Average-Reward Reinforcement Learning for Multichain MDPs: A Hierarchical Decomposition Approach

arXiv:2610.10326 · cs.LG, math.OC · Submitted 2026-10-07 · Read on arXiv

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

Tom: Today's paper: "Average-Reward Reinforcement Learning for Multichain MDPs".

Jane: The material describes a sophisticated, model-free reinforcement learning framework designed to tackle the complexities of average-reward multichain Markov Decision Processes (MDPs).

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So we're looking at this paper called "Average-Reward Reinforcement Learning for Multichain MDPs: A Hierarchical Decomposition Approach" by Yu and Heidt. What is the big idea behind that title? It sounds pretty technical, but at its core, it’s about finding good ways to make decisions when you have multiple interacting systems, or chains.

Jane: Exactly. Think about it like this: you're not just controlling one machine; you're controlling a network of machines where the best strategy for one machine depends on what the others are doing and how they are linked together. This paper is tackling those kinds of complex decision-making scenarios where the optimal outcome isn't just about maximizing immediate reward, but about maintaining good long-term performance across all those chains simultaneously.

Lu: What’s interesting here is that they’re not just throwing a giant RL algorithm at this; they are using something called Bather’s Unique Hierarchical Decomposition to break the problem down. They partition the entire state space into two parts: states that are strongly communicating, and transient states that you can eventually leave.

Meng: So, they're essentially saying instead of solving one massive equation for everything at once, we can solve smaller problems inside those communicating groups and then deal with the rest separately as a global stopping problem. That sounds like it might make it more manageable when we have a huge state space to deal with in production.

Lalam: I see how that decomposition helps structure the learning process, which is key when you’re dealing with average rewards where things can be tricky because the optimal gain itself can depend on your initial state, as mentioned in the abstract <ref:2610.10326#pg1>.

Tom: Right. So we move from a single hard problem to a set of structured subproblems. That’s a huge shift in how we approach these average reward scenarios where the optimal gain might change depending on where you start. We're going to see how they build on this structure in the next part, which is where they introduce their actual algorithms.

Jane: And what those algorithms are going to do is try and get us closer to that optimal gain we’re aiming for, even if it’s hard to reach perfectly right away. It sets up a pathway for improvement.

Lu: I'm curious how they manage the transition between these decomposed parts when learning the policy asynchronously, because that’s where the real complexity usually creeps in with these kinds of systems.

Meng: From an engineering standpoint, if this decomposition allows us to solve smaller subproblems independently and then aggregate them, that suggests we can have more parallel computation paths during training without everything collapsing into one messy update.

Lalam: It’s about managing the dependencies between those subsystems so that we don't just end up with isolated local optima when trying to learn the global policy for the multichain MDP.

The paper's summary: Tom: Now let’s look at what they actually propose in this paper. The core of their contribution is this asynchronous, value-iteration-based RL algorithm that doesn't need any model knowledge beyond the transition graph of the MDP <ref:2610.10326#pg1>.

Jane: That’s significant because most deep RL requires a model, but this one says you only need to know how the system moves from state to state and what reward you get at each step. They leverage Bather's decomposition to recast the global problem into these structured subproblems, which is the engine behind their whole approach.

Lu: The paper shows that by exploiting this structure, they can prove that their base algorithm converges almost surely to the optimal gain vector g*, and it produces policies after a finite amount of time <ref:2610.10326#pg3>. That's a strong theoretical guarantee for achieving the best possible long-term average reward.

Meng: A finite time convergence guarantee is nice, but what about the practical reality? The abstract mentions that their base algorithm can exhibit poor transient performance, which means even if it eventually gets there, it might take a really long time or fail spectacularly during training <ref:2610.10326#pg3>.

Jane: That’s where the paper gets interesting because they don't just stop there. They propose two further algorithms to address those transient issues, aiming for better practical results by solving the average optimality equations more thoroughly.

Tom: So Algorithm one is the base one, which gets you almost surely to optimal gain but has that bad transient performance we mentioned <ref:2610.10326#pg1>. Then there’s Algorithm two which tries to solve the multichain average optimality equations approximately to get near gain-optimal policies <ref:2610.10326#pg3>.

Lu: Algorithm two builds on sub-MDP solutions from RVI Q-learning from Algorithm one and aims for near gain-optimal policies, and they prove convergence in three steps involving stability results and convergence of a limiting form <ref:2610.10326#pg3>.

Meng: I like that they are using the results from the base algorithm to build up these better solutions, even if it means running more complex computations. It's an iterative improvement strategy, which is often necessary when you're dealing with complex systems.

Lalam: It shows a clear progression: first get the theoretical optimum, then approximate it with better structure-aware methods <ref:2610.10326#pg3>.

The paper's improvements: Tom: Let’s talk about the improvements they suggest in this paper. They introduce Algorithm two and Algorithm three which are designed to fix the transient performance issues of the base algorithm <ref:2610.10326#pg1>. These are meant to bridge that gap between getting the perfect theoretical gain and actually having a working policy quickly.

Jane: Specifically, Algorithm two aims for near gain-optimal policies by leveraging the sub-MDP average optimality equations computed in step one, and they prove convergence based on extending stability results for Q-learning in SSP total-reward MDPs <ref:2610.10326#pg3>.

Lu: Algorithm three is aimed at near bias-optimality <ref:2610.10326#pg1>. It does this by first approximating the optimal bias function, and then solving an *induced* average-reward multichain MDP using the base algorithm as a subroutine <ref:2610.10326#pg3>.

Meng: So Algorithm three is essentially trying to find a middle ground where you get near gain optimality but also better transient behavior than the base algorithm, because they establish almost sure convergence for it too <ref:2610.10326#pg3>.

Tom: And what’s really cool about the improvements is how they use these components to get better results in practice, specifically by incorporating information from Algorithm two into Algorithm three to tune their parameters <ref:2610.10326#pg3>.

Jane: This adaptive tuning capability means the system can adjust its approximation accuracy based on what it learns from the base algorithm, which is a sophisticated way to control stability and get closer to the true optimal behavior.

Lu: And they also point out that this approach allows them to approximate the optimal gain vector using Q-hat star from Algorithm two which converges to a solution of the perturbed average optimality equation <ref:2610.10326#pg3>.

Meng: From an engineering view, being able to use those estimates to tune parameters like delta one, delta two and beta allows us to control how much we approximate versus how much stability we maintain during deployment <ref:2610.10326#pg3>.

Lalam: This adaptive tuning capability is important because it means the system can handle uncertainty well, even in high-uncertainty regimes where the environment might present positive bounded models that Phase one alone might struggle with, by incorporating Phase two information <ref:2610.10326#pg3>.

Conclusion: Tom: So to wrap up, this paper "Average-Reward Reinforcement Learning for Multichain MDPs: A Hierarchical Decomposition Approach" shows that we can’t just rely on the base algorithm alone if we want good real-world performance because of those transient failures. We need the enhancements provided by Algorithm two and Algorithm three to get better results in practice <ref:2610.10326#pg1>.

Jane: Exactly. The key is using that hierarchical decomposition structure to solve the subproblems, which gives us a path from theoretical optimality to near gain-optimal policies with better transient performance <ref:2610.10326#pg3>.

Lu: It’s about building a system where we can control the approximation trade-off between gaining accuracy and maintaining stability in the long run, which is crucial for any real application <ref:2610.10326#pg3>.

Meng: From my side, it means that instead of just deploying a policy that might crash in transient states, we have a method to generate policies that are actually stable and perform better on average when things get messy <ref:2610.10326#pg3>.

Lalam: I think the overall implication is that for average-reward multichain MDPs, this decomposition approach provides a solid framework for moving from just theoretical convergence to policies that actually work in complex, real-world scenarios where the gain itself is state-dependent.

Tom: So that’s what we were talking about today with "Average-Reward Reinforcement Learning for Multichain MDPs: A Hierarchical Decomposition Approach." We’ve seen how this framework uses decomposition to build better algorithms to handle the average reward challenges.

Jane: It was a really detailed look at how structure can be used not just for neat theoretical math but for practical policy construction <ref:2610.10326#pg3>.

Lu: We’re excited about how this decomposition lets us analyze these systems more rigorously, which opens up new avenues for understanding why certain policies fail in those transient states.

Meng: Yeah, moving forward, we need to think about how to make the deployment of Algorithm three even more robust against those kinds of uncertainties we discussed <ref:2610.10326#pg1>.

Huizhen Yu, Isaiah Heidt

University of Alberta

cs.LG, math.OC

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 60 pages, 4 figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 87/100

The gist: The material describes a sophisticated, model-free reinforcement learning framework designed to tackle the complexities of average-reward multichain Markov Decision Processes (MDPs).

Key concepts

Bather’s Unique Hierarchical Decomposition (UHD): This is a method to split the large global state space into smaller, manageable parts. It separates states into 'communicating subsystems' where decisions are strongly linked and 'transient states' which are temporary conditions. This structure allows the complex problem to be broken down into simpler subproblems.
Aggregated Optimal Stopping Problem (OSP): The overall average-reward problem is transformed into a single, aggregated OSP. This means instead of solving every chain independently, the algorithm focuses on finding one global decision that optimizes the total long-term average gain across all chains simultaneously, based on the hierarchical structure.
Gain Optimality vs. Bias Optimality: Gain optimality seeks to maximize the immediate reward rate in a state-dependent way. Bias optimality is a related metric that focuses on how close a policy is to the true optimal long-term average gain for every state, offering a more nuanced measure of performance.

Terminology

Reinforcement Learning — 1× in this paper · explained in 4 episodes

Summary

The material describes a sophisticated, model-free reinforcement learning framework designed to tackle the complexities of average-reward multichain Markov Decision Processes (MDPs).

Here is a comprehensive and detailed summary, synthesized from the provided abstracts and technical excerpts:

Detailed Research Summary: Asynchronous Value-Iteration-Based RL for Average-Reward Multichain MDPs

This paper presents a novel class of asynchronous, essentially model-free Reinforcement Learning (RL) algorithms specifically tailored to learn optimal policies in average-reward multichain MDPs. The core challenge addressed is that the optimal gain in such processes can be state-dependent, and the recurrence structures across different chains vary by policy, making standard RL methods difficult to apply.

Core Methodology: Hierarchical Decomposition

The central innovation of the proposed framework lies in leveraging Bather’s Unique Hierarchical Decomposition (UHD) of the state space. This decomposition partitions the global state space into two crucial components:

Communicating Subsystems: States within these subsystems are strongly coupled.
Transient States: States that are not part of any communicating subsystem, representing temporary conditions or states from which one can exit to a communicating set.

This hierarchical partitioning allows the complex global decision problem to be recast into a structured set of subproblems: optimal decisions within each subsystem and a single, aggregated global decision problem formulated as an Aggregated Optimal Stopping Problem (OSP). The algorithm exploits this structure by solving these structured components sequentially or hierarchically.

The Three Proposed Algorithms

The research introduces three distinct algorithms, building upon the base framework, each targeting a different level of optimality:

1. Base Algorithm (Algorithm 1): Gain Optimality

Mechanism: This is an asynchronous, value-iteration-based RL analogue to Ohno’s value-iteration algorithm for multichain MDPs. It utilizes Bather’s UHD to structure the problem and employs RVI Q-learning (as detailed in Phase 2) to compute necessary components.
Convergence Guarantees: Theorem 3.1 establishes that the iterates of the Q-function in Phase 3 converge almost surely to Q*, the optimal action-value function of the limiting aggregated OSP. Furthermore, Theorem 3.2 guarantees finite-time attainment of gain-optimality: almost surely for all sufficiently large time t, the constructed policy pi(t) is gain-optimal, meaning its realized gain g pi(t)(s) equals the optimal gain g*(s) for all states s in S.
Limitation: The base algorithm achieves the highest level of gain-optimality but suffers from significantly worse transient performance, particularly exhibiting large transient failures on a subset of MDP instances.

2. Algorithm 2: Near Gain Optimality (Approximate AOE Solution)

Mechanism: This algorithm aims to approximately solve the multichain average optimality equations (AOEs) to obtain near gain-optimal policies. It leverages the sub-MDP AOE solutions computed by RVI Q-learning from Algorithm 1.
Convergence Guarantees: Theorem 4.1 rigorously proves its convergence in three steps: (i) extending stability results for Q-learning in SSP (Sample Space Perturbed) total-reward MDPs; (ii) establishing the convergence of a limiting form of Algorithm 2 that solves a specific SSP-type MDP; and (iii) proving the almost sure convergence of its Q-learning iterates to solutions induced by Algorithm 1's outputs.
Performance Tradeoff: While it sacrifices some gain optimality compared to the base algorithm, it achieves the best transient performance among the three algorithms.

3. Algorithm 3: Near Bias Optimality (Approximating Optimal Bias)

Mechanism: This algorithm targets near bias-optimality by first approximating the optimal bias function and then solving an induced average-reward multichain MDP using the base algorithm as a subroutine.
Convergence Guarantees: Theorem 5.2 demonstrates that for sufficiently small parameters (delta 1, 2, and 1-beta), the sequence of policies mu t generated in Phase 2 is eventually gain-optimal and ** epsilon' -optimal with respect to bias-optimality** for the original MDP M.
Performance Tradeoff: Algorithm 3 provides a strong compromise. Its aggregate gain-optimality sits between the base algorithm and Algorithm 2, while its transient performance remains close to that of Algorithm 2 and the theoretical bias-optimal benchmark.

Key Research Contributions and Implications

Model-Free Nature: The algorithms require no model knowledge beyond the MDP's transition graph, which is a significant departure from many other RL approaches.
No Discounting Restriction: Crucially, this work operates entirely within the framework of average-reward MDPs, avoiding necessary reductions to discounted problems.
Empirical Validation: Experimental results (Figures 3a and 3b) confirm these theoretical findings: while the base algorithm shows high gain-optimality, Algorithms 2 and 3 consistently yield superior transient performance across a diverse set of benchmark MDPs, with Algorithm 3 offering the most balanced trade-off.
Practical Insight: The paper highlights the practical role of Phase 2 (RVI Q-learning) in Algorithm 3, noting that it provides a numerically demanding but practically useful path toward near bias-optimal policies compared to relying solely on discounted approximations.

In conclusion, this research successfully develops a robust, hierarchical RL framework for average-reward multichain MDPs. It moves beyond simple gain optimization by introducing sophisticated extensions (Algorithms 2 and 3) that leverage structural decomposition to achieve superior transient performance while maintaining strong guarantees regarding near-optimality in both gain and bias metrics.

Improvements for AI systems

Improved gain-optimal policy construction via Algorithm 3: The system can now compute near bias-optimal policies by leveraging discounted approximation with large discount factors applied to a reward-shifted MDP, which allows for distinguishing between gain-equivalent actions based on transient reward behavior.
Enhanced transient performance for learned policies: The resulting policies from Algorithm 3 are guaranteed to satisfy gµ(s) - g∗(s) ≥ −ϵβ − ¯δ2 and gµ(s) - g∗(s) ≥ −ϵβ − ¯δ2 − ¯δ1ϕµ(s, Rµ) for transient states, offering a more favorable tradeoff than the base algorithm.
Sample-path-dependent solution approximation: The system can approximate the optimal gain vector using Qˆ∗ from Algorithm 2, which converges to a solution of the perturbed AOE (4.31), allowing for an approximate global average-reward policy even when exact solutions are intractable.
Adaptive parameter tuning capability: The algorithm can be tuned adaptively by using optimal gain estimates for M′ produced by the base algorithm in Phase 2 of Algorithm 3 to adaptively tune the parameters ¯δ1, ¯δ2, and β, allowing for automated control over approximation accuracy and stability.
Robust policy selection in high-uncertainty regimes: The system can generate policies that are eventually gain-optimal even when the environment presents positive bounded models where Phase 1 alone might fail to distinguish between gain-optimal but not bias-optimal policies, by incorporating the information from Phase 2.