On the Convergence and Stability of Upside-Down Reinforcement Learning, Goal-Conditioned Supervised Learning, and Online Decision Transformers (2502.05672v1)

Abstract: 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.

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

The paper at hand explores a rigorous and comprehensive analysis of convergence and stability for a set of supervised learning-based algorithms in the field of reinforcement learning (RL), namely, Episodic Upside-Down Reinforcement Learning (eUDRL), Goal-Conditioned Supervised Learning (GCSL), and Online Decision Transformers (ODTs). Historically, these algorithms have shown competitive performance across a range of RL benchmarks, such as games and robotic tasks. However, their theoretical underpinnings have been largely unexplored, particularly when it comes to their performance across different environmental conditions. This paper fills a critical gap by establishing a theoretical framework to understand these algorithms, particularly through their convergence and stability in various reinforcement learning settings.

The research is structured around two fundamental questions:

1. Under what conditions do UDRL, GCSL, and ODT converge in Markov Decision Processes (MDPs) with a specified transition kernel?
2. How stable are these algorithms when small perturbations are introduced to the environment?

Key Results

The authors begin by establishing the unique role of deterministic transition kernels in ensuring convergence to optimal policies. When environments are deterministic, eUDRL naturally converges to these optimal solutions by iterating over trajectory samples and updating policies through supervised learning. The paper rigorously proves this convergence for deterministic environments, affirming that eUDRL yields optimal behavior in such settings.

However, the challenge emerges when dealing with non-deterministic environments. The paper demonstrates that even minor levels of non-determinism can cause instability in eUDRL. This is due to the algorithm's inherent assumption of environment determinism, which may not hold in practice. A detailed analysis reveals that eUDRL's goal-reaching objectives can exhibit discontinuity in relation to environment perturbations—highlighted by the use of example environments and numerical experiments within the paper.

To address such instability, the authors explore regularization techniques applied during the eUDRL recursive process. Regularization introduces stability by utilizing convex combinations, essentially blending updated policies with uniform distributions akin to ε-greedy strategies. This transformation not only accommodates discontinuities but also aids in managing relative continuity, aligning closer to the stability of ODT, known for its successful online fine-tuning ability even under stochastic conditions.

Implications

The findings have several profound implications:

• Practical Deployment: By identifying conditions under which eUDRL and its variants remain stable, this paper offers useful insights for deploying such algorithms in real-world systems. The conditions outlined in this paper help ensure that these models can cope with the inherent randomness present in various real-world tasks.
• Algorithm Improvement: The topological approach leveraged in this paper helps in identifying how regularization strategies, like entropy regularization in ODT, contribute to the stability and performance of RL algorithms even in less deterministic conditions.
• Broader Adoption and Adaptation: The theoretical insights provided in this paper may stimulate broader adoption of supervised learning techniques in reinforcement learning. By understanding their limitations and ensuring mathematical rigor, researchers can confidently apply these techniques to novel and diverse applications.

Future Directions

While this paper lays a foundational understanding of convergence and stability for eUDRL, GCSL, and ODT, several avenues for future research remain open. These include:

• Further exploration of discontinuity boundaries and the development of techniques to mitigate their impact in high-dimensional or highly stochastic environments.
• Application of these algorithms in broader RL scenarios, such as those involving dynamic and continuously evolving domains.
• Refinement of regularization techniques to ensure even more robust and scalable performance across a variety of RL benchmarks.

In summary, this paper elevates the theoretical discourse around key algorithms in the RL domain, presenting a nuanced understanding of their behavior under distinct experimental conditions. This contributes significantly to the general understanding of supervised-learning influenced RL algorithms and their real-world applicabilities.