Optimization objective

The goal in a Markov decision process is to find a good "policy" for the decision maker: a function ${\displaystyle \pi }$ that specifies the action ${\displaystyle \pi (s)}$ that the decision maker will choose when in state ${\displaystyle s}$. Once a Markov decision process is combined with a policy in this way, this fixes the action for each state and the resulting combination behaves like a Markov chain (since the action chosen in state ${\displaystyle s}$ is completely determined by ${\displaystyle \pi (s)}$ and ${\displaystyle \Pr(s_{t+1}=s'\mid s_{t}=s,a_{t}=a)}$ reduces to ${\displaystyle \Pr(s_{t+1}=s'\mid s_{t}=s)}$, a Markov transition matrix).

The objective is to choose a policy ${\displaystyle \pi }$ that will maximize some cumulative function of the random rewards, typically the expected discounted sum over a potentially infinite horizon:

${\displaystyle E\left[\sum _{t=0}^{\infty }{\gamma ^{t}R_{a_{t}}(s_{t},s_{t+1})}\right]}$ (where we choose ${\displaystyle a_{t}=\pi (s_{t})}$, i.e. actions given by the policy). And the expectation is taken over ${\displaystyle s_{t+1}\sim P_{a_{t}}(s_{t},s_{t+1})}$

where ${\displaystyle \ \gamma \ }$ is the discount factor satisfying ${\displaystyle 0\leq \ \gamma \ \leq \ 1}$, which is usually close to 1 (for example, ${\displaystyle \gamma =1/(1+r)}$ for some discount rate r). A lower discount factor motivates the decision maker to favor taking actions early, rather than postpone them indefinitely.

A policy that maximizes the function above is called an optimal policy and is usually denoted ${\displaystyle \pi ^{*}}$. A particular MDP may have multiple distinct optimal policies. Because of the Markov property, it can be shown that the optimal policy is a function of the current state, as assumed above.

Simulator models

In many cases, it is difficult to represent the transition probability distributions, ${\displaystyle P_{a}(s,s')}$, explicitly. In such cases, a simulator can be used to model the MDP implicitly by providing samples from the transition distributions. One common form of implicit MDP model is an episodic environment simulator that can be started from an initial state and yields a subsequent state and reward every time it receives an action input. In this manner, trajectories of states, actions, and rewards, often called episodes may be produced.

Another form of simulator is a generative model, a single step simulator that can generate samples of the next state and reward given any state and action.^[4] (Note that this is a different meaning from the term generative model in the context of statistical classification.) In algorithms that are expressed using pseudocode, ${\displaystyle G}$ is often used to represent a generative model. For example the expression ${\displaystyle s',r\gets G(s,a)}$ might denote the action of sampling from the generative model where ${\displaystyle s}$ and ${\displaystyle a}$ are the current state and action, and ${\displaystyle s'}$ and ${\displaystyle r}$ are the new state and reward. Compared to an episodic simulator, a generative model has the advantage that it can yield data from any state, not only those encountered in a trajectory.

These model classes form a hierarchy of information content: an explicit model trivially yields a generative model through sampling from the distributions, and repeated application of a generative model yields an episodic simulator. In the opposite direction, it is only possible to learn approximate models through regression. The type of model available for a particular MDP plays a significant role in determining which solution algorithms are appropriate. For example, the dynamic programming algorithms described in the next section require an explicit model, and Monte Carlo tree search requires a generative model (or an episodic simulator that can be copied at any state), whereas most reinforcement learning algorithms require only an episodic simulator.

Notable variants

Partial observability

The solution above assumes that the state ${\displaystyle s}$ is known when action is to be taken; otherwise ${\displaystyle \pi (s)}$ cannot be calculated. When this assumption is not true, the problem is called a partially observable Markov decision process or POMDP.

Reinforcement learning

Reinforcement learning uses MDPs where the probabilities or rewards are unknown.^[13]

For this purpose it is useful to define a further function, which corresponds to taking the action ${\displaystyle a}$ and then continuing optimally (or according to whatever policy one currently has):

${\displaystyle \ Q(s,a)=\sum _{s'}P_{a}(s,s')(R_{a}(s,s')+\gamma V(s')).\ }$

While this function is also unknown, experience during learning is based on ${\displaystyle (s,a)}$ pairs (together with the outcome ${\displaystyle s'}$; that is, "I was in state ${\displaystyle s}$ and I tried doing ${\displaystyle a}$ and ${\displaystyle s'}$ happened"). Thus, one has an array ${\displaystyle Q}$ and uses experience to update it directly. This is known as Q-learning.

Reinforcement learning can solve Markov-Decision processes without explicit specification of the transition probabilities; the values of the transition probabilities are needed in value and policy iteration. In reinforcement learning, instead of explicit specification of the transition probabilities, the transition probabilities are accessed through a simulator that is typically restarted many times from a uniformly random initial state. Reinforcement learning can also be combined with function approximation to address problems with a very large number of states.

Learning automata

Another application of MDP process in machine learning theory is called learning automata. This is also one type of reinforcement learning if the environment is stochastic. The first detail learning automata paper is surveyed by Narendra and Thathachar (1974), which were originally described explicitly as finite state automata.^[14] Similar to reinforcement learning, a learning automata algorithm also has the advantage of solving the problem when probability or rewards are unknown. The difference between learning automata and Q-learning is that the former technique omits the memory of Q-values, but updates the action probability directly to find the learning result. Learning automata is a learning scheme with a rigorous proof of convergence.^[15]

In learning automata theory, a stochastic automaton consists of:

• a set x of possible inputs,
• a set Φ = { Φ₁, ..., Φ_s } of possible internal states,
• a set α = { α₁, ..., α_r } of possible outputs, or actions, with r ≤ s,
• an initial state probability vector p(0) = ≪ p₁(0), ..., p_s(0) ≫,
• a computable function A which after each time step t generates p(t + 1) from p(t), the current input, and the current state, and
• a function G: Φ → α which generates the output at each time step.

The states of such an automaton correspond to the states of a "discrete-state discrete-parameter Markov process".^[16] At each time step t = 0,1,2,3,..., the automaton reads an input from its environment, updates P(t) to P(t + 1) by A, randomly chooses a successor state according to the probabilities P(t + 1) and outputs the corresponding action. The automaton's environment, in turn, reads the action and sends the next input to the automaton.^[15]

Category theoretic interpretation

Other than the rewards, a Markov decision process ${\displaystyle (S,A,P)}$ can be understood in terms of Category theory. Namely, let ${\displaystyle {\mathcal {A}}}$ denote the free monoid with generating set A. Let Dist denote the Kleisli category of the Giry monad. Then a functor ${\displaystyle {\mathcal {A}}\to \mathbf {Dist} }$ encodes both the set S of states and the probability function P.

In this way, Markov decision processes could be generalized from monoids (categories with one object) to arbitrary categories. One can call the result ${\displaystyle ({\mathcal {C}},F:{\mathcal {C}}\to \mathbf {Dist} )}$ a context-dependent Markov decision process, because moving from one object to another in ${\displaystyle {\mathcal {C}}}$ changes the set of available actions and the set of possible states.

Definition

In order to discuss the continuous-time Markov decision process, we introduce two sets of notations:

If the state space and action space are finite,

• ${\displaystyle {\mathcal {S}}}$: State space;
• ${\displaystyle {\mathcal {A}}}$: Action space;
• ${\displaystyle q(i\mid j,a)}$: ${\displaystyle {\mathcal {S}}\times {\mathcal {A}}\rightarrow \triangle {\mathcal {S}}}$, transition rate function;
• ${\displaystyle R(i,a)}$: ${\displaystyle {\mathcal {S}}\times {\mathcal {A}}\rightarrow \mathbb {R} }$, a reward function.

If the state space and action space are continuous,

• ${\displaystyle {\mathcal {X}}}$: state space;
• ${\displaystyle {\mathcal {U}}}$: space of possible control;
• ${\displaystyle f(x,u)}$: ${\displaystyle {\mathcal {X}}\times {\mathcal {U}}\rightarrow \triangle {\mathcal {X}}}$, a transition rate function;
• ${\displaystyle r(x,u)}$: ${\displaystyle {\mathcal {X}}\times {\mathcal {U}}\rightarrow \mathbb {R} }$, a reward rate function such that ${\displaystyle r(x(t),u(t))\,dt=dR(x(t),u(t))}$, where ${\displaystyle R(x,u)}$ is the reward function we discussed in previous case.

Problem

Like the discrete-time Markov decision processes, in continuous-time Markov decision processes we want to find the optimal policy or control which could give us the optimal expected integrated reward:

${\displaystyle \max \operatorname {E} _{u}\left[\left.\int _{0}^{\infty }\gamma ^{t}r(x(t),u(t))\,dt\;\right|x_{0}\right]}$

where ${\displaystyle 0\leq \gamma <1.}$

Linear programming formulation

If the state space and action space are finite, we could use linear programming to find the optimal policy, which was one of the earliest approaches applied. Here we only consider the ergodic model, which means our continuous-time MDP becomes an ergodic continuous-time Markov chain under a stationary policy. Under this assumption, although the decision maker can make a decision at any time at the current state, they could not benefit more by taking more than one action. It is better for them to take an action only at the time when system is transitioning from the current state to another state. Under some conditions,(for detail check Corollary 3.14 of Continuous-Time Markov Decision Processes), if our optimal value function ${\displaystyle V^{*}}$ is independent of state ${\displaystyle i}$, we will have the following inequality:

${\displaystyle g\geq R(i,a)+\sum _{j\in S}q(j\mid i,a)h(j)\quad \forall i\in S{\text{ and }}a\in A(i)}$

If there exists a function ${\displaystyle h}$, then ${\displaystyle {\bar {V}}^{*}}$ will be the smallest ${\displaystyle g}$ satisfying the above equation. In order to find ${\displaystyle {\bar {V}}^{*}}$, we could use the following linear programming model:

• Primal linear program(P-LP)

${\displaystyle {\begin{aligned}{\text{Minimize}}\quad &g\\{\text{s.t}}\quad &g-\sum _{j\in S}q(j\mid i,a)h(j)\geq R(i,a)\,\,\forall i\in S,\,a\in A(i)\end{aligned}}}$

• Dual linear program(D-LP)

${\displaystyle {\begin{aligned}{\text{Maximize}}&\sum _{i\in S}\sum _{a\in A(i)}R(i,a)y(i,a)\\{\text{s.t.}}&\sum _{i\in S}\sum _{a\in A(i)}q(j\mid i,a)y(i,a)=0\quad \forall j\in S,\\&\sum _{i\in S}\sum _{a\in A(i)}y(i,a)=1,\\&y(i,a)\geq 0\qquad \forall a\in A(i){\text{ and }}\forall i\in S\end{aligned}}}$

${\displaystyle y(i,a)}$ is a feasible solution to the D-LP if ${\displaystyle y(i,a)}$ is nonnative and satisfied the constraints in the D-LP problem. A feasible solution ${\displaystyle y^{*}(i,a)}$ to the D-LP is said to be an optimal solution if

${\displaystyle {\begin{aligned}\sum _{i\in S}\sum _{a\in A(i)}R(i,a)y^{*}(i,a)\geq \sum _{i\in S}\sum _{a\in A(i)}R(i,a)y(i,a)\end{aligned}}}$

for all feasible solution ${\displaystyle y(i,a)}$ to the D-LP. Once we have found the optimal solution ${\displaystyle y^{*}(i,a)}$, we can use it to establish the optimal policies.

Hamilton–Jacobi–Bellman equation

In continuous-time MDP, if the state space and action space are continuous, the optimal criterion could be found by solving Hamilton–Jacobi–Bellman (HJB) partial differential equation. In order to discuss the HJB equation, we need to reformulate our problem

${\displaystyle {\begin{aligned}V(x(0),0)={}&\max _{u}\int _{0}^{T}r(x(t),u(t))\,dt+D[x(T)]\\{\text{s.t.}}\quad &{\frac {dx(t)}{dt}}=f[t,x(t),u(t)]\end{aligned}}}$

${\displaystyle D(\cdot )}$ is the terminal reward function, ${\displaystyle x(t)}$ is the system state vector, ${\displaystyle u(t)}$ is the system control vector we try to find. ${\displaystyle f(\cdot )}$ shows how the state vector changes over time. The Hamilton–Jacobi–Bellman equation is as follows:

${\displaystyle 0=\max _{u}(r(t,x,u)+{\frac {\partial V(t,x)}{\partial x}}f(t,x,u))}$

We could solve the equation to find the optimal control ${\displaystyle u(t)}$, which could give us the optimal value function ${\displaystyle V^{*}}$

Application

Continuous-time Markov decision processes have applications in queueing systems, epidemic processes, and population processes.