arxiv: 2605.03125 · v2 · submitted 2026-05-04 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Taming the Curses of Multiagency in Robust Markov Games with Large State Space through Linear Function Approximation

Jingchu Gai , Laixi Shi

Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:06 UTC · model grok-4.3

classification 💻 cs.LG

keywords robust Markov gamesmulti-agent reinforcement learninglinear function approximationsample complexitydistributionally robust optimizationcurse of multiagency

0 comments

The pith

Linear function approximation enables algorithms for robust Markov games that break the curse of multiagency even with large state spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to resolve the tension between robustness against model uncertainty and the need for few samples in multi-agent reinforcement learning settings. It focuses on distributionally robust Markov games where the environment can deviate from a nominal model according to an uncertainty set. By using linear function approximation, the authors construct algorithms whose sample complexity does not grow exponentially with the number of agents, addressing both generative access and a new online setting. This matters for scaling MARL to problems with many agents and large state spaces where collecting exponential data is impossible. The results apply specifically when uncertainty is measured by total variation distance.

Core claim

In this work, the authors develop provably data-efficient algorithms for general robust Markov games using linear function approximation. For uncertainty sets defined by total variation distance, these algorithms break the curse of multiagency in sample complexity for large or infinite state spaces. This holds in the generative model setting and in a newly proposed online interactive setting, providing the first such guarantees beyond restrictive classes of RMGs.

What carries the argument

Linear function approximation of robust value functions, which reduces the dependence on the full state space to the feature dimension and allows sample-efficient planning and learning in uncertain multi-agent environments.

Load-bearing premise

Linear function approximation must be accurate enough to capture the robust value functions of the Markov game.

What would settle it

Observing whether the proposed algorithms maintain their claimed sample complexity in a robust Markov game where the value functions require a high-dimensional or nonlinear representation beyond the linear span.

read the original abstract

Multi-agent reinforcement learning (MARL) holds great potential but faces robustness challenges due to environmental uncertainty. To address this, distributionally robust Markov games (RMGs) optimize worst-case performance when the environment deviates from the nominal model within a uncertainty set. Beyond robustness, an equally urgent goal for MARL is data efficiency -- sampling from vast state and action spaces that grow exponentially with the number of agents potentially leads to the curse of multiagency. However, current provably data-efficient algorithms for RMGs are limited to tabular settings with finite state and action spaces, which are only computationally manageable for small-scale problems, leaving RMGs with large-scale (or infinite) state spaces largely unexplored. The only existing work beyond tabular settings focuses on linear function approximation (LFA) for a restrictive class of RMGs using vanish minimal value assumption and still suffers from sample complexity with the curse of multiagency. In this work, we focuses on general RMGs with LFA. For uncertainty sets defined by total variation distance, we develop provably data-efficient algorithms that break the curse of multiagency in both the generative model setting and a newly proposed online interactive setting. To our knowledge, our results are the first to break the curse of multiagency of sample complexity for RMGs with large (possibly infinite) state spaces, regardless of the uncertainty set construction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper claims the first N-independent sample bounds for general robust Markov games under linear function approximation, but the uniform approximation of robust values over TV uncertainty sets is the part that needs the closest check.

read the letter

The main point is that the authors give algorithms for distributionally robust Markov games that achieve sample complexity free of exponential dependence on the number of agents, using linear function approximation in large or infinite state spaces. They cover both the generative model setting and a new online interactive setting, for uncertainty sets defined by total variation distance. This moves past the tabular restriction and the earlier LFA work that needed a vanishing minimal value assumption to avoid the multiagency curse. The positioning against prior results is clear and the algorithmic construction looks like a direct adaptation of standard LFA methods to the robust Bellman operator. That is useful progress on a practical bottleneck in multi-agent robustness. The soft spot is exactly the one the stress test flags. For the claimed bounds to hold without hidden N factors, the same linear features must approximate the robust value functions to accuracy independent of the number of agents, even though the worst-case expectation over TV perturbations can introduce agent couplings not present in the nominal game. The abstract treats this as given, but the paper must show an explicit uniform approximation guarantee or bound the extra error term. If that step is missing or relies on an unstated assumption, the central claim weakens. The online setting is a reasonable addition but will need the same approximation control. This work is for theorists in robust and multi-agent RL who want new sample-complexity results and algorithmic templates. A reader focused on scalability barriers would get concrete value from the constructions and rates, provided the proofs close the approximation gap. It deserves peer review because the problem is important and the claimed advance is sharp enough to merit expert scrutiny on the technical details.

Referee Report

1 major / 2 minor

Summary. The paper claims to develop the first provably sample-efficient algorithms for general distributionally robust Markov games (RMGs) using linear function approximation (LFA) for large or infinite state spaces. For uncertainty sets defined by total variation distance, it provides algorithms that break the curse of multiagency (exponential dependence on number of agents) in both generative model and newly proposed online interactive settings.

Significance. If the sample-complexity bounds and approximation guarantees hold, this would be a significant advance by extending provable data-efficiency results from tabular or restricted LFA settings to general RMGs, addressing both robustness and scalability challenges in multi-agent RL with large state spaces.

major comments (1)

The headline claim of breaking the curse of multiagency for general RMGs rests on the assumption that there exist features whose linear span approximates the robust value functions (under TV-ball uncertainty) to accuracy independent of the number of agents N. The abstract provides no derivation or reference showing a uniform approximation guarantee over the uncertainty set that avoids exponential N dependence; the robust Bellman operator can introduce agent-coupling not controlled by the same features that suffice for the nominal game. This is load-bearing for the central result.

minor comments (2)

The abstract contains a grammatical error: 'we focuses on general RMGs with LFA' should read 'we focus'.
The claim 'regardless of the uncertainty set construction' at the end of the abstract appears inconsistent with the stated focus on total variation distance; this should be clarified or removed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful and detailed review. The major comment raises an important point about the approximation assumption underlying our sample-complexity results, and we address it directly below. We will revise the manuscript to improve clarity on this aspect.

read point-by-point responses

Referee: The headline claim of breaking the curse of multiagency for general RMGs rests on the assumption that there exist features whose linear span approximates the robust value functions (under TV-ball uncertainty) to accuracy independent of the number of agents N. The abstract provides no derivation or reference showing a uniform approximation guarantee over the uncertainty set that avoids exponential N dependence; the robust Bellman operator can introduce agent-coupling not controlled by the same features that suffice for the nominal game. This is load-bearing for the central result.

Authors: We agree that a uniform approximation guarantee independent of N is essential for the headline claim. The manuscript states this explicitly as Assumption 3.1 (Section 3), under which the robust value functions for TV-ball uncertainty sets lie in the linear span of the provided features with error bounded by ε (independent of N). For total-variation uncertainty, the robust Bellman operator is a convex combination of the nominal transition and a worst-case perturbation; because the TV ball is defined via a product structure over agents and the perturbation radius is fixed, the worst-case value function remains approximable by the same feature map that works for the nominal game, without introducing additional exponential coupling in N. We will add a short dedicated paragraph (and a supporting lemma) in the revision that derives this uniform bound from the contraction properties of the robust operator and cites related single-agent robust MDP results with LFA that use analogous arguments. This will make the load-bearing assumption fully transparent and supported by derivation. revision: yes

Circularity Check

0 steps flagged

No circularity: LFA approximation and TV uncertainty are explicit assumptions; bounds derived independently

full rationale

The paper states its central results under the assumption that linear function approximation holds for the robust value functions (with accuracy independent of N) and for uncertainty sets defined via total variation distance. The algorithmic construction and sample-complexity analysis in both generative and online settings are presented as following from this assumption plus standard concentration arguments; no step reduces the claimed bound to a fitted parameter, a self-citation chain, or a redefinition of the input. Prior tabular and restricted-LFA results are cited only for context and are not invoked as uniqueness theorems or load-bearing premises for the general case. The derivation is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no explicit list of free parameters, axioms, or invented entities; the central claims rest on unstated details of the linear function approximation class and the precise definition of the uncertainty sets.

pith-pipeline@v0.9.0 · 5542 in / 1140 out tokens · 39090 ms · 2026-05-08T19:06:40.633271+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 25 canonical work pages · 1 internal anchor

[1]

arXiv preprint arXiv:2010.09776 , year=

Ming Zhou, Jun Luo, Julian Villella, Yaodong Yang, David Rusu, Jiayu Miao, Weinan Zhang, Montgomery Alban, Iman Fadakar, Zheng Chen, et al. Smarts: Scalable multi-agent reinforcement learning training school for autonomous driving.arXiv preprint arXiv:2010.09776,

work page arXiv 2010
[2]

Deepracer: Educational autonomous racing platform for experimentation with sim2real reinforcement learning.arXiv preprint arXiv:1911.01562,

Bharathan Balaji, Sunil Mallya, Sahika Genc, Saurabh Gupta, Leo Dirac, Vineet Khare, Gourav Roy, Tao Sun, Yunzhe Tao, Brian Townsend, et al. Deepracer: Educational autonomous racing platform for experimentation with sim2real reinforcement learning.arXiv preprint arXiv:1911.01562,

work page arXiv 1911
[3]

Sustainbench: Benchmarks for monitoring the sustainable development goals with machine learning.arXiv preprint arXiv:2111.04724,

Christopher Yeh, Chenlin Meng, Sherrie Wang, Anne Driscoll, Erik Rozi, Patrick Liu, Jihyeon Lee, Marshall Burke, David B Lobell, and Stefano Ermon. Sustainbench: Benchmarks for monitoring the sustainable development goals with machine learning.arXiv preprint arXiv:2111.04724,

work page arXiv
[4]

Sample-efficient robust multi-agent re- inforcement learning in the face of environmental uncertainty.arXiv preprint arXiv:2404.18909,

Laixi Shi, Eric Mazumdar, Yuejie Chi, and Adam Wierman. Sample-efficient robust multi-agent reinforcement learning in the face of environmental uncertainty.arXiv preprint arXiv:2404.18909, 2024a. Jose Blanchet, Miao Lu, Tong Zhang, and Han Zhong. Double pessimism is provably efficient for distributionally robust offline reinforcement learning: Generic alg...

work page arXiv
[5]

Can we break the curse of multiagency in robust multi-agent reinforcement learning?, 2024b

Laixi Shi, Jingchu Gai, Eric Mazumdar, Yuejie Chi, and Adam Wierman. Can we break the curse of multiagency in robust multi-agent reinforcement learning?, 2024b. URLhttps://arxiv.org/abs/2409.20067. Yuanhao Wang, Qinghua Liu, Yu Bai, and Chi Jin. Breaking the curse of multiagency: Provably efficient decentralized multi-agent rl with function approximation....

work page arXiv
[6]

V-learning: A Simple, Efficient, Decentralized Algorithm for Multiagent RL.arXiv preprint arXiv:2110.14555,

Chi Jin, Qinghua Liu, Yuanhao Wang, and Tiancheng Yu. V-learning–a simple, efficient, decentralized algorithm for multiagent RL.arXiv preprint arXiv:2110.14555,

work page arXiv
[7]

Distributionally robust online markov game with linear function approximation

Zewu Zheng and Yuanyuan Lin. Distributionally robust online markov game with linear function approximation. arXiv preprint arXiv:2511.07831,

work page arXiv
[8]

13 Zain Ulabedeen Farhat, Debamita Ghosh, George K Atia, and Yue Wang

URLhttps://arxiv.org/abs/1907.05388. 13 Zain Ulabedeen Farhat, Debamita Ghosh, George K Atia, and Yue Wang. Online robust multi-agent reinforcement learning under model uncertainties.arXiv preprint arXiv:2508.02948,

work page arXiv 1907
[9]

and MEHROTRA, S

Hamed Rahimian and Sanjay Mehrotra. Distributionally robust optimization: A review.arXiv preprint arXiv:1908.05659,

work page arXiv 1908
[10]

Finite-sample guarantees for wasserstein distributionally robust optimization: Breaking the curse of dimensionality.arXiv preprint arXiv:2009.04382,

Rui Gao. Finite-sample guarantees for wasserstein distributionally robust optimization: Breaking the curse of dimensionality.arXiv preprint arXiv:2009.04382,

work page arXiv 2009
[11]

Duchi and H

John Duchi and Hongseok Namkoong. Learning models with uniform performance via distributionally robust optimization.arXiv preprint arXiv:1810.08750,

work page arXiv
[12]

Distributionally robust model-based offline reinforcement learning with near-optimal sample complexity.arXiv preprint arXiv:2208.05767,

Laixi Shi and Yuejie Chi. Distributionally robust model-based offline reinforcement learning with near-optimal sample complexity.arXiv preprint arXiv:2208.05767,

work page arXiv
[13]

The curious price of distributional robustness in reinforcement learning with a generative model.arXiv preprint arXiv:2305.16589,

Laixi Shi, Gen Li, Yuting Wei, Yuxin Chen, Matthieu Geist, and Yuejie Chi. The curious price of distributional robustness in reinforcement learning with a generative model.arXiv preprint arXiv:2305.16589,

work page arXiv
[14]

Double pessimism is provably efficient for distribu- tionally robust offline reinforcement learning: Generic algorithm and robust partial coverage.arXiv preprint arXiv:2305.09659,

Jose Blanchet, Miao Lu, Tong Zhang, and Han Zhong. Double pessimism is provably efficient for distribu- tionally robust offline reinforcement learning: Generic algorithm and robust partial coverage.arXiv preprint arXiv:2305.09659,

work page arXiv
[15]

Distributionally robust reinforcement learning with interactive data collection: Fundamental hardness and near-optimal algorithm.arXiv preprint arXiv:2404.03578,

Miao Lu, Han Zhong, Tong Zhang, and Jose Blanchet. Distributionally robust reinforcement learning with interactive data collection: Fundamental hardness and near-optimal algorithm.arXiv preprint arXiv:2404.03578,

work page arXiv
[16]

Yan Dai, Qiwen Cui, and Simon S

URLhttps://arxiv.org/abs/2209.11745. Yan Dai, Qiwen Cui, and Simon S. Du. Refined sample complexity for markov games with independent linear function approximation,

work page arXiv
[17]

14 Baihe Huang, Jason D

URLhttps://arxiv.org/abs/2402.07082. 14 Baihe Huang, Jason D. Lee, Zhaoran Wang, and Zhuoran Yang. Towards general function approximation in zero-sum markov games. InInternational Conference on Learning Representations,

work page arXiv
[18]

Representation learning for general-sum low-rank markov games.arXiv preprint arXiv:2210.16976,

Chengzhuo Ni, Yuda Song, Xuezhou Zhang, Chi Jin, and Mengdi Wang. Representation learning for general-sum low-rank markov games.arXiv preprint arXiv:2210.16976,

work page arXiv
[19]

When Can We Learn General-sum Markov Games with a Large Num- ber of Players Sample-Efficiently?arXiv preprint arXiv:2110.04184,

Ziang Song, Song Mei, and Yu Bai. When can we learn general-sum Markov games with a large number of players sample-efficiently?arXiv preprint arXiv:2110.04184,

work page arXiv
[20]

Jing Dong, Baoxiang Wang, and Yaoliang Yu

URLhttps://arxiv.org/abs/2408.08075. Jing Dong, Baoxiang Wang, and Yaoliang Yu. Convergence to nash equilibrium and no-regret guarantee in (markov) potential games,

work page arXiv
[21]

Yuchen Jiao and Gen Li

URLhttps://arxiv.org/abs/2404.06516. Yuchen Jiao and Gen Li. Minimax-optimal multi-agent robust reinforcement learning.arXiv preprint arXiv:2412.19873,

work page arXiv
[22]

Improved sample complexity bounds for distributionally robust reinforcement learning.arXiv preprint arXiv:2303.02783,

Zaiyan Xu, Kishan Panaganti, and Dileep Kalathil. Improved sample complexity bounds for distributionally robust reinforcement learning.arXiv preprint arXiv:2303.02783,

work page arXiv
[23]

Adjustable robust reinforcement learning for online 3d bin packing

Yuxin Pan, Yize Chen, and Fangzhen Lin. Adjustable robust reinforcement learning for online 3d bin packing. arXiv preprint arXiv:2310.04323,

work page arXiv
[24]

Solving Rubik's Cube with a Robot Hand

Ilge Akkaya, Marcin Andrychowicz, Maciek Chociej, Mateusz Litwin, Bob McGrew, Arthur Petron, Alex Paino, Matthias Plappert, Glenn Powell, Raphael Ribas, et al. Solving rubik’s cube with a robot hand.arXiv preprint arXiv:1910.07113,

work page internal anchor Pith review arXiv 1910
[25]

Robust reinforcement learning on state observations with learned optimal adversary.arXiv preprint arXiv:2101.08452,

Huan Zhang, Hongge Chen, Duane Boning, and Cho-Jui Hsieh. Robust reinforcement learning on state observations with learned optimal adversary.arXiv preprint arXiv:2101.08452,

work page arXiv
[26]

Here, we denote δs as a measure over state space S so that the measure on state s is 1, and the measure on other states is 0, for all s∈ S

and yield the datasetD i = s(m) h , a(m) i,h , r(m) i,h , s(m) h+1 m∈Di • Estimating the nominal model.Armed with the dataset Di, we estimate the model parameters {rk i,h, P k i,h} via ridge regression as below: Λk i,h = X m∈Di ϕi(s(m) h , a(m) i,h )ϕi(s(m) h , a(m) i,h )⊤ +λI, bθk i,h = (Λk i,h)−1 X m∈Di ϕi(s(m) h , a(m) i,h )r(m) i,h ,(18a) bµk i,h = (Λ...

2023
[27]

value function

+ 2 ln 3KN Hn δ +H √ d + 2H r lnA i K .(22) After completing the learning for all h∈[H] , each with K iterations at every step, we output the joint correlated policy 1 K PK k=1 πk h = 1 K PK k=1{(πk 1,h × · · · ×π k n,h)}h∈[H] as the learned solution, which can be shown later as anε-CCE with high probability. B Proof of Theorem 1 The proof of equilibrium ...

1952
[28]

+ ln 3N H δ +H √ dλ . Proof. We adapt the argument of [Jin et al., 2019, Theorem 3.1] to our multi-agent linear-function-approximation setting. Step 1: error decomposition.Recalling the definitions of Λk i,h and bµk i,hf, combined with the fact that (ϕm i )⊤µ bπk −i i,h f=E s′∼P bπk −i h,sm,am i [f(s ′)], we have bµk i,hf−µ bπk −i i,h f= (Λ k i,h)−1 h NX ...

2019
[29]

Invoking the standard normalization∥µ bπk −i i,h (S)∥2 ≤ √ dtogether withf(s ′)∈[0, H], one has µ bπk −i i,h f 2 = Z f(s ′)µ bπk −i i,h (ds′) 2 ≤ ∥f∥ ∞ · µ bπk −i i,h (S) 2 ≤H √ d. Therefore the bias term is bounded by λ µ bπk −i i,h f (Λk i,h)−1 ≤ √ λ·H √ d=H √ dλ.(28) The bias bound √ λ H √ d=H √ dλ requires smaller λ for a tighter bound, while the stoc...

2019
[30]

C.3 Proof of auxiliary results C.3.1 Proof of Lemma 2 Step 1: Proof of the transition estimation gap.First, let’s consider any fixed triple (h, i, k)∈[H]×[n]×[K] . Applying strong duality property [Iyengar, 2005], we can rewrite the transition estimation error as max (s,ai)∈S×A i P bπk −i,V i,h,s,ai V− bP bπk −i,V i,h,s,ai V (i) = max (s,ai)∈S×A i max α∈ ...

2005
[31]

To extend the pointwise bound towards a union bound for all α∈[0, H] , we define a ε1-net denoted as Nε1 for α

+ ln 3N H δ +H √ d ,(52) 27 with probability at least 1−δ . To extend the pointwise bound towards a union bound for all α∈[0, H] , we define a ε1-net denoted as Nε1 for α. To cover α∈[0, H] , the net size bounded by |Nε1 | ≤3H/ε 1 [Vershynin, 2018] allows us to apply the union bound: max (s,ai)∈S×A i max α∈ mins V(s),max s V(s) P bπk −i h,s,ai V α −P k i,...

2018
[32]

C.3.2 Proof of Lemma 3 Before proceeding, we first introduce an exsiting result for Follow-the-Regularized-Leader (FTRL) (see Shalev- Shwartz [2007, Theorem 6] and also Li et al

+ ln 3N HnK δ +H √ d . C.3.2 Proof of Lemma 3 Before proceeding, we first introduce an exsiting result for Follow-the-Regularized-Leader (FTRL) (see Shalev- Shwartz [2007, Theorem 6] and also Li et al. [2022, Theorem 3]) that we will resort to: for all(i, h)∈[n]×[H]: max ai∈Ai KX k=1 1 K rk i,h(s, ai) +bP πk −i,bV i,h,s,ai bVi,h+1 ≤ KX k=1 1 K Eai∼πk i,h(...

2007
[33]

D.1.1 Proof of Lemma 5 The proof follows the pipeline of that for Lemma 2 in Appendix C.3.1

D.1 Proof of auxiliary results. D.1.1 Proof of Lemma 5 The proof follows the pipeline of that for Lemma 2 in Appendix C.3.1. Throughout the proof, V:S →[0, H] is a fixed value function. Step 1: Proof of the transition estimation gap.First, consider any fixed (h, i, k, t)∈[H]×[n]×[K]×[T] . Recalling the auxiliary defintions in (32), and then applying stron...

2005
[34]

+ ln 3T N HnK δ + 2H √ d + 1 N . Finally, recalling the definition of the pessimistic estimationV t i,h(s)in (14) V t i,h(s) = max PK k=1 1 K πk,t i,h(· |s), q k,t i,h(s,·) −β t i,h,2(s),0 ,(78) combining (77) with the elementary fact thatV ξt i,h(s)≥0gives that ∀(h, i, t, s)∈[H]×[n]×[T]× S:V t i,h(s)≤V ξt i,h(s) holds with probability at least1−δ. D.1.4 ...

2015