# Markov Decision Process

A sequential decision problem for a fully observable, stochastic environment with a Markovian transition model and additive rewards is called a Markov decision process. It consists of a set of states, with initial state $$s_0$$, a set $$ACTIONS(s)$$ of actions in each state, a transition model $$P(s'|s, a)$$, and a reward function $$R(s)$$.

A policy, denoted $$\pi$$, specifies what the agent should do in any state $$s$$. This action is denoted by $$\pi(s)$$. The optimal policy $$\pi^*$$ yields the highest expected utility.

The careful balancing of risk and reward is a characteristic of MDPs that does not arise in deterministic search problems.

## Optimality in Markov Decision Processes

### Finite Horizon

\begin{equation} E\left( \sum_{t=0}^{h} r_t \right) \end{equation}

### Infinite Horizon

\begin{equation} E\left( \sum_{t=0}^{\infty} \gamma^t r_t \right) \end{equation}

### Average-reward

\begin{equation} \lim_{h \rightarrow \infty} E\left( \sum_{t=0}^{h} \frac{1}{h} r_t \right) \end{equation}

## Learning Performance (Kaelbling, Littman, and Moore, n.d.)

1. Asymptotic convergence:

\begin{equation} \pi_n \rightarrow \pi^* \text { as } n \rightarrow \infty \end{equation}

1. PAC:

\begin{equation} P(N_{errors} > F(\cdot, \epsilon, \delta)) \le \delta \end{equation}

Does not give any guarantee about the policy while it is learning

1. Regret (e.g. bound $$B$$ on total regret):

\begin{equation} \mathrm{max} \sum_{t=0}^{T} r_{tj} - r_t < B \end{equation}

No notion of “many small mistakes”, or “few major mistakes”.

1. Uniform-PAC

unifies notion of PAC and regret (Dann, Lattimore, and Brunskill, n.d.)

## Bibliography

Dann, Christoph, Tor Lattimore, and Emma Brunskill. n.d. “Unifying PAC and Regret: Uniform PAC Bounds for Episodic Reinforcement Learning.” In Advances in Neural Information Processing Systems, 5713–23.

Kaelbling, Leslie Pack, Michael L Littman, and Andrew W Moore. n.d. “Reinforcement Learning: A Survey” 4:237–85.