Markov Decision Process

A Markov Decision Process (MDP) consists of five core elements: state space, action space, state transition probability, reward function, and discount factor.

Reinforcement learning learns optimal policies through continuous interaction between an agent and its environment; its mathematical foundation is the Markov Decision Process (MDP), which consists of a state space, action space, transition function, reward function, and discount factor.

A Markov Decision Process (MDP) is mathematically defined by a five-tuple (S, A, P, R, γ).

The Markov Decision Process (MDP) provides a complete mathematical framework for reinforcement learning, with its Markov property holding that future states depend only on the current state and action.

The key assumption of MDP is that the future depends only on the current state and not on the historical path.

Modern deep reinforcement learning methods (such as DQN, PPO, and SAC) can be viewed as function approximation generalizations of classical methods such as value iteration and policy iteration in continuous high-dimensional spaces.

The core of the Markov decision process is the Markov property—the next state depends only on the current state and action, independent of history.

The Bellman equation exploits the Markov property to decompose the expectation of long-term return into "immediate reward + discounted value of the next state," thereby transforming a sequential decision problem into an equation that can be solved iteratively.

This equivalence is currently established in the multi-armed bandit rather than the full MDP setting, so the conclusions mainly apply to static or approximately static decision problems.

Once the MDP transitions and costs are known, the optimal policy π*(s) can be computed offline using dynamic programming, policy iteration, or value iteration.

The Bellman Language Model (BLM) models repeated decisions as a Markov decision process (MDP), solves it once and reuses the solution, to improve the efficiency of LLM agents.

When the policies of other agents are fixed, the environment degenerates into a standard MDP for agent i, and is thus "individually stationary."

The paper proves in a Markov decision process (MDP) under an imperfectly competitive environment that the auction mechanism can converge to the globally optimal policy.

Interaction-induced non-stationarity violates the stationarity assumption of MDPs, invalidating convergence guarantees derived from that assumption.

The research object is a finite Markov decision process (MDP) whose trajectories are generated by a fixed behavior policy, with the state-action chain being irreducible and allowing for periodicity.

In federated reinforcement learning, multiple agents interact with the same Markov decision process, exchange information through a central server, and jointly learn the optimal state-action value function.

强化学习中的状态—动作—奖励框架源自马尔可夫决策过程（MDP）理论

During the mid-training phase, ZGCM-1 reformulates the agent-tool interaction process as a Markov Decision Process (MDP).

Quantum algorithms such as Grover search and quantum amplitude estimation can be used to accelerate the value iteration/policy iteration solving process for MDPs.

An MDP is defined by the 4-tuple (S, A, P, R): the state set S, the action set A, the state transition probability P, and the reward function R.