On the Convergence and Stability of Upside-Down Reinforcement Learning, Goal-Conditioned Supervised Learning, and Online Decision Transformers
🤖 AI Summary
This work establishes, for the first time, a unified theoretical framework for convergence and noise robustness of three emerging sequence-based RL paradigms—Episodic Upside-Down RL, Goal-Conditioned Supervised Learning, and Online Decision Transformers—under Markovian dynamics. Methodologically, it introduces episodic state-space modeling, quotient-topological continuity analysis, and dynamical systems fixed-point theory, integrated with transition kernel perturbation analysis to derive explicit kernel-dependent bounds on policy performance, value functions, and goal-reaching capability. Theoretically, it proves that algorithms converge to near-optimal policies as the transition kernel approaches determinism, and that solutions are continuous and asymptotically stable in the quotient topology. Numerical experiments empirically validate these theoretical guarantees. This work provides the first rigorous, unifying foundation for supervised and sequence-based RL, bridging theoretical analysis with practical algorithm design.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Analyzes convergence and stability of reinforcement learning algorithms.
Investigates optimal solutions under specific environmental conditions.
Assesses policy stability in environments with minimal noise.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Analyzes convergence in reinforcement learning
Studies stability under environmental noise
Introduces new theoretical concepts
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