendix C reviews this as a known result from [Sun+21a] and does not prove it; authors of this paper overlap with [Sun+21a].
domain assumption The optimal competitive ratio for OC-Notice is 1 + W((theta-1)/e).
Appendix G says optimality is established in prior work such as [Lec+24]; the paper supplies a new upper-bound route but does not re-prove the lower bound.
ad hoc to paper Worst-case instances for known-horizon with box constraints can be reduced to sequences whose prices after the switching step are all pmin.
Proposition 1 states this structure and its proof is a sketch; the reduction is load-bearing for the competitive ratio in Eq. (7).
ad hoc to paper For the reduced worst-case sequence sigma(tau+1), the offline optimum OPT equals the objective of the particular dual solution k phi_{tau+1} + b sum mu_t.
Used in Lemma 8 (Appendix F.2, Eq. 39) without proving dual optimality; only feasibility plus weak duality is established elsewhere.
standard math Gronwall's inequality and the AM-GM inequality can be applied to the lower-bound and dual-feasibility derivations.
Appendix B.2 and B.3 use these standard inequalities; they are cited but not proved.

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Cite this review

Pith. "Pith review of Knowing When to Stop Matters: A Unified Algorithm for Online Conversion under Horizon Uncertainty." pith.science (2026). https://pith.science/paper/BLLOQOML

@misc{pith2026250203817,
  author       = {Pith},
  title        = {Pith review of: Knowing When to Stop Matters: A Unified Algorithm for Online Conversion under Horizon Uncertainty},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLLOQOML}},
  note         = {Machine review of arXiv:2502.03817}
}

read the original abstract

This paper investigates the online conversion problem, which involves sequentially trading a divisible resource (e.g., energy) under dynamically changing prices to maximize profit. A key challenge in online conversion is managing decisions under horizon uncertainty, where the duration of trading is either known, revealed partway, or entirely unknown. We propose a unified algorithm that achieves optimal competitive guarantees across these horizon models, accounting for practical constraints such as box constraints, which limit the maximum allowable trade per step. Additionally, we extend the algorithm to a learning-augmented version, leveraging horizon predictions to adaptively balance performance: achieving near-optimal results when predictions are accurate while maintaining strong guarantees when predictions are unreliable. These results advance the understanding of online conversion under various degrees of horizon uncertainty and provide more practical strategies to address real world constraints.

Figures

Figures reproduced from arXiv: 2502.03817 by the authors.

**Figure 1.** Comparison of CR across different settings. OC-Known refers to the setting without box constraints, while OC-Known-b represents the same setting but with non-trivial box constraints. knowledge, PseudoMax𝝓 is the first to achieve a tight competitive ratio for OC-Known under non-trivial box constraints.1 Comparisons of optimal CRs in different settings [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

**Figure 2.** Illustration of the two-phase trading with box constraints. Step [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Comparison of the unified algorithm across three settings with rate limit. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Comparison of the learning-augmented algorithm across different settings of [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: Performance of Algorithm 1 under various 𝜆 and 𝑇pred. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: Cumulative revenue for the learning-augmented algorithm vs time slot [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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Online bin packing with predic- tions

[AKS23] Spyros Angelopoulos, Shahin Kamali, and Kimia Shadkami. “Online bin packing with predic- tions”. In: Journal of Artificial Intelligence Research 78 (2023), pp. 1111–1141. [AKZ22] Spyros Angelopoulos, Shahin Kamali, and Dehou Zhang. “Online search with best-price and query-based predictions”. In: Proceedings of the AAAI Conference on Artificial Int...

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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Online bin packing with predic- tions

[AKS23] Spyros Angelopoulos, Shahin Kamali, and Kimia Shadkami. “Online bin packing with predic- tions”. In: Journal of Artificial Intelligence Research 78 (2023), pp. 1111–1141. [AKZ22] Spyros Angelopoulos, Shahin Kamali, and Dehou Zhang. “Online search with best-price and query-based predictions”. In: Proceedings of the AAAI Conference on Artificial Int...

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Overcoming Brittleness in Pareto-Optimal Learning-Augmented Algorithms

2022, pp. 9652–9660. [Ang+24] Spyros Angelopoulos et al. “Overcoming Brittleness in Pareto-Optimal Learning-Augmented Algorithms”. In: arXiv preprint arXiv:2408.04122 (2024). 13 [BN09] Niv Buchbinder and Joseph (Seffi) Naor. “The Design of Competitive Online Algorithms via a Primal–Dual Approach”. In: Foundations and Trends® in Theoretical Computer Scienc...

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Optimal search and one-way trading online algorithms

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Online Conversion with Switching Costs: Robust and Learning-augmented Algorithms

[Lec+24] Adam Lechowicz et al. “Online Conversion with Switching Costs: Robust and Learning-augmented Algorithms”. In: ACM SIGMETRICS Performance Evaluation Review 52.1 (2024), pp. 45–46. [Lee+24] Russell Lee et al. “Online Search with Predictions: Pareto-optimal Algorithm and its Applica- tions in Energy Markets”. In: Proceedings of the 15th ACM Internat...

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Algorithms with predictions

[MV22] Michael Mitzenmacher and Sergei Vassilvitskii. “Algorithms with predictions”. In: Communi- cations of the ACM 65.7 (2022), pp. 33–35. [PJM] PJM. Regulation Market Clearing Prices . https://dataminer2.pjm.com/feed/reg_prices. [PSK18] Manish Purohit, Zoya Svitkina, and Ravi Kumar. “Improving online algorithms via ML pre- dictions”. In: Advances in Ne...

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