# Scalable Principal-Agent Contract Design via Gradient-Based Optimization

arXiv: https://arxiv.org/abs/2510.21177 · HTML: https://mlsanity.com/paper/2510.21177/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 39 references that are in the map.
- First posted on arXiv 24 Oct 2025.

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

- [Learning to Reweight Examples for Robust Deep Learning](https://mlsanity.com/paper/1803.09050/) (2018)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization](https://mlsanity.com/paper/1607.05447/) (2016)
- [Adaptive Contract Design for Crowdsourcing Markets: Bandit Algorithms for Repeated Principal-Agent Problems](https://mlsanity.com/paper/1405.2875/) (2014)
- [Optimal Auctions through Deep Learning: Advances in Differentiable Economics](https://mlsanity.com/paper/1706.03459/) (2017)
- [Simple versus Optimal Contracts](https://mlsanity.com/paper/1808.03713/) (2018)
- [The Complexity of Contracts](https://mlsanity.com/paper/2002.12034/) (2020)
- [Contracts with Private Cost per Unit-of-Effort](https://mlsanity.com/paper/2111.09179/) (2021)
- [Deep Contract Design via Discontinuous Networks](https://mlsanity.com/paper/2307.02318/) (2023)
- [On the (In)approximability of Combinatorial Contracts](https://mlsanity.com/paper/2311.18425/) (2023)
- [Information Design in the Principal-Agent Problem](https://mlsanity.com/paper/2209.13688/) (2022)
- [Multi-Agent Contract Design beyond Binary Actions](https://mlsanity.com/paper/2402.13824/) (2024)
- [Ambiguous Contracts](https://mlsanity.com/paper/2302.07621/) (2023)

## Abstract (arXiv)

We study a bilevel max-max optimization framework for principal-agent contract design, in which a principal chooses incentives to maximize utility while anticipating the agent's best response. This problem, central to moral hazard and contract theory, underlies applications ranging from market design to delegated portfolio management, hedge fund fee structures, and executive compensation. While linear-quadratic models such as Holmstr"om-Milgrom admit closed-form solutions, realistic environments with nonlinear utilities, stochastic dynamics, or high-dimensional actions generally do not. We introduce a generic algorithmic framework that removes this reliance on closed forms. Our method adapts modern machine learning techniques for bilevel optimization -- using implicit differentiation with conjugate gradients (CG) -- to compute hypergradients efficiently through Hessian-vector products, without ever forming or inverting Hessians. In benchmark CARA-Normal (Constant Absolute Risk Aversion with Gaussian distribution of uncertainty) environments, the approach recovers known analytical optima and converges reliably from random initialization. More broadly, because it is matrix-free, variance-reduced, and problem-agnostic, the framework extends naturally to complex nonlinear contracts where closed-form solutions are unavailable, such as sigmoidal wage schedules (logistic pay), relative-performance/tournament compensation with common shocks, multi-task contracts with vector actions and heterogeneous noise, and CARA-Poisson count models with mathbbE[Xmid a]=ea. This provides a new computational tool for contract design, enabling systematic study of models that have remained analytically intractable.