Towards Formalizing Reinforcement Learning Theory

arXiv:2511.03618 · cs.LG, stat.ML · Submitted 2026-08-20 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Next we'll be talking about the paper "Towards Formalizing Reinforcement Learning Theory".

Jane: The paper was written by Liu, J. and Yuan, Y. from.

Tom: Stay tuned as we take you through the paper and discuss its implications.

Title: Tom: So, looking at the title itself, "Towards Formalizing Reinforcement Learning Theory," it suggests a massive effort to solidify the mathematical underpinnings of these algorithms.

Jane: It’s about moving away from relying on informal intuition and toward establishing precise proofs that guarantee almost sure convergence for both Q-learning and linear TD.

Lu: The title implies they are closing gaps left by previous attempts, which were often too delicate or prone to errors, as the paper notes in its introduction.

Meng: I’m interested in how they handle the scope—are these proofs only for simple scenarios or does it suggest a practical application framework?

Lalam: The title suggests that our entire body of knowledge regarding RL is about to become more robust and dependable, which is a huge cultural win.

Summary: Tom: Now, the paper summarizes its findings by showing exactly how these two core RL algorithms behave over time. They aren't just talking about theoretical concepts; they are making concrete claims about trajectories.

Jane: They’ve managed to prove that for both Q-learning and linear TD, the iterates will eventually converge to their fixed points—that q t goes to q* and w t goes to w*. It’s a clean, definitive result.

Lu: What's truly impressive is that they achieve this convergence using a unified framework based on the Robbins-Siegmund theorem, which is far more powerful than just relying on approximations of ODE solutions. The math gives us much more leverage here.

Meng: The unification of these two distinct algorithms suggests that they share a common mathematical structure when we look at them through this specific lens, which simplifies implementation in large systems because the underlying principles are consistent.

Lalam: Reliability is what I see in this summary; the ability to predict with certainty that an autonomous system will reliably reach its intended goal is the ultimate benefit of this paper for ensuring dependable AI operation.

Improvements: Tom: The paper makes significant improvements by focusing on a unified framework, especially when dealing with Markovian samples. It’s not just about making the math cleaner; it' is about handling real-world data flow.

Jane: They are using a sophisticated "skeleton iterates" technique to convert the randomness from a Markov Chain into what they call Martingale difference noise, which is absolutely key for analyzing convergence in a stochastic environment.

Lu: This approach allows them to handle the complexity of real-world sequential data while maintaining mathematical rigor, unlike earlier methods that often struggled with such inherent stochasticity. It’s allowing us to do complex math on messy data.

Meng: The way they manage this randomness means that the algorithm can be adapted to process continuous streams of data without losing its theoretical guarantee of reaching a solution, which is a huge operational win for me.

Lalam: I think this provides a much better path toward future work; we now have a stable, rigorous foundation upon which to build much more complex and dynamic AI systems that operate over time reliably.

Conclusion: Tom: So, to wrap up the discussion, we've seen how "Towards Formalizing Reinforcement Learning Theory" provides a rigorous proof of convergence for Q-learning and linear TD using modern techniques.

Jane: It’s a massive step toward making RL theory trustworthy by using these advanced mathematical methods that ensure mathematical certainty in stochastic systems.

Lu: I am particularly excited about the path forward, like extending this framework to tackle more complex issues such as eligibility traces or other challenging off-policy methods. The potential for expansion is huge.

Meng: My main concern is how quickly we can take these proofs and apply them to large-scale production environments, given the formal guarantees they provide us regarding stability and accuracy.

Lalam: I believe that this work will help us define the boundaries of reliable AI, allowing us to design systems whose ultimate goals are guaranteed to be met through verifiable logic.

Tom: This is a truly fantastic achievement in theoretical computer science. Thank you all for breaking down "Towards Formalizing Reinforcement Learning Theory" with me today.

Jane: It’s been a pleasure talking through these complex ideas and the rigor of this paper with everyone on the team.

Lu: I hope this initial success leads to huge collaborative opportunities in the future, pushing the boundaries of formal methods globally across all researchers.

Meng: My hope is that this translates into practical, robust algorithms for real-world deployment very quickly and that our implementation efforts can match the theory.

Lalam: My final thought is that this work contributes significantly to a culture of verifiable science, ensuring we are building AI on solid mathematical ground.

Liu, J., Yuan, Y.

cs.LG, stat.ML

Submitted: 2026-08-20

Updated: 2026-08-21

Code: https://github.com/ShangtongZhang/rl-theory-in-lean

Importance score: 91/100

The gist: In this paper, we formalize the almost sure convergence of Q-learning and linear temporal difference (TD) learning with Markovian samples using the Lean 4 theorem prover based on the Mathlib library.

Key concepts

Formalizing Reinforcement Learning Theory: This refers to the effort to solidify the mathematical foundations of RL algorithms. The paper aims to move away from informal intuition and establishing precise proofs that guarantee almost sure convergence for Q-learning and linear TD.
Q-learning and Linear TD: These are two core reinforcement learning algorithms discussed in the paper. The findings prove that when these algorithms run over time, their iterates will eventually converge to specific fixed points (q_t goes to q* and w_t goes to w*).
Robbins-Siegmund theorem: This is a unified mathematical framework used in the paper. It allows the authors to prove convergence using a powerful method that is more effective than relying on approximations of ODE solutions.

Terminology

Q-learning — 44× in this paper · explained in 3 episodes
Reinforcement Learning — 15× in this paper · explained in 4 episodes
Unified Framework — 6× in this paper · explained in 5 episodes
Actor — 4× in this paper · explained in 3 episodes
Benchmark — 4× in this paper · explained in 2 episodes
Lyapunov function — 4× in this paper · explained in 2 episodes
Trace — 4× in this paper · explained in 2 episodes
Reinforcement Learning (RL) — 3× in this paper · explained in 25 episodes
alpha — 3× in this paper · explained in 1 episode
Benchmarking — 2× in this paper · explained in 3 episodes
Generalization — 2× in this paper · explained in 13 episodes
Markov Decision Process — 2× in this paper
Accuracy — 1× in this paper · explained in 1 episode
Action Space — 1× in this paper · explained in 1 episode
Adaptive Algorithm — 1× in this paper · explained in 1 episode
Core — 1× in this paper · explained in 2 episodes
Formal Methods — 1× in this paper · explained in 3 episodes
Hallucination — 1× in this paper · explained in 3 episodes
Markov Decision Process (MDP) — 1× in this paper · explained in 11 episodes
Markov Reward Process — 1× in this paper
Sample Complexity — 1× in this paper · explained in 15 episodes
Verification — 1× in this paper · explained in 2 episodes

Summary

In this paper, we formalize the almost sure convergence of Q-learning and linear temporal difference (TD) learning with Markovian samples using the Lean 4 theorem prover based on the Mathlib library.

Motivation and Scope

Q-learning and linear TD are among the earliest and most influential reinforcement learning (RL) algorithms. While their convergence properties have been a major research topic, we argue that the convergence proofs are usually delicate for two reasons. First, the standard ODE approach is full of details and bug-prone, with numerous documented gaps in existing literature. Second, RL theory is typically formulated within the Markov Decision Process (MDP) framework, which requires complex measure theory—specifically using the Ionescu-Tulcea theorem to construct a probability space for infinite length trajectories of the MDP—to rigorously study stochastic iterates.

Formalization Methodology

This work develops the first formalization of the almost sure convergence of Q-learning and linear TD with Markovian samples on finite state action MDP, utilizing Lean 4. The methodology is centered around modern techniques that combine Lyapunov functions and the Robbins-Siegmund theorem, employing a skeleton iterates technique to convert Markovian noise into Martingale difference noise. This provides a unified framework for formalizing not only almost sure convergence but also high probability concentration, Lp convergence, and the corresponding convergence rates.

Formal Theorem Statements

The paper presents several key theorems regarding the almost sure convergence of these algorithms:

Linear TD Convergence (Markovian Samples):

Theorem 3.1 states that for a finite Markov chain S t that is irreducible and aperiodic, if the step size is alpha t = (t+2) nu with nu in (2/3, 1), then the iterates w t generated by (Linear TD) with R t+1 = r pi(S t satisfy that t to infinity w t = w* a.s.

Linear TD Convergence (i.i.d. Samples):

Theorem 3.2 shows that if the Markov chain is irreducible and aperiodic, considering (Linear TD) but replacing (S t, S t+1) with (S t,0, S t,1) where S t,0 about d pi and S t,1 about P pi(S t,0, times), and if the step size alpha t satisfies the Robbins-Monro condition, the iterates w t with R t+1 = r pi(S t) satisfy that t to infinity w t = w* a.s.

Q-Learning Convergence (Markovian Samples):

Theorem 3.3 asserts that for any fixed policy pi, if the induced finite Markov chain is irreducible and aperiodic, and the step size is alpha t = (t+2) nu with nu in (2/3, 1), then the iterates q t generated by (Q-learning) satisfy that t to infinity q t = q* a.s.

Future Extensions and Limitations

The framework is designed to be extended to other modes of convergence, including L2 convergence rates with i.i.d. samples and almost sure convergence rates under Markovian samples (which the authors believe will be straightforward). The paper also notes that the most technically challenging aspect is computing conditional expectation in Lean, which requires going into detail about the construction generated by the Ionescu-Tulcea theorem.

The formalization effort also serves as a high-quality dataset for benchmarking LLM’s reasoning and coding capability, with the authors noting that LLMs have been instrumental in this project as a personalized tutor and a powerful search engine.

Improvements for AI systems

(The following improvements are derived from synthesizing the theoretical convergence analysis, formal verification methodologies, and advanced stochastic approximation techniques found within the provided bibliography.)

Improvement: Development of a full-stack system that embeds formal proof assistants (like Lean/Mathlib or Coq) directly into the policy optimization loop. Instead of merely outputting weights, the CRLE outputs a set of mathematically verified invariants and performance bounds for the resulting policy function pi(s). This addresses the critical gap between theoretical convergence guarantees and verifiable deployment safety.

What the Improved AI System Can Do:

Guaranteed Safety Constraints: The system can be formally constrained to ensure that, within a defined operational envelope (e.g., state space S safe), the expected cumulative reward E[sum R t] will never violate pre-set safety thresholds, even during periods of high stochastic noise or model uncertainty.
Proof of Stability: It provides mathematical proof that the policy iteration process converges to a fixed point pi* within a verifiable number of steps, eliminating the risk associated with unproven asymptotic stability in complex environments.
Regulatory Compliance: It generates auditable, machine-readable proofs suitable for high-stakes domains (e.g., autonomous vehicles, medical robotics) where failure is unacceptable and regulatory proof is mandatory.