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

TL;DR

This paper introduces a unified primal-dual framework for learning in budget-constrained first-price auctions, addressing the challenge of inferring private values from censored data while controlling regret and violations.

Contribution

It develops a novel adaptive primal-dual method with regret guarantees for constrained bidding with latent valuations in first-price auctions.

Findings

Achieves near-optimal regret bounds.

First theoretically grounded solution for constrained bidding with latent valuations.

Effectively stabilizes dual variables using a new burn-in procedure.

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.