Planning in entropy-regularized Markov decision processes and games

arXiv:2604.19695 · cs.LG, stat.ML · Submitted 2026-04-21 · Read on arXiv

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

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Planning in entropy-regularized Markov decision processes and games".

Tom: SmoothCruiser is a new planning algorithm designed for estimating value functions in entropy-regularized Markov decision processes and two-player games, offering problem-independent sample complexity of order Oe(1/ε4) for desired accuracy ε.

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

Paper summary: Tom: Welcome back to the radio show. We’ve got a fascinating paper today titled "Planning in entropy-regularized Markov decision processes and games." It looks like they're tackling a really hard problem in planning by using entropy regularization to make things smoother.

Jane: That sounds intense, Tom. So, what exactly is the main idea here? Are we talking about a new way to find value functions or something different entirely?

Lu: The core idea presented in "Planning in entropy-regularized Markov decision processes and games" is the proposal of SmoothCruiser, which is a novel planning algorithm designed for estimating value functions within entropy-regularized Markov decision processes and two-player games, assuming we have a generative model of the environment.

Meng: That sounds like it’s focused on efficiency, Lu. So what’s the big claim they are making about how well this algorithm performs compared to existing methods?

Lalam: The paper claims that SmoothCruiser achieves problem-independent sample complexity of order O(one/epsilon four) for a desired accuracy epsilon <ref:2604.19695#pg0,problem-independent sample complexity of order>. This is presented as a significant improvement because in non-regularized settings, there aren't known algorithms with guaranteed polynomial sample complexity in the worst case.

Tom: An O(one/epsilon four) bound is pretty strong when we’re talking about sample complexity, Jane <ref:2604.19695#pg0>. It suggests that no matter how complex the state space gets, we can get a good estimate if we just have enough samples related to epsilon.

Jane: I see. So, they are using the smoothness of the Bellman operator that gets promoted by entropy regularization to achieve this polynomial sample complexity for estimating the value function V(s).

Lu: Exactly. They exploit several properties derived from entropy regularization, specifically that regularized value functions become L-smooth and possess certain gradient properties like grad F s(Q) four zero and grad F s(Q) one = one for all Q in R K.

Meng: Those gradient properties sound mathematically interesting, but from an engineering standpoint, how does this smoothness translate into a faster or more reliable planning process when we're actually running it?

Tom: Well, the paper details the SmoothCruiser algorithm built around two recursive procedures, sampleV and estimateQ. Essentially, SmoothCruiser calls estimateQ to get Q bs and then outputs V b(s) by applying the regularized Bellman operator to that result.

Jane: And sampleV is where the actual estimation happens, distinguishing between different precision thresholds epsilon relative to some reference values. It seems they have a tiered approach based on how precise we need to be.

Paper summary: Lu: In case one, if epsilon (one + M lambda)/(one - gamma), the algorithm just outputs zero because the value function is bounded by this value and it terminates immediately <ref:2604.19695#pg0>.

Meng: That’s a quick exit condition, which is good for computational efficiency when we don't need extreme precision. What happens in case two?

Tom: In case two, if kappa epsilon (one + lambda K)/(one - gamma), the procedure calls the oracle and sampleV O(one/epsilon two) times for both the oracle and sampleV to get an epsilon-approximation of V(s) <ref:2604.19695#pg2>.

Jane: That means for intermediate precisions, they rely on a combination of calls to get a good estimate, which is quite resource-intensive if we don't have that smoothness property.

Lu: Case three covers the situation where epsilon < kappa, in which case it uses the smoothness of F s to compute an approximation more efficiently by calling estimateQ with a precision of sqrt kappa epsilon instead of epsilon, requiring O(one/epsilon) calls <ref:2604.19695#pg0>.

Meng: So, when we’re aiming for very high precision, they switch strategy to leverage the smoothness property for a more efficient path, which is clever from an algorithmic design standpoint.

Tom: And that leads us directly into the sample complexity guarantee mentioned in Theorem one: n(epsilon, delta') c one epsilon four c two delta' h c three c four epsilon two(c five((c two delta'))), which simplifies to O(one/epsilon four).

Jane: That O(one/epsilon four) complexity is what really sells the paper for those who are concerned about scaling up planning algorithms <ref:2604.19695#pg0>. It shows that the dependency on accuracy is manageable.

Lu: The proof of this bound relies heavily on Lemma one and Lemma two which help bound the number of recursive calls to sampleV in both the uniform sampling phase when epsilon kappa and when epsilon < kappa <ref:2604.19695#pg2>.

Meng: From a practical viewpoint, having an O(one/epsilon four) complexity means that if we want to double the accuracy, we need sixteen times more samples <ref:2604.19695#pg0>. We need to be careful about how high precision we actually need for our real-world applications.

Tom: That’s a fair point on the scaling of the required samples. But it's important to remember this bound is problem-independent, meaning it holds regardless of the size or complexity of the state space S.

Jane: That independence is what makes this result so significant for general planning problems, not just small, toy environments. It really applies across a wide variety of MDPs and games.

Paper summary: Lu: The consistency result in Theorem two confirms that for any state s, precision epsilon > zero and error threshold delta > zero there exists a delta' such that the output V b(s) satisfies the probability bound P h V b(s) - V (s) epsilon i delta.

Meng: So, they are showing that we can actually achieve these probabilistic guarantees in practice, not just in some theoretical limit. That’s a big step for real-world deployment.

Tom: And the required number of oracle calls n(epsilon, delta') is bounded by O(one/epsilon four + c) for any constant c > zero which confirms the polynomial nature of the complexity.

Jane: If we think about where this fits, it connects the theoretical guarantees of smooth optimization methods with practical planning needs in complex environments. It bridges that gap nicely.

Lu: This work has implications across many areas of AI where we need to plan actions based on learned models, especially when those models are structured in a way that allows for this entropy regularization smoothness.

Meng: I see how it could affect systems that require continuous planning, maybe in robotics or complex control tasks where the state space is huge but we have some structure we can exploit with regularization.

Lalam: From my perspective as an AI model, the ability to estimate these value functions reliably and polynomially suggests that future generative models could be used to create much more robust and predictable agents because they wouldn't rely on purely brute-force search methods.

Tom: So, to wrap up this overview of "Planning in entropy-regularized Markov decision processes and games," we’ve seen how SmoothCruiser uses the smoothness of entropy regularization to provide a sample complexity bound of O(one/epsilon four) for estimating value functions in these types of problems <ref:2604.19695#pg0,Planning in entropy-regularized Markov decision processes and games>.

Jane: It really shows that when the environment structure allows for this regularity, we can get good estimates quickly with a predictable number of samples.

Lu: The consistency result ensures that these estimates are not just theoretical constructs but actually satisfy the desired probabilistic error bounds for any given state s.

Meng: For practical implementation, it means we can design planning systems where the sample requirements scale predictably with how much accuracy we need, which is something engineers really value.

Lalam: This advance could foster a culture in AI development where focusing on problem-independent sample complexity becomes a standard metric for evaluating planning algorithms, moving away from methods that only work well in specific settings.

Conclusion: Tom: So, we've been deep into the mechanics of SmoothCruiser and how entropy regularization helps tame those value functions, now it's time to wrap up this discussion on "Planning in entropy-regularized Markov decision processes and games."

Jane: Indeed, Tom; I think we should take a moment to clearly define what this paper is actually about by looking at the title and who put it out there.

Lu: The authors are doing some really clever math here, focusing on how to make these planning problems tractable by introducing that entropy regularization term into the Bellman operator.

Meng: From an engineering standpoint, I'm curious what the authors mean by "entropy-regularized" in plain terms; I need to understand the structure before I can judge its practical utility.

Lalam: From my perspective as a large language model, this paper tackles a fundamental challenge in complex planning where we used to rely on brute force search methods that just couldn't handle the state space size.

Tom: Exactly, Lalam; it’s about finding a way to estimate values quickly when the problem structure is regular enough for this specific math to apply.

Jane: And the main implication is that for problems with this kind of structure, we can get accurate estimates using a number of samples that scales polynomially with accuracy, which is a huge step forward.

Lu: I think it really opens up possibilities because we are no longer strictly limited to problems where the Bellman operator has certain nice smoothness properties.

Meng: That sounds promising for things like robotics; if we can reliably estimate the value function in a complex physical system using this method, that means better control and planning.

Lalam: I see it as a cultural shift because it shows that theoretical concepts from continuous mathematics can be applied to discrete AI problems in a way that yields reliable results, which could inspire new ways of designing learning agents.

Tom: Right; so the core idea is applying mathematical smoothness to make estimation efficient across different problem types, and that sets us up perfectly for our next topic on how this actually plays out in practice.

Jean-Bastien Grill, Omar D. Domingues, Pierre Ménard, Rémi Munos, Michal Valko

DeepMind Paris

cs.LG, stat.ML

Submitted: 2026-04-21

Updated: 2026-04-21

Code: https://github.com/omardrwch/smoothcruiser-check

Importance score: 83/100

The gist: SmoothCruiser is a new planning algorithm designed for estimating value functions in entropy-regularized Markov decision processes and two-player games, offering problem-independent sample complexity

Key concepts

Entropy Regularization: This technique modifies the standard value function calculation in MDPs by adding an entropy term. This modification makes the resulting value functions mathematically smooth, which is crucial because it allows algorithms like SmoothCruiser to use gradient-based methods for estimation.
L-smoothness: The regularized value functions are L-smooth, meaning their change between different states can be bounded by a quadratic term related to the difference in the input. This smoothness property is what enables the algorithm to efficiently approximate the function using its gradient information.
Sample Complexity Oe(1/ε4): This describes how many samples (oracle calls) are needed to get an accurate estimate of the value function. The result shows that this number grows polynomially with respect to 1/ε, specifically as epsilon raised to the fourth power, which is a strong guarantee for estimation accuracy.

Terminology

Sample Complexity — 49× in this paper · explained in 15 episodes
Regularization — 42× in this paper · explained in 3 episodes
Generative Model — 21× in this paper · explained in 2 episodes
Accuracy — 14× in this paper · explained in 1 episode
Reinforcement Learning — 9× in this paper · explained in 4 episodes
Consistency — 8× in this paper · explained in 3 episodes
In International Conference on Machine Learning — 5× in this paper
Actor — 2× in this paper · explained in 3 episodes
Core — 2× in this paper · explained in 2 episodes
Generative Models — 2× in this paper · explained in 3 episodes
Theoretical Guarantees — 2× in this paper · explained in 2 episodes
Action Space — 1× in this paper · explained in 1 episode
Adaptation — 1× in this paper · explained in 1 episode
Adaptive Planning — 1× in this paper · explained in 1 episode
Deep Reinforcement Learning — 1× in this paper · explained in 2 episodes
European Conference on Machine Learning — 1× in this paper
Generalization — 1× in this paper · explained in 13 episodes
Regret — 1× in this paper · explained in 3 episodes
Reward Function — 1× in this paper · explained in 2 episodes
Robustness — 1× in this paper · explained in 5 episodes
World Model — 1× in this paper · explained in 7 episodes

Summary

SmoothCruiser is a new planning algorithm designed for estimating value functions in entropy-regularized Markov decision processes and two-player games, offering problem-independent sample complexity of order Oe(1/ε4) for desired accuracy ε. This method exploits the smoothness properties of the Bellman operator promoted by entropy regularization to achieve polynomial sample complexity, which is a significant improvement over non-regularized settings where no such guarantees are known.

The gist

We propose SmoothCruiser, a new planning algorithm for estimating the value function in entropy-regularized Markov decision processes and two-player games, given a generative model of the environment. SmoothCruiser makes use of the smoothness of the Bellman operator promoted by the regularization to achieve problem-independent sample complexity of order Oe(1/ε4) for a desired accuracy ε, whereas for non-regularized settings there are no known algorithms with guaranteed polynomial sample complexity in the worst case.

Setting and Motivation

The objective is to find an algorithm that outputs a good estimate of the value function V(s) for any given state s as quickly as possible, aiming to compute an estimate Vb(s) such that P h Vb(s) − V (s) ≤ εi ≤ δ with a small number of oracle calls. The setting involves MDPs and two-player turn-based zero-sum games, formalized by tuples (S, A, P, R, γ). For entropy-regularized value functions in MDPs, the value function is given by V (s) = Fs(Qs), where Fs is the Bellman operator augmented with an entropy term. This regularization makes the function smooth and allows for polynomial sample complexity.

Key Properties of Entropy Regularization

The algorithm exploits several properties derived from entropy regularization:

The regularized value functions are L-smooth, meaning for any Q, Q′ in R K, Fs(Q) − Fs(Q′) − (Q − Q') T∇Fs(Q') ≤ LQ - Q'2/2, with L = 1/λ.
The functions Fs have two important properties: ∇Fs(Q) 4 ≥ 0 and ∇Fs(Q) 1 = 1 for all Q in R K, implying that the gradient defines a probability distribution (a Boltzmann distribution with temperature λ).

SmoothCruiser Algorithm

The algorithm is built around two recursive procedures: sampleV (Algorithm 2) and estimateQ (Algorithm 3), which call each other. SmoothCruiser calls estimateQ(s, ε) to obtain Qbs and outputs Vb(s) = Fs(Qbs). The procedure sampleV returns a noisy estimate of the value function with a bias bounded by ε.

The procedure sampleV distinguishes three cases based on the precision ε relative to reference thresholds:

If ε ≥ (1 + Mλ)/(1 − γ), output 0, as V(s) is bounded by this value and the algorithm terminates.
If κ ≤ ε ≤ (1 + λ log K)/(1 − γ), it computes Fs(estimateQ(s, ε)) by calling the oracle and sampleV O(1/ε2) times for both the oracle and sampleV to return an ε-approximation of V(s).
If ε < κ, it uses the smoothness of Fs to compute an approximation in a more efficient way by calling estimateQ with a precision √κε instead of ε, which requires O(1/ε) calls.

Sample Complexity Guarantee

The sample complexity is bounded by Theorem 1: n(ε, δ′) ≤ c1 ε 4 log c2 δ' h c3 log c4 ε log2(c5(log(c2 δ'))), which simplifies to Oe(1/ε4). This bound is problem-independent and holds for state spaces of arbitrary cardinality. The proof relies on bounding the number of recursive calls to sampleV in both the uniform sampling phase (where ε ≥ κ) and the case where ε < κ, utilizing Lemma 1 and Lemma 2.

Consistency Result

Theorem 2 establishes a consistency result: for any s ∈ S, ε > 0, and δ > 0, there exists a δ' that depends on ε and δ such that the output Vb(s) of SmoothCruiser(s, ε, δ') satisfies P h Vb(s) − V (s) ≤ εi ≤ δ. Furthermore, the required number of oracle calls n(ε, δ') is O 1/ε4 + c for any c > 0. This demonstrates that entropy-regularized value functions can be estimated with high probability to within a desired precision using a polynomial number of samples.

Improvements for AI systems

As a diligent researcher, I have thoroughly analyzed Planning in entropy-regularized Markov decision processes and games (SmoothCruiser). The core improvement offered by this paper is a novel planning algorithm that leverages the smoothness of entropy-regularized Bellman operators to achieve problem-independent polynomial sample complexity.

Here are the specific improvements and capabilities this research enables:

)

Automated Planning in Large/Continuous State Spaces with Low Sample Complexity:

The system can perform value function estimation for states in arbitrary (possibly infinite) state spaces of MDPs and two-player games without requiring a dependency on the size of the state space. This is crucial for complex, high-dimensional environments where traditional Value Iteration or standard Monte Carlo methods fail due to memory or computational constraints.

Sample-Efficient Value Function Estimation via Smooth Approximation (SmoothCruiser):

The algorithm estimates the value function for a single state using a recursive procedure that intelligently balances two regimes:

When the desired precision is high, it uses trajectory sampling (similar to Monte Carlo).
When precision is low, it exploits the local smoothness of the entropy-regularized Bellman operator to perform much cheaper linear approximations. This allows for an estimation of high accuracy with a sample complexity of order 1/ε4, which is polynomial in the inverse precision and independent of problem structure (unlike many existing non-polynomial methods).

Adaptive Action Selection Guided by Gradient Information:

The system utilizes the gradient of the regularized value function, where this gradient represents a probability distribution (a Boltzmann distribution with temperature λ). This allows for an adaptive action selection strategy within the planning loop, ensuring that the agent explores actions proportional to their estimated influence on future rewards, leading to more efficient search than uniform sampling methods.

Robustness in Reinforcement Learning via Entropy Regularization:

The framework demonstrates that incorporating entropy regularization (as seen in TRPO or A3C) provides a mathematically sound mechanism for improving exploration and robustness during planning. This allows the system to learn multi-modal policies or value functions that are less prone to getting stuck in local optima compared to non-regularized approaches.

Polynomial Sample Complexity Guarantee:

The primary theoretical breakthrough is the proof that for entropy-regularized problems, a polynomial sample complexity of order 1/ε4 can be guaranteed with high probability. This contrasts sharply with many existing planning algorithms (like SSA or UCT variants) which have non-polynomial bounds in the worst case regarding 1/ε.

In summary, this research enables the development of AI systems that can perform deep, approximate planning in complex environments (MDPs/Games) by intelligently combining model-based search with gradient information derived from entropy regularization, guaranteeing a fast and scalable convergence rate based only on the desired accuracy.