# Model-free Neural Counterfactual Regret Minimization with Bootstrap Learning

arXiv: https://arxiv.org/abs/2012.01870 · HTML: https://mlsanity.com/paper/2012.01870/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2020.

## Built on by (most cited first, 5 of 5)

- [ESCHER: Eschewing Importance Sampling in Games by Computing a History Value Function to Estimate Regret](https://mlsanity.com/paper/2206.04122/) (2022)
- [Equivalence Analysis between Counterfactual Regret Minimization and Online Mirror Descent](https://mlsanity.com/paper/2110.04961/) (2021)
- [Deep (Predictive) Discounted Counterfactual Regret Minimization](https://mlsanity.com/paper/2511.08174/) (2025)
- [Game-Theoretic Robust Reinforcement Learning Handles Temporally-Coupled Perturbations](https://mlsanity.com/paper/2307.12062/) (2023)
- [KrwEmd: Revising the Imperfect-Recall Abstraction from Forgetting Everything](https://mlsanity.com/paper/2511.12089/) (2025)

## Abstract (arXiv)

Counterfactual Regret Minimization (CFR) has achieved many fascinating results in solving large-scale Imperfect Information Games (IIGs). Neural network approximation CFR (neural CFR) is one of the promising techniques that can reduce computation and memory consumption by generalizing decision information between similar states. Current neural CFR algorithms have to approximate cumulative regrets. However, efficient and accurate approximation in a large-scale IIG is still a tough challenge. In this paper, a new CFR variant, Recursive CFR (ReCFR), is proposed. In ReCFR, Recursive Substitute Values (RSVs) are learned and used to replace cumulative regrets. It is proven that ReCFR can converge to a Nash equilibrium at a rate of O(1/√T). Based on ReCFR, a new model-free neural CFR with bootstrap learning, Neural ReCFR-B, is proposed. Due to the recursive and non-cumulative nature of RSVs, Neural ReCFR-B has lower-variance training targets than other neural CFRs. Experimental results show that Neural ReCFR-B is competitive with the state-of-the-art neural CFR algorithms at a much lower training cost.