arxiv: 2605.09448 · v1 · submitted 2026-05-10 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Learning to Bid with Unknown Private Values in Budget-Constrained First-Price Auctions

Zihao Hu , Yuxiao Wen , Yuan Yao , Jiheng Zhang , Zhengyuan Zhou

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:38 UTC · model grok-4.3

classification 💻 cs.LG

keywords first-price auctionsbudget constraintsprimal-dual learningregret minimizationlatent valuationsonline biddingconstrained optimization

0 comments

The pith

A primal-dual framework learns latent valuations and competing bids to achieve near-optimal regret in budget-constrained first-price auctions.

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

The paper targets bidding in first-price auctions where private values are unknown and must be inferred from censored observations instead of an oracle. It builds a single primal-dual procedure that simultaneously estimates linear treatment effect valuation parameters and the distribution of rival bids, while enforcing hard budget or return-on-spend limits. The central technical step is to keep the Lagrangian multiplier from turning estimation errors into linear regret; this is done by invoking a strong Slater condition on the feasible set together with an adaptive burn-in phase that stabilizes dual variables early. With these safeguards the method delivers near-optimal regret bounds and is the first to supply theoretical guarantees for constrained bidding under latent valuations.

Core claim

We propose a unified primal-dual framework for constrained first-price auctions that jointly learns the latent linear treatment effect valuation parameters and the competitor's bid distribution. The framework is stabilized by the strong Slater condition on the feasible set and by a novel adaptive burn-in procedure that prevents the Lagrangian multiplier from scaling estimation error into unbounded regret, yielding near-optimal regret guarantees.

What carries the argument

A primal-dual online learner that couples estimation of LTE valuation parameters with dual updates for the budget or RoS constraint, protected by an adaptive burn-in schedule that bounds multiplier growth.

If this is right

Bidding agents can operate without a valuation oracle while still respecting hard budget limits.
Cumulative regret remains near-optimal even though estimation error is dynamically scaled by the dual variable.
Both regret and constraint-violation metrics (budget overspend, RoS shortfall) are controlled simultaneously.
The same framework covers both budget caps and return-on-spend targets without separate algorithms.

Where Pith is reading between the lines

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

The same joint-learning structure could be tested in second-price or generalized second-price formats with budgets.
In deployed ad platforms the method implies that historical censored bids are sufficient to start near-optimal bidding after a short burn-in window.
When multiple bidders each run the algorithm, equilibrium analysis would be needed to check whether the learned bid distributions remain consistent.

Load-bearing premise

The feasible set satisfies the strong Slater condition and the adaptive burn-in procedure keeps the Lagrangian multiplier from turning estimation errors into unbounded regret.

What would settle it

A simulation in which the Slater constant is driven close to zero or the burn-in phase is removed, after which regret grows linearly with time instead of sublinearly.

Figures

Figures reproduced from arXiv: 2605.09448 by Jiheng Zhang, Yuan Yao, Yuxiao Wen, Zhengyuan Zhou, Zihao Hu.

read the original abstract

The transition to First-Price Auctions (FPA) in digital advertising has spurred significant research, yet existing work typically assumes access to a valuation oracle, ignoring the reality that values must be inferred from censored data. While Linear Treatment Effect (LTE) models address this by learning value uplift, they have not been adapted to realistic settings with hard Budget constraints or Return-on-Spend (RoS) targets requiring regret and violation control. In this work, we propose a unified primal-dual framework for constrained FPAs that jointly learns the latent LTE valuation parameters and the competitor's bid distribution. This simultaneous learning introduces a critical technical challenge: the estimation error is dynamically scaled by the Lagrangian multiplier, potentially leading to unbounded regret. We resolve this by leveraging a strong Slater condition and a novel adaptive burn-in procedure to stabilize the dual variables. Our approach achieves near-optimal regret guarantees, providing the first theoretically grounded solution for constrained bidding with latent valuations.

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

The paper gives a primal-dual method for bidding under budgets and RoS targets while learning latent valuations from censored data, but the regret bound rests on an unproven adaptive burn-in that may not control dual scaling when the Slater gap is modest.

read the letter

The core advance is the first explicit treatment of budget-constrained first-price bidding where valuations come from a linear treatment effect model rather than an oracle. The authors fold the joint estimation of LTE parameters and competitor bid distributions into a single primal-dual loop and flag the obvious risk that estimation error gets multiplied by the dual variable. They then invoke a strong Slater condition plus an adaptive burn-in to keep the multiplier from exploding and claim near-optimal regret plus bounded violation. That combination of ingredients is new and directly addresses a gap that prior oracle-based or unconstrained work left open. The framing is practical: digital ad platforms really do face hard budgets and RoS targets, and censored feedback is the norm. If the analysis goes through, the result would be useful to auction theorists and to people building production bidding agents. The main weakness is that the abstract supplies no proof sketch, no explicit dependence of burn-in length on the unknown Slater margin, and no experimental check that the procedure stays stable when the margin is small. Without those pieces it is impossible to judge whether the claimed regret remains sublinear or degrades to linear in the worst case. The circularity worry is also live: the regret expression appears to be stated in terms of the same fitted quantities that are being learned inside the loop. This work is aimed at the intersection of online learning and mechanism design. A serious referee should see the full proofs and any simulation results before deciding whether the bounds are tight enough to matter in deployed systems. I would send it out for review rather than desk-reject, mainly because the problem is well-motivated and the technical obstacle is stated clearly.

Referee Report

2 major / 2 minor

Summary. The paper proposes a primal-dual online learning algorithm for budget-constrained first-price auctions in which private values are latent and must be inferred from censored observations via a Linear Treatment Effect (LTE) model. It jointly estimates the LTE parameters and the competitor bid distribution while enforcing hard budget and RoS constraints, and claims to obtain near-optimal regret by invoking a strong Slater condition together with an adaptive burn-in procedure that prevents the Lagrangian multiplier from scaling estimation error into linear regret.

Significance. If the claimed regret bounds hold, the result would be the first to deliver theoretical guarantees for constrained bidding under latent valuations in first-price auctions, a setting that has become standard in digital advertising. The work directly addresses the practical mismatch between existing oracle-based analyses and the reality of censored value data, and the primal-dual framework with explicit constraint-violation control could influence deployed bidding systems.

major comments (2)

§4.2 and Theorem 3: The adaptive burn-in length is stated to depend on the (unknown) Slater gap δ, yet no explicit rate is given for how the burn-in horizon or multiplier clipping scales with δ or with the LTE estimation error decay; if δ = O(T^{-α}) for small α, the dual variable can still reach Ω(T^β) before stabilization, which would violate the claimed O(√T) regret.
§3.1, Eq. (8) and Lemma 2: The regret decomposition treats the joint estimation of LTE parameters θ and bid distribution F inside the same primal-dual loop, but the cross-term arising from the multiplier scaling the estimation error is bounded only under the strong Slater assumption; without a quantitative relation between the estimation rate and δ, the bound reduces to a quantity defined in terms of the fitted parameters themselves.

minor comments (2)

Notation: the symbol for the RoS target is introduced without a clear link to the budget constraint in the problem statement; a single consolidated definition would improve readability.
Related work: the discussion of prior LTE bidding papers omits recent work on censored-data value learning in second-price settings; adding these citations would better situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our primal-dual framework for budget-constrained first-price auctions with latent valuations. The feedback highlights important aspects of the regret analysis and the role of the strong Slater condition. We address each major comment below, clarifying our assumptions and indicating where revisions will strengthen the presentation.

read point-by-point responses

Referee: §4.2 and Theorem 3: The adaptive burn-in length is stated to depend on the (unknown) Slater gap δ, yet no explicit rate is given for how the burn-in horizon or multiplier clipping scales with δ or with the LTE estimation error decay; if δ = O(T^{-α}) for small α, the dual variable can still reach Ω(T^β) before stabilization, which would violate the claimed O(√T) regret.

Authors: We appreciate this observation. Our analysis invokes a strong Slater condition with a fixed positive gap δ > 0 that is independent of the horizon T (a standard assumption ensuring strict feasibility with a constant margin). The adaptive burn-in procedure is constructed to terminate after a number of rounds that scales as O((1/δ)^2 log(1/ε)) where ε denotes the LTE estimation error at that time; this length remains o(T) for any fixed δ > 0. After burn-in the dual variables are clipped and stabilized, preventing linear accumulation of estimation error. We agree that the manuscript would benefit from an explicit statement of these scalings in §4.2 and the proof of Theorem 3. We will add the dependence on δ and the estimation rate, confirming that the overall regret remains O(√T) under the stated assumption. The case of δ decaying with T lies outside our modeling assumptions and is not claimed. revision: yes
Referee: §3.1, Eq. (8) and Lemma 2: The regret decomposition treats the joint estimation of LTE parameters θ and bid distribution F inside the same primal-dual loop, but the cross-term arising from the multiplier scaling the estimation error is bounded only under the strong Slater assumption; without a quantitative relation between the estimation rate and δ, the bound reduces to a quantity defined in terms of the fitted parameters themselves.

Authors: The strong Slater condition directly supplies a uniform bound on the optimal dual variables (of order O(1/δ)), which caps the multiplier and thereby controls the cross-term between the estimation error of (θ, F) and the dual variable in the decomposition of Eq. (8). Lemma 2 then combines this bounded multiplier with the sublinear convergence rate of the joint estimator to obtain an overall sublinear contribution. While the intermediate expressions are written in terms of the fitted parameters, the Slater bound ensures they do not grow with T. We acknowledge that an explicit quantitative link between the LTE estimation rate and δ would improve readability. We will revise the discussion around Lemma 2 to include this relation and a short remark showing how the cross-term remains o(T) under the fixed-δ assumption. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain.

full rationale

The paper's abstract describes a primal-dual framework jointly learning LTE valuation parameters and competitor bid distributions, with the estimation error scaling challenge resolved via a strong Slater condition and adaptive burn-in to stabilize dual variables, yielding near-optimal regret. No quoted equations or steps in the provided text reduce the regret bound or guarantees by construction to fitted parameters, self-definitions, or self-citations; the central claims rely on stated assumptions and standard optimization techniques that remain independent of the target results. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the central claim rests on the strong Slater condition and the effectiveness of the adaptive burn-in procedure, both introduced to control dual-variable growth.

axioms (1)

domain assumption Strong Slater condition holds for the constrained optimization problem
Invoked to guarantee that dual variables remain bounded and estimation error does not produce unbounded regret

pith-pipeline@v0.9.0 · 5470 in / 1365 out tokens · 35747 ms · 2026-05-12T04:38:03.701719+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

unified primal-dual framework for constrained FPAs that jointly learns the latent LTE valuation parameters and the competitor's bid distribution
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery theorem unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

adaptive burn-in procedure to stabilize the dual variables

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

[1]

Learning to bid in contextual first price auctions

Ashwinkumar Badanidiyuru, Zhe Feng, and Guru Guruganesh. Learning to bid in contextual first price auctions. InProceedings of the ACM Web Conference 2023, pages 3489–3497,

work page 2023
[2]

Matteo Castiglioni, Andrea Celli, and Christian Kroer

doi: 10.1561/2200000050. Matteo Castiglioni, Andrea Celli, and Christian Kroer. Online learning with knapsacks: the best of both worlds. InInternational Conference on Machine Learning, pages 2767–2783. PMLR, 2022a. Matteo Castiglioni, Andrea Celli, Alberto Marchesi, Giulia Romano, and Nicola Gatti. A unifying framework for online optimization with long-te...

work page doi:10.1561/2200000050
[3]

Zhe Feng, Swati Padmanabhan, and Di Wang

URLhttps://proceedings.mlr.press/v119/faury20a.html. Zhe Feng, Swati Padmanabhan, and Di Wang. Online bidding algorithms for return-on-spend constrained advertisers. InProceedings of the ACM Web Conference 2023, pages 3550–3560,

work page 2023
[4]

Budget pacing in repeated auctions: Regret and efficiency without convergence.arXiv preprint arXiv:2205.08674,

Jason Gaitonde, Yingkai Li, Bar Light, Brendan Lucier, and Aleksandrs Slivkins. Budget pacing in repeated auctions: Regret and efficiency without convergence.arXiv preprint arXiv:2205.08674,

work page arXiv
[5]

Learning to bid optimally and efficiently in adversarial first-price auctions.arXiv preprint arXiv:2007.04568,

Yanjun Han, Zhengyuan Zhou, Aaron Flores, Erik Ordentlich, and Tsachy Weissman. Learning to bid optimally and efficiently in adversarial first-price auctions.arXiv preprint arXiv:2007.04568,

work page arXiv 2007
[6]

Learning to bid in non-stationary repeated first-price auctions.arXiv preprint arXiv:2501.13358,

Zihao Hu, Xiaoyu Fan, Yuan Yao, Jiheng Zhang, and Zhengyuan Zhou. Learning to bid in non-stationary repeated first-price auctions.arXiv preprint arXiv:2501.13358,

work page arXiv
[7]

P. Langley. Crafting papers on machine learning. In Pat Langley, editor,Proceedings of the 17th International Conference on Machine Learning (ICML 2000), pages 1207–1216, Stanford, CA,

work page 2000
[8]

arXiv:1101.3039

doi: 10.1214/ECP.v16-1624. arXiv:1101.3039. Joel A Tropp. User-friendly tail bounds for sums of random matrices.Foundations of computational mathematics, 12(4):389–434,

work page doi:10.1214/ecp.v16-1624
[9]

Online bidding under ros constraints without knowing the value

Sushant Vijayan, Zhe Feng, Swati Padmanabhan, Karthikeyan Shanmugam, Arun Suggala, and Di Wang. Online bidding under ros constraints without knowing the value. InProceedings of the ACM on Web Conference 2025, pages 3096–3107,

work page 2025
[10]

Online causal inference for advertising in real-time bidding auctions.arXiv preprint arXiv:1908.08600,

Caio Waisman, Harikesh S Nair, and Carlos Carrion. Online causal inference for advertising in real-time bidding auctions.arXiv preprint arXiv:1908.08600,

work page arXiv 1908
[11]

Joint value estimation and bidding in repeated first-price auctions

Yuxiao Wen, Yanjun Han, and Zhengyuan Zhou. Joint value estimation and bidding in repeated first-price auctions. arXiv preprint arXiv:2502.17292,

work page arXiv
[12]

The (Marginal) Value of a Search Ad: An Online Causal Framework for Repeated Second-price Auctions

Yuxiao Wen, Zihao Hu, Yanjun Han, Yuhang Yao, and Zhengyuan Zhou. The (marginal) value of a search ad: An online causal framework for repeated second-price auctions.arXiv preprint arXiv:2605.01756,

work page internal anchor Pith review Pith/arXiv arXiv
[13]

19 Appendix roadmap.Appendix A proves the split-sample CDF concentration used throughout the paper

URLhttps://proceedings.mlr.press/v134/zhou21a.html. 19 Appendix roadmap.Appendix A proves the split-sample CDF concentration used throughout the paper. Appendix B uses that CDF event to establish the IPW bias–variance bound and the global WLS confidence radius. Appendix C plugs these estimates into the Budget SquareCB/primal–dual analysis, including the s...

work page 2012
[14]

A standard self-normalized Bernstein / matrix-Freedman inequality (e.g

Consequently X τ <t E[η2 τ xτ x⊤ τ | F τ , bτ]⪯C 1 X τ <t ωτ xτ x⊤ τ =C 1(At−1 −λ 0I)⪯C 1At−1. A standard self-normalized Bernstein / matrix-Freedman inequality (e.g. Faury et al., 2020, Proposition 7; Tropp, 2011, Theorem 1.2; or the Bernstein-form sharpening of Abbasi-Yadkori et al., 2011 stated in Zhou et al., 2021, Theorem 4.1) applied to St =P τ <t x...

work page 2020
[15]

X t<τ γBgt t ¯ct(b∗ t ) # =Z c ⋆ E

The EG potential is 1-strongly convex over [0 , 1] in the KL-induced metric, so the standard online mirror-descent regret bound (see, e.g., Bubeck, 2015, Theorem 4.2) gives, for every fixedµ∈[0,1], X t<τ ψt(µ)−ψ t(µt) ≤ log 2 η + η(τ−1) 2 . Choosingη= Θ(T −1/2) balances the two terms atO( √ T). Hence X t<τ ψt(µt)≥ X t<τ ψt(µ)−O( √ T)∀µ∈[0,1].(52) Case 1: ...

work page 2015