# Learning Concave Bid Shading Strategies in Online Auctions via Measure-valued Proximal Optimization

arXiv: https://arxiv.org/abs/2509.10693 · HTML: https://mlsanity.com/paper/2509.10693/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 12 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [Deep & Cross Network for Ad Click Predictions](https://mlsanity.com/paper/1708.05123/) (2017)
- [Field-weighted Factorization Machines for Click-Through Rate Prediction in Display Advertising](https://mlsanity.com/paper/1806.03514/) (2018)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [A Smoothed Dual Approach for Variational Wasserstein Problems](https://mlsanity.com/paper/1503.02533/) (2015)
- [A Nonparametric Delayed Feedback Model for Conversion Rate Prediction](https://mlsanity.com/paper/1802.00255/) (2018)
- [Gradient Flows in Filtering and Fisher-Rao Geometry](https://mlsanity.com/paper/1710.00064/) (2017)
- [Bid Shading in The Brave New World of First-Price Auctions](https://mlsanity.com/paper/2009.01360/) (2020)
- [Gradient Flow Algorithms for Density Propagation in Stochastic Systems](https://mlsanity.com/paper/1908.00533/) (2019)
- [Generalized Sinkhorn iterations for regularizing inverse problems using optimal mass transport](https://mlsanity.com/paper/1612.02273/) (2016)
- [Proximal Recursion for Solving the Fokker-Planck Equation](https://mlsanity.com/paper/1809.10844/) (2018)
- [Gradient Flows in Uncertainty Propagation and Filtering of Linear Gaussian Systems](https://mlsanity.com/paper/1704.00102/) (2017)
- [Hopfield Neural Network Flow: A Geometric Viewpoint](https://mlsanity.com/paper/1908.01270/) (2019)
- [Proximal Recursion for the Wonham Filter](https://mlsanity.com/paper/1909.05827/) (2019)
- [Stochastic Uncertainty Propagation in Power System Dynamics using Measure-valued Proximal Recursions](https://mlsanity.com/paper/2108.13405/) (2021)
- [Wasserstein Consensus ADMM](https://mlsanity.com/paper/2309.07351/) (2023)

## Abstract (arXiv)

This work proposes a bid shading strategy for first-price auctions as a measure-valued optimization problem. We consider a standard parametric form for bid shading and formulate the problem as convex optimization over the joint distribution of shading parameters. After each auction, the shading parameter distribution is adapted via a regularized Wasserstein-proximal update with a data-driven energy functional. This energy functional is conditional on the context, i.e., on publisher/user attributes such as domain, ad slot type, device, or location. The proposed algorithm encourages the bid distribution to place more weight on values with higher expected surplus, i.e., where the win probability and the value gap are both large. We show that the resulting measure-valued convex optimization problem admits a closed form solution. A numerical example illustrates the proposed method.