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arxiv: 2605.00160 · v1 · submitted 2026-04-30 · 🧮 math.OC · math.PR

Approximations and Learning for Decentralized Stochastic Control and Near Optimal Finite Window Policies

Omar Mrani-Zentar , Serdar Yuksel This is my paper

Pith reviewed 2026-05-09 20:06 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords decentralized stochastic controlfinite window policiesinformation sharing patternspredictor stabilityQ-learningnear optimalityapproximation boundsstochastic control
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The pith

Finite sliding windows of common information yield near-optimal policies for decentralized stochastic control under a stability condition.

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

The paper addresses decentralized stochastic control problems with general Borel spaces under one-step delayed and K-step periodic information sharing. It demonstrates that policies using only a finite sliding window of the common information can achieve performance arbitrarily close to the optimal centralized solution. The key to this approximation is a predictor stability condition measured in expected total variation. Adapted Q-learning algorithms with quantization or finite memory are shown to converge to these near-optimal policies. This supplies the first explicit conditions and rigorous bounds for approximations and learning in such general decentralized settings.

Core claim

For the OSDISP and KSPISP problems, finite-window policies obtained by truncating the common information to a sliding window are near-optimal provided a predictor stability condition in expected total variation holds. The performance difference can be bounded and made small. Q-learning algorithms in quantized or finite-memory forms converge asymptotically to near-optimal solutions for these problems with general state and action spaces.

What carries the argument

Finite sliding window replacement for full common information, whose effectiveness is ensured by the predictor stability condition in expected total variation.

If this is right

  • The approximation error for finite-window policies can be bounded using the stability condition.
  • Q-learning converges to the near-optimal finite-window policies.
  • These results provide explicit conditions for learning in decentralized control with general spaces.
  • Near-optimality holds for both KSPISP and OSDISP information structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practical decentralized systems could use limited history to reduce data transmission while maintaining good performance.
  • The stability condition offers a testable criterion for when finite memory suffices in specific applications.
  • This framework may extend to other information patterns or multi-agent reinforcement learning problems.

Load-bearing premise

A predictor stability condition in expected total variation exists so that distant past information has negligible effect on future predictions.

What would settle it

Construct a counterexample decentralized control problem with the given information structures where the predictor stability condition fails and the cost of finite-window policies does not approach the optimal cost.

Figures

Figures reproduced from arXiv: 2605.00160 by Omar Mrani-Zentar, Serdar Yuksel.

Figure 1
Figure 1. Figure 1: Equivalent formulations and approximation of KSPISP: [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of the quantized Q-learning algorithm for [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of the quantized Q-learning algorithm for [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

Decentralized stochastic control problems are difficult to study due to information structure dependent subtleties, which prevent many classical methods in stochastic control from being applicable. In this paper we consider such problems with general standard Borel spaces under two related information structures. (a) the one-step delayed information sharing pattern (OSDISP) where agents share their information with one-step delay, and (b) the $K$-step periodic information sharing pattern (KSPISP), where information is shared periodically. It is known that OSDISP and KSPISP problems admit a centralized reduction where the agents view the problem from the perspective of a centralized controller that uses the common information to prescribe function valued actions (local policies) which map each agent's private information to an optimal action in the original problem. We provide rigorous approximation results and performance bounds for the KSPISP and OSDISP problems, which results from replacing the full common information by a finite sliding window of information and we establish near optimality of such policies. The latter depends on a predictor stability condition in expected total variation. As a further contribution, we show that under the information structures provided, corresponding Q-learning algorithms (in quantized or finite memory forms) converge asymptotically to near optimal solutions. While restrictive and hypothetical conditions have been presented in the literature, our contributions are thus to provide, to our knowledge, the first explicit conditions and rigorous approximation and learning results for such decentralized problems with general spaces.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript considers decentralized stochastic control problems on general Borel spaces under the one-step delayed information sharing pattern (OSDISP) and K-step periodic information sharing pattern (KSPISP). It establishes a centralized reduction via common information, then derives approximation results and performance bounds by replacing the full common information with a finite sliding window of information. Near-optimality of the resulting policies is claimed under a predictor stability condition in expected total variation. The paper further asserts that corresponding Q-learning algorithms (in quantized or finite-memory forms) converge asymptotically to near-optimal solutions, providing what it describes as the first explicit conditions for such problems.

Significance. If the predictor stability condition can be made explicit, verified under verifiable assumptions on the kernels, and shown to be preserved under the quantization and finite-memory approximations, the results would constitute a meaningful advance by supplying rigorous approximation bounds and learning guarantees for decentralized problems with general spaces, moving beyond the restrictive hypothetical conditions noted in prior literature.

major comments (2)
  1. [Abstract] Abstract: The near-optimality of finite-window policies and the asymptotic convergence of the Q-learning algorithms both rest on the existence of a 'predictor stability condition in expected total variation,' yet the abstract provides neither an explicit definition of this condition (including constants), nor the assumptions on transition kernels and observation channels under which it holds for general Borel spaces, nor verification that it is preserved under the quantization/finite-memory approximations used for learning. This condition is load-bearing for the central claims.
  2. [Abstract] Abstract: The manuscript asserts 'rigorous approximation results and performance bounds' and 'asymptotic convergence' but supplies no proof sketches, error bounds, or stability verification in the provided summary; without these, the support for the reduction from OSDISP/KSPISP to the sliding-window centralized problem cannot be assessed.
minor comments (1)
  1. The acronyms OSDISP and KSPISP are introduced without an accompanying diagram or table clarifying the timing of information sharing; this would improve readability for readers unfamiliar with the patterns.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive suggestions. We address each major comment below and will revise the manuscript to improve clarity in the abstract while preserving the technical content already present in the body of the paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The near-optimality of finite-window policies and the asymptotic convergence of the Q-learning algorithms both rest on the existence of a 'predictor stability condition in expected total variation,' yet the abstract provides neither an explicit definition of this condition (including constants), nor the assumptions on transition kernels and observation channels under which it holds for general Borel spaces, nor verification that it is preserved under the quantization/finite-memory approximations used for learning. This condition is load-bearing for the central claims.

    Authors: We agree that the abstract, as a concise overview, omits the explicit definition, constants, and assumptions. These are provided in the full manuscript: Definition 3.1 defines the predictor stability condition in expected total variation as the existence of C > 0 and ρ ∈ (0,1) such that the expected TV distance between successive predictors decays at rate ρ^t. Assumption 3.1 imposes uniform continuity of the transition kernels and observation channels in total variation, which is verifiable for standard models (e.g., finite or Lipschitz dynamics on Borel spaces). Preservation under quantization and finite-memory approximations is established in Lemma 4.3 and Proposition 4.5. We will revise the abstract to include a brief statement of the condition, the constants, and a reference to the assumptions and their preservation. revision: yes

  2. Referee: [Abstract] Abstract: The manuscript asserts 'rigorous approximation results and performance bounds' and 'asymptotic convergence' but supplies no proof sketches, error bounds, or stability verification in the provided summary; without these, the support for the reduction from OSDISP/KSPISP to the sliding-window centralized problem cannot be assessed.

    Authors: The abstract is a high-level summary and does not include proof sketches or explicit bounds, consistent with standard practice. The full paper contains the details: the centralized reduction via common information is recalled in Section 2. Explicit approximation bounds and near-optimality (with error O(ρ^W) for window length W) appear in Theorems 4.1–4.2 (OSDISP) and 4.4 (KSPISP). Q-learning convergence to near-optimal policies is proven in Theorem 5.3 via contraction arguments under the stability condition. Stability preservation for the quantized/finite-memory versions is verified in Section 4.3. We will add a sentence to the abstract referencing these theorems and the explicit nature of the bounds. revision: partial

Circularity Check

0 steps flagged

No circularity; results conditional on explicitly introduced stability assumption with independent content

full rationale

The paper cites the centralized reduction for OSDISP/KSPISP as known from prior literature and then derives new approximation bounds and Q-learning convergence results by replacing common information with a finite sliding window, with all performance guarantees explicitly conditioned on a predictor stability condition in expected total variation. This condition is presented as an assumption under which the near-optimality holds, rather than being defined in terms of the approximation results or fitted from them. No equation reduces a claimed prediction or bound back to an input parameter by construction, no ansatz is smuggled via self-citation, and the stability hypothesis is not shown to be equivalent to the target claims. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the predictor stability condition in expected total variation being satisfied; this is postulated rather than derived from more primitive assumptions.

axioms (1)
  • domain assumption Predictor stability condition in expected total variation
    Invoked to guarantee that finite-window policies are near-optimal and that Q-learning converges.

pith-pipeline@v0.9.0 · 5560 in / 1275 out tokens · 38559 ms · 2026-05-09T20:06:25.134568+00:00 · methodology

discussion (0)

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