# On the Convergence and Stability of Upside-Down Reinforcement Learning, Goal-Conditioned Supervised Learning, and Online Decision Transformers

arXiv: https://arxiv.org/abs/2502.05672 · HTML: https://mlsanity.com/paper/2502.05672/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Offline RL algorithms, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 7 of its 43 references that are in the map.
- First posted on arXiv 8 Feb 2025.

## Placed in

- Offline RL and imitation learning from data / [Offline RL algorithms](https://mlsanity.com/offlinerl/offline-rl-algorithms/)
- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 7 of 7)

- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [D4RL: Datasets for Deep Data-Driven Reinforcement Learning](https://mlsanity.com/paper/2004.07219/) (2020)
- [Reinforcement Learning Upside Down: Don't Predict Rewards -- Just Map Them to Actions](https://mlsanity.com/paper/1912.02875/) (2019)
- [Training Agents using Upside-Down Reinforcement Learning](https://mlsanity.com/paper/1912.02877/) (2019)
- [Generalized Decision Transformer for Offline Hindsight Information Matching](https://mlsanity.com/paper/2111.10364/) (2021)
- [Reward-Conditioned Policies](https://mlsanity.com/paper/1912.13465/) (2019)
- [ARCHER: Aggressive Rewards to Counter bias in Hindsight Experience Replay](https://mlsanity.com/paper/1809.02070/) (2018)

## Abstract (arXiv)

This article provides a rigorous analysis of convergence and stability of Episodic Upside-Down Reinforcement Learning, Goal-Conditioned Supervised Learning and Online Decision Transformers. These algorithms performed competitively across various benchmarks, from games to robotic tasks, but their theoretical understanding is limited to specific environmental conditions. This work initiates a theoretical foundation for algorithms that build on the broad paradigm of approaching reinforcement learning through supervised learning or sequence modeling. At the core of this investigation lies the analysis of conditions on the underlying environment, under which the algorithms can identify optimal solutions. We also assess whether emerging solutions remain stable in situations where the environment is subject to tiny levels of noise. Specifically, we study the continuity and asymptotic convergence of command-conditioned policies, values and the goal-reaching objective depending on the transition kernel of the underlying Markov Decision Process. We demonstrate that near-optimal behavior is achieved if the transition kernel is located in a sufficiently small neighborhood of a deterministic kernel. The mentioned quantities are continuous (with respect to a specific topology) at deterministic kernels, both asymptotically and after a finite number of learning cycles. The developed methods allow us to present the first explicit estimates on the convergence and stability of policies and values in terms of the underlying transition kernels. On the theoretical side we introduce a number of new concepts to reinforcement learning, like working in segment spaces, studying continuity in quotient topologies and the application of the fixed-point theory of dynamical systems. The theoretical study is accompanied by a detailed investigation of example environments and numerical experiments.