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arxiv: 2606.04335 · v2 · pith:FUN5GJNVnew · submitted 2026-06-03 · 💻 cs.LG · cs.SY· eess.SY

Policy Gradient for Continuous-Time Robust Markov Decision Processes

Tanya Veeravalli , David M. Bossens , Atsushi Nitanda This is my paper

Pith reviewed 2026-06-28 07:17 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SY
keywords continuous-time RMDPpolicy gradientrobust reinforcement learningadversarial gradientsdouble-loop optimizationmean-field optimizationsample complexityneural ODE
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The pith

Policy gradients and adversarial gradients for continuous-time robust MDPs are derived from pathwise and adjoint formulas on differential equations.

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

This paper develops a continuous-time version of robust Markov decision processes to design reinforcement learning agents with performance guarantees under worst-case dynamics. It derives policy and adversarial gradients using pathwise and adjoint-based methods for stochastic and ordinary differential equations. Double-loop optimizers are proposed that achieve linear convergence with access to oracles and Õ(1/ε²) sample complexity in the sample-based case for undiscounted total cost problems. Mean-field optimizers are also introduced for distributional optimization with their own convergence guarantees. The approach is demonstrated on problems with neural ordinary differential equation dynamics.

Core claim

In the continuous-time RMDP framework, policy gradients and adversarial gradients are obtained through pathwise differentiation and adjoint methods applied to SDEs and ODEs. These gradients enable double-loop optimization algorithms that converge linearly in the oracle model and attain Õ(1/ε²) sample complexity in the sample-based model for undiscounted total-cost MDPs. Mean-field versions of the optimizers are shown to converge at Õ(1/K) in the oracle setting and Õ(N²/ε) under N-particle approximation. The methods are validated on continuous-time RMDPs whose dynamics are given by neural ODEs.

What carries the argument

Pathwise and adjoint-based gradient formulas for SDEs and ODEs within the continuous-time robust MDP model.

Load-bearing premise

The continuous-time RMDP framework allows derivation of policy and adversarial gradients via pathwise and adjoint methods for SDEs and ODEs, and that the proposed optimizers achieve the stated convergence rates under the undiscounted total cost MDP analysis.

What would settle it

Running the double-loop optimizer on a benchmark continuous-time RMDP and observing sublinear convergence instead of linear convergence would falsify the claimed rates.

Figures

Figures reproduced from arXiv: 2606.04335 by Atsushi Nitanda, David M. Bossens, Tanya Veeravalli.

Figure 1
Figure 1. Figure 1: Histogram of the test performance of the policies obtained by the distributional optimisers. The [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
read the original abstract

The framework of robust Markov decision processes (RMDPs) allows the design of reinforcement learning agents that satisfy performance guarantees under worst-case transition dynamics. Traditional RMDPs consider discrete-time dynamics and recently, sample-efficient policy gradient algorithms have been considered in this context. This paper investigates policy gradient algorithms within a continuous-time RMDP framework. Policy gradients and adversarial gradients are derived using pathwise and adjoint-based formulas for stochastic and ordinary differential equations. We propose double-loop optimisers to obtain linear convergence in the oracle-based setting and an $\tilde{\mathcal{O}}(\frac{1}{\epsilon^2})$ sample complexity in the sample-based setting in an analysis which also derives novel tools for the framework of undiscounted total cost MDPs. Additionally, we propose mean-field optimisers as distributional optimisers with an $\tilde{\mathcal{O}}(\frac{1}{K})$ oracle-based convergence rate and an $\tilde{\mathcal{O}}(\frac{N^2}{\epsilon})$ sample complexity under $N$-particle approximation. The effectiveness of continuous-time policy gradient algorithms is confirmed for both optimisers on continuous-time RMDPs with neural ordinary differential equation dynamics.

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

1 major / 2 minor

Summary. The paper develops policy gradient algorithms for continuous-time robust Markov decision processes (RMDPs). It derives policy and adversarial gradients via pathwise and adjoint-based formulas for SDEs and ODEs. Double-loop optimizers are proposed to achieve linear convergence in the oracle-based setting and Õ(1/ε²) sample complexity in the sample-based setting, supported by novel tools for undiscounted total-cost MDPs; mean-field optimizers are also introduced with Õ(1/K) oracle convergence and Õ(N²/ε) sample complexity under N-particle approximation. The methods are validated empirically on continuous-time RMDPs with neural ODE dynamics.

Significance. If the gradient derivations and convergence rates hold, the work extends robust RL to continuous time with explicit rates and novel analysis tools for undiscounted total-cost settings, which could enable more reliable policy optimization in control applications. The provision of both oracle and sample-based guarantees plus mean-field variants strengthens the contribution if the undiscounted analysis is complete.

major comments (1)
  1. [Analysis of undiscounted total-cost MDPs (referenced in abstract)] The central convergence claims (linear oracle convergence and Õ(1/ε²) sample complexity) rest on novel tools for undiscounted total-cost MDPs to justify derivative-expectation interchange and contraction arguments. The provided analysis must explicitly state global boundedness, uniform integrability, or absorbing-state conditions; local Lipschitz assumptions alone are insufficient to guarantee the rates outside a neighborhood of the optimum, as noted in the stress-test concern.
minor comments (2)
  1. Clarify the precise form of the pathwise and adjoint gradient formulas (e.g., any regularity conditions on the SDE/ODE coefficients) to allow direct verification of the derivations.
  2. The experimental section should report the specific neural ODE architectures, discretization schemes, and robustness parameters used in the validation to improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our work extending policy gradients to continuous-time robust MDPs. We address the major comment below and will revise the manuscript accordingly to strengthen the theoretical foundations.

read point-by-point responses

  1. Referee: [Analysis of undiscounted total-cost MDPs (referenced in abstract)] The central convergence claims (linear oracle convergence and Õ(1/ε²) sample complexity) rest on novel tools for undiscounted total-cost MDPs to justify derivative-expectation interchange and contraction arguments. The provided analysis must explicitly state global boundedness, uniform integrability, or absorbing-state conditions; local Lipschitz assumptions alone are insufficient to guarantee the rates outside a neighborhood of the optimum, as noted in the stress-test concern.

    Authors: We agree that the convergence analysis for the undiscounted total-cost setting requires explicit global conditions to rigorously justify the interchange of derivative and expectation as well as the contraction mapping arguments. While the manuscript introduces novel tools under local Lipschitz continuity of the dynamics and running costs (standard for SDE/ODE control problems), these alone do not suffice for global rates. In the revised version we will augment the standing assumptions (currently in Section 3) with either (i) global boundedness of the value functions together with uniform integrability of the cost processes, or (ii) an absorbing-state condition ensuring almost-sure termination in finite time. The corresponding modifications will be inserted into the statements of Theorems 4.1 and 4.3 and their proofs, thereby extending the linear oracle convergence and Õ(1/ε²) sample-complexity guarantees from a neighborhood of the optimum to the full parameter space. We believe this clarification directly resolves the concern while preserving the novelty of the undiscounted analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rely on standard SDE/ODE techniques and external MDP analysis

full rationale

The paper's central claims derive policy/adversarial gradients via pathwise and adjoint formulas for SDEs/ODEs and obtain convergence rates for double-loop and mean-field optimizers in continuous-time RMDPs. These build on established differential equation methods and introduce novel tools for undiscounted total-cost MDPs without any quoted step reducing by construction to a fitted parameter, self-citation chain, or renamed input. No self-definitional loops, fitted-input predictions, or load-bearing self-citations appear in the provided abstract or description; the framework is self-contained against external stochastic control benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no specific information on free parameters, axioms, or new entities; full text required for assessment.

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    2024