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Analysis of approximate linear programming solution to Markov decision problem with log barrier function

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Markov decision programmingreinforcement learninglinear programmingdynamic programming
Abstract:

There are two primary approaches to solving Markov decision problems (MDPs): dynamic programming based on the Bellman equation and linear programming (LP). Dynamic programming methods are the most widely used and form the foundation of both classical and modern reinforcement learning (RL). By contrast, LP-based methods have been less commonly employed, although they have recently gained attention in contexts such as offline RL. The relative underuse of the LP-based methods stems from the fact that it leads to an inequality-constrained optimization problem, which is generally more challenging to solve effectively compared with Bellman-equation-based methods. The purpose of this paper is to establish a theoretical foundation for solving LP-based MDPs in a more effective and practical manner. Our key idea is to leverage the log-barrier function, widely used in inequality-constrained optimization, to transform the LP formulation of the MDP into an unconstrained optimization problem. This reformulation enables approximate solutions to be obtained easily via gradient descent. While the method may appear naive, to the best of our knowledge, a thorough theoretical interpretation of this approach has not yet been developed. This paper aims to bridge this gap.

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Overview

Overall Novelty Assessment

The paper proposes using log-barrier functions to transform the LP formulation of MDPs into an unconstrained optimization problem solvable via gradient descent, with rigorous error bounds and convergence guarantees. Within the taxonomy, it resides in the 'Log Barrier Methods for LP-Based MDP Solutions' leaf under 'Linear Programming Formulations for MDPs'. This leaf contains only two papers total, indicating a sparse and relatively unexplored research direction. The sibling paper appears to be a related work on approximate linear programming, suggesting this is an emerging niche rather than a crowded subfield.

The taxonomy reveals that neighboring leaves focus on polynomial-time LP algorithms and geometric perspectives on MDP optimization, both emphasizing theoretical foundations for LP-based methods. The broader 'Linear Programming Formulations for MDPs' branch contrasts with the more populated 'Reinforcement Learning with Regularization' branch, which contains multiple leaves addressing barrier methods in online learning contexts (infinite-horizon average-reward, adversarial settings, high-probability bounds). The paper's approach bridges classical LP optimization with modern gradient-based techniques, positioning it at the intersection of exact methods and learning paradigms.

Among seventeen candidates examined across three contributions, none were found to clearly refute the proposed approach. The core log-barrier LP formulation examined nine candidates with zero refutations, the analytical properties contribution examined six with none refutable, and the deep RL variant examined two with none refutable. This suggests that within the limited search scope—primarily top-K semantic matches and citation expansion—the specific combination of log-barrier transformation with rigorous theoretical analysis for LP-based MDPs appears relatively novel, though the small candidate pool limits definitive conclusions about field-wide novelty.

Based on the limited literature search of seventeen papers, the work appears to occupy a sparsely populated research direction with minimal direct prior overlap among examined candidates. However, the analysis does not cover exhaustive searches across all MDP or interior-point optimization literature, and the small taxonomy leaf size may reflect either genuine novelty or incomplete field coverage in the search methodology.

Taxonomy

Core-task Taxonomy Papers
16
3
Claimed Contributions
17
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Solving Markov decision problems using linear programming with log barrier functions. The field of Markov decision processes encompasses diverse solution methodologies that can be organized into several main branches. Linear Programming Formulations for MDPs explore direct optimization approaches, including interior-point and barrier methods that reformulate sequential decision problems as convex programs. Constrained Markov Decision Processes address settings where agents must satisfy safety or resource constraints alongside reward maximization, often requiring specialized algorithms that balance feasibility and optimality. Reinforcement Learning with Regularization investigates how entropy or other penalty terms shape exploration and convergence, bridging classical dynamic programming with modern learning paradigms. Finally, Integrated Prediction and Optimization examines end-to-end frameworks that combine forecasting with decision-making, particularly relevant when model parameters must be learned from data. These branches reflect complementary perspectives: some emphasize exact or approximate solution techniques for known models, while others focus on learning under uncertainty or incorporating side constraints. Within the Linear Programming Formulations branch, a small cluster of works investigates barrier-based methods for solving MDPs efficiently. Analysis of approximate linear[0] contributes to this line by examining how log barrier functions can be integrated into LP-based MDP solvers, building on earlier heuristic approaches such as An interior point heuristic[13] that applied interior-point techniques to large-scale problems. Compared to Model-free reinforcement learning in[1] or Learning infinite-horizon average-reward Markov[4], which prioritize sample-based learning without explicit model knowledge, the barrier-method cluster assumes access to transition dynamics and leverages convex optimization machinery. Meanwhile, works like A learning-based approach to[3] blend optimization with learning, suggesting potential synergies. The main trade-offs revolve around computational scalability, the need for accurate models, and the handling of constraints—questions that remain active as researchers seek to unify classical LP techniques with modern data-driven paradigms.

Claimed Contributions

Log-barrier LP formulation for MDPs with rigorous error bounds

The authors introduce a novel log-barrier reformulation of the linear programming approach to Markov decision problems, converting the inequality-constrained LP into an unconstrained optimization problem. They derive rigorous upper and lower bounds on the approximation error that scale linearly with the barrier parameter η.

9 retrieved papers
Analytical properties and convergence guarantees for the barrier-based objective

The authors prove structural properties of the log-barrier objective function, including convexity, local strong convexity, local Lipschitz continuity, and properties of the feasible domain and sublevel sets. They also establish exponential convergence of deterministic gradient descent in the tabular setting.

6 retrieved papers
Deep RL variant with log-barrier loss function

The authors propose a new deep reinforcement learning algorithm that replaces the conventional mean-squared-error Bellman loss with a log-barrier-based loss function. This yields modified DQN and DDPG algorithms, which are empirically evaluated on OpenAI Gym benchmark tasks.

2 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Log-barrier LP formulation for MDPs with rigorous error bounds

The authors introduce a novel log-barrier reformulation of the linear programming approach to Markov decision problems, converting the inequality-constrained LP into an unconstrained optimization problem. They derive rigorous upper and lower bounds on the approximation error that scale linearly with the barrier parameter η.

Contribution

Analytical properties and convergence guarantees for the barrier-based objective

The authors prove structural properties of the log-barrier objective function, including convexity, local strong convexity, local Lipschitz continuity, and properties of the feasible domain and sublevel sets. They also establish exponential convergence of deterministic gradient descent in the tabular setting.

Contribution

Deep RL variant with log-barrier loss function

The authors propose a new deep reinforcement learning algorithm that replaces the conventional mean-squared-error Bellman loss with a log-barrier-based loss function. This yields modified DQN and DDPG algorithms, which are empirically evaluated on OpenAI Gym benchmark tasks.

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