# Tropical Reinforcement Learning

arXiv: https://arxiv.org/abs/2610.02478 · HTML: https://mlsanity.com/paper/2610.02478/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL training of language and agent models, one of 15 directions in Deep reinforcement learning methods.
- Builds on 35 of its 43 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Placed in

- Deep reinforcement learning methods / [RL training of language and agent models](https://mlsanity.com/deeprl/rl-training-of-language-and-agent-models/)

## Builds on (most cited first, 30 of 35)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [WebShop: Towards Scalable Real-World Web Interaction with Grounded Language Agents](https://mlsanity.com/paper/2207.01206/) (2022)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Measuring and Narrowing the Compositionality Gap in Language Models](https://mlsanity.com/paper/2210.03350/) (2022)
- [DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence](https://mlsanity.com/paper/2606.19348/) (2026)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Group-in-Group Policy Optimization for LLM Agent Training](https://mlsanity.com/paper/2505.10978/) (2025)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [ReST-MCTS*: LLM Self-Training via Process Reward Guided Tree Search](https://mlsanity.com/paper/2406.03816/) (2024)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [Measuring Compositional Generalization: A Comprehensive Method on Realistic Data](https://mlsanity.com/paper/1912.09713/) (2019)
- [VinePPO: Refining Credit Assignment in RL Training of LLMs](https://mlsanity.com/paper/2410.01679/) (2024)
- [Imagination-Augmented Agents for Deep Reinforcement Learning](https://mlsanity.com/paper/1707.06203/) (2017)
- [Pass@K Policy Optimization: Solving Harder Reinforcement Learning Problems](https://mlsanity.com/paper/2505.15201/) (2025)
- [Optimizing Language Models for Inference Time Objectives using Reinforcement Learning](https://mlsanity.com/paper/2503.19595/) (2025)
- [From f(x) and g(x) to f(g(x)): LLMs Learn New Skills in RL by Composing Old Ones](https://mlsanity.com/paper/2509.25123/) (2025)
- [Maximum Likelihood Reinforcement Learning](https://mlsanity.com/paper/2602.02710/) (2026)
- [Revisiting Entropy in Reinforcement Learning for Large Reasoning Models](https://mlsanity.com/paper/2511.05993/) (2025)
- [RAGEN-2: Reasoning Collapse in Agentic RL](https://mlsanity.com/paper/2604.06268/) (2026)
- [Save the Good Prefix: Precise Error Penalization via Process-Supervised RL to Enhance LLM Reasoning](https://mlsanity.com/paper/2601.18984/) (2026)
- [Seemingly Simple Planning Problems are Computationally Challenging: The Countdown Game](https://mlsanity.com/paper/2508.02900/) (2025)
- [Reason in Chains, Learn in Trees: Self-Rectification and Grafting for Multi-turn Agent Policy Optimization](https://mlsanity.com/paper/2604.07165/) (2026)

## Abstract (arXiv)

Reinforcement learning for large language models typically maximizes expected return, adding up the probabilities of all successful trajectories. However, the classical sum formulation can only report how often the model policy succeeds, not which solution actually worked, and because probabilities sum to one, reinforcing one solution can make the model forget another that was never shown to be wrong. This makes expected return a poor fit for compositional reasoning, where a solution must be assembled from reasoning steps that the model produces in separate, often failed, attempts but rarely produces together. To address this, we propose Tropical Reinforcement Learning, which rests on a simple change of algebra: instead of adding the probabilities of alternative solutions, we take their maximum, which yields the tropical semiring. The value of a state then becomes the log-probability of its most likely verified solution, together with an explicit path that can be replayed and reused. This enables true composition, since the best prefix and the best suffix meeting at a shared state can be joined even when they come from different rollouts. To put this into practice, we introduce TROPIC, a training algorithm for deterministic, resettable environments with verifiable outcomes. On four agentic tasks (Sokoban, Countdown, FrozenLake, WebShop), TROPIC outperforms the strongest on-policy baselines by up to 16 percentage points. Changing the algebra of reinforcement learning, not just its estimators, can thus substantially improve compositional reasoning in language models