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

2026-05-10

AI Summary

This work addresses the challenge of bidding in first-price digital ad auctions under budget or return-on-spend (RoS) constraints, where private valuations are unobservable. The authors propose the first theoretically grounded solution by developing a unified primal-dual online learning framework that jointly estimates linear treatment effect (LTE) valuation parameters and the bidder’s bid distribution while respecting hard constraints. They innovatively extend the LTE model to constrained first-price auction settings and introduce a strong Slater condition together with an adaptive warm-up mechanism to mitigate unbounded regret caused by error amplification in Lagrange multipliers. The resulting algorithm achieves near-optimal regret bounds, offering both solid theoretical guarantees and a practical approach for bidding under implicit valuations and complex constraints.

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.

Problem

• First-Price Auctions
• Budget Constraints
• Unknown Private Values
• Return-on-Spend
• Constrained Bidding

Innovation

• primal-dual framework
• latent valuation learning
• budget-constrained bidding
• adaptive burn-in
• first-price auctions