07-09-2026, 09:53 PM
Mathematical methods of reinforcement learning
by Denis Belomestny
Summary
This comprehensive literature review organizes the foundational mathematical frameworks that support modern reinforcement learning (RL) algorithms. Moving beyond simple heuristic designs, the authors trace how probability theory, optimization, and operator theory provide rigorous convergence guarantees and performance bounds across various learning paradigms.
By grounding sequential decision-making in the foundational theory of Markov decision processes (MDPs), the survey unifies classical dynamic programming with modern stochastic approximation. It thoroughly explores diverse structural environments—ranging from tabular representations to complex function approximation using deep neural networks—and analyzes finite-horizon, discounted, and average-reward settings.
A major contribution of this work is its mathematical harmonization of the core exploration-exploitation dilemma. The review connects classical statistical estimation with online learning, highlighting how a regret-based viewpoint drives advanced exploration strategies like optimistic exploration. Furthermore, the survey evaluates both model-based approaches, which explicitly map out transition dynamics, and model-free techniques, which optimize policies directly through environment interactions.
Ultimately, this work provides a rigorous theoretical backbone for the AI community, demonstrating how analytical tools like concentration inequalities and entropy-regularized policy gradients translate abstract mathematical concepts into stable, scalable, and highly efficient real-world artificial intelligence applications.
ARTICLE [PDF]
by Denis Belomestny
Summary
This comprehensive literature review organizes the foundational mathematical frameworks that support modern reinforcement learning (RL) algorithms. Moving beyond simple heuristic designs, the authors trace how probability theory, optimization, and operator theory provide rigorous convergence guarantees and performance bounds across various learning paradigms.
By grounding sequential decision-making in the foundational theory of Markov decision processes (MDPs), the survey unifies classical dynamic programming with modern stochastic approximation. It thoroughly explores diverse structural environments—ranging from tabular representations to complex function approximation using deep neural networks—and analyzes finite-horizon, discounted, and average-reward settings.
A major contribution of this work is its mathematical harmonization of the core exploration-exploitation dilemma. The review connects classical statistical estimation with online learning, highlighting how a regret-based viewpoint drives advanced exploration strategies like optimistic exploration. Furthermore, the survey evaluates both model-based approaches, which explicitly map out transition dynamics, and model-free techniques, which optimize policies directly through environment interactions.
Ultimately, this work provides a rigorous theoretical backbone for the AI community, demonstrating how analytical tools like concentration inequalities and entropy-regularized policy gradients translate abstract mathematical concepts into stable, scalable, and highly efficient real-world artificial intelligence applications.
ARTICLE [PDF]
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