# Maximum Risk Minimization with Random Forests

arXiv: https://arxiv.org/abs/2512.10445 · HTML: https://mlsanity.com/paper/2512.10445/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 64 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 30 of 33)

- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Just Train Twice: Improving Group Robustness without Training Group Information](https://mlsanity.com/paper/2107.09044/) (2021)
- [Towards Out-Of-Distribution Generalization: A Survey](https://mlsanity.com/paper/2108.13624/) (2021)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Distributionally Robust Language Modeling](https://mlsanity.com/paper/1909.02060/) (2019)
- [Invariant Causal Prediction for Nonlinear Models](https://mlsanity.com/paper/1706.08576/) (2017)
- [Anchor regression: heterogeneous data meets causality](https://mlsanity.com/paper/1801.06229/) (2018)
- [A causal framework for distribution generalization](https://mlsanity.com/paper/2006.07433/) (2020)
- [Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application](https://mlsanity.com/paper/1309.1541/) (2013)
- [Clarabel: An interior-point solver for conic programs with quadratic objectives](https://mlsanity.com/paper/2405.12762/) (2024)
- [Nonlinear Invariant Risk Minimization: A Causal Approach](https://mlsanity.com/paper/2102.12353/) (2021)
- [Towards Unsupervised Domain Generalization](https://mlsanity.com/paper/2107.06219/) (2021)
- [Extragradient Method: O(1/K) Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity](https://mlsanity.com/paper/2110.04261/) (2021)
- [Maximin effects in inhomogeneous large-scale data](https://mlsanity.com/paper/1406.0596/) (2014)
- [Minimax Regret Optimization for Robust Machine Learning under Distribution Shift](https://mlsanity.com/paper/2202.05436/) (2022)
- [A Universal Algorithm for Variational Inequalities Adaptive to Smoothness and Noise](https://mlsanity.com/paper/1902.01637/) (2019)
- [Statistical Inference for Maximin Effects: Identifying Stable Associations across Multiple Studies](https://mlsanity.com/paper/2011.07568/) (2020)
- [On Convergence of Gradient Descent Ascent: A Tight Local Analysis](https://mlsanity.com/paper/2207.00957/) (2022)
- [Magging: maximin aggregation for inhomogeneous large-scale data](https://mlsanity.com/paper/1409.2638/) (2014)
- [Minimax Regret Learning for Data with Heterogeneous Subgroups](https://mlsanity.com/paper/2405.01709/) (2024)
- [Invariant Ancestry Search](https://mlsanity.com/paper/2202.00913/) (2022)
- [Minimax Regret Estimation for Generalizing Heterogeneous Treatment Effects with Multisite Data](https://mlsanity.com/paper/2412.11136/) (2024)
- [Switching Regression Models and Causal Inference in the Presence of Discrete Latent Variables](https://mlsanity.com/paper/1808.05541/) (2018)
- [Stochastic Approximation Approaches to Group Distributionally Robust Optimization and Beyond](https://mlsanity.com/paper/2302.09267/) (2023)

## Built on by (most cited first, 3 of 3)

- [Anchor PCA](https://mlsanity.com/paper/2606.06233/) (2026)
- [Anti-causal domain generalization: Leveraging unlabeled data](https://mlsanity.com/paper/2602.17187/) (2026)
- [Prediction-Intervention Games and Invariant Sets](https://mlsanity.com/paper/2605.16828/) (2026)

## Abstract (arXiv)

We consider a regression setting where observations are collected in different environments modeled by different data distributions. The field of out-of-distribution (OOD) generalization aims to design methods that generalize better to test environments whose distributions differ from those observed during training. One line of such works has proposed to minimize the maximum risk across environments, a principle that we refer to as MaxRM (Maximum Risk Minimization). In this work, we introduce variants of random forests based on the principle of MaxRM. We provide computationally efficient algorithms and prove statistical consistency for our primary method. Our proposed method can be used with each of the following three risks: the mean squared error, the negative reward, and the regret (which quantifies the excess risk relative to the best predictor). For MaxRM with regret as the risk, we prove a novel out-of-sample guarantee over unseen test distributions. Finally, we evaluate the proposed methods on both simulated and real-world data.