https://introml.mit.edu/

  Lecture 10: Markov Decision Processes   

 

Shen Shen

Apr 13, 2026

3pm, Room 10-250

Slides and Lecture Recording

Intro to Machine Learning

Toddler demo, Russ Tedrake thesis, 2004

uses vanilla policy gradient (actor-critic)

Reinforcement Learning with Human Feedback

Outline

  • Markov Decision Processes Definition

  • Policy Evaluation

    • State value functions: \(\mathrm{V}^{\pi}\)

    • Bellman recursions and Bellman equations

  • Policy Optimization

    • Optimal policies \(\pi^*\)

    • Optimal action value functions: \(\mathrm{Q}^*\)

    • Value iteration

  • Markov Decision Processes Definition

  • Policy Evaluation

    • State value functions: \(\mathrm{V}^{\pi}\)

    • Bellman recursions and Bellman equations

  • Policy Optimization

    • Optimal policies \(\pi^*\)

    • Optimal action value functions: \(\mathrm{Q}^*\)

    • Value iteration

Markov Decision Processes

  • Research area initiated in the 50s by Bellman, known under various names:

    • Stochastic optimal control (Control theory)

    • Stochastic shortest path (Operations research)

    • Sequential decision making under uncertainty (Economics)

    • Reinforcement learning (Artificial intelligence, Machine learning)

  • A rich variety of elegant theory, mathematics, algorithms, and applications, but also considerable variation in notation.

  • We will use the most RL-flavored notations.

  • (state, action) results in a transition \(\mathrm{T}\) into a next state:

    • Normally, we get to the “intended” state;

      • E.g., in state (7), action “↑” gets to state (4)

    • If an action would take Mario out of the grid world, stay put;

      • E.g., in state (9), “→” gets back to state (9)

    • In state (6), action “↑” leads to two possibilities:

      • 20% chance to (2)

      • 80% chance to (3).

80\%
20\%

Running example: Mario in a grid-world

  • 9 possible states \(s\)
  • 4 possible actions \(a\): {Up ↑, Down ↓, Left ←, Right →}
1
1
1
1
-10
-10
-10
-10

reward of (3, \(\downarrow\))

reward of \((3,\uparrow\))

reward of \((6, \downarrow\))

reward of \((6,\rightarrow\))

  • (state, action) pairs give rewards:
    • in state 3, any action gives reward 1
    • in state 6, any action gives reward -10
    • any other (state, action) pair gives reward 0
  • discount factor: a scalar of 0.9 that reduces the 'worth' of future rewards depending on when Mario receives them.
    • So, e.g., for (3, \(\leftarrow\)) pair, Mario gets
      • at the start of the game, a reward of 1
      • at the 2nd time step, a discounted reward of 0.9
      • at the 3rd time step, a further discounted reward of \((0.9)^2\) ... and so on

Mario in a grid-world, cont'd

  • \(\mathcal{S}\) : state space, contains all possible states \(s\).
  • \(\mathcal{A}\) : action space, contains all possible actions \(a\).
  • \(\mathrm{T}\left(s, a, s^{\prime}\right)\) : the probability of transition from state \(s\) to \(s^{\prime}\) when action \(a\) is taken.

Markov Decision Processes - Definition and terminologies

80\%
20\%

\(\mathrm{T}\left(7, \uparrow, 4\right) = 1\)

\(\mathrm{T}\left(9, \rightarrow, 9\right) = 1\)

\(\mathrm{T}\left(6, \uparrow, 3\right) = 0.8\)

\(\mathrm{T}\left(6, \uparrow, 2\right) = 0.2\)

In 6.390,

  • \(\mathcal{S}\) and \(\mathcal{A}\) are small discrete sets, unless otherwise specified.
  • \(s^{\prime}\) and \(a^{\prime}\) are short-hand for the next-timestep state and action.
  • \(\mathcal{S}\) : state space, contains all possible states \(s\).
  • \(\mathcal{A}\) : action space, contains all possible actions \(a\).
  • \(\mathrm{T}\left(s, a, s^{\prime}\right)\) : the probability of transition from state \(s\) to \(s^{\prime}\) when action \(a\) is taken.
  • \(\mathrm{R}(s, a)\) : reward, takes in a (state, action) pair and returns a reward.

Markov Decision Processes - Definition and terminologies

reward of \((3,\uparrow\))

reward of \((6,\rightarrow\))

\(\mathrm{R}\left(3, \uparrow \right) = 1\)

\(\mathrm{R}\left(6, \rightarrow \right) = -10\)

In 6.390,

  • \(\mathcal{S}\) and \(\mathcal{A}\) are small discrete sets, unless otherwise specified.
  • \(s^{\prime}\) and \(a^{\prime}\) are short-hand for the next-timestep state and action.
  • \(\mathrm{R}(s, a)\) is deterministic and bounded.
  • \(\mathcal{S}\) : state space, contains all possible states \(s\).
  • \(\mathcal{A}\) : action space, contains all possible actions \(a\).
  • \(\mathrm{T}\left(s, a, s^{\prime}\right)\) : the probability of transition from state \(s\) to \(s^{\prime}\) when action \(a\) is taken.
  • \(\mathrm{R}(s, a)\) : reward, takes in a (state, action) pair and returns a reward.
  • \(\gamma \in [0,1]\): discount factor, a scalar.
  • \(\pi{(s)}\) : policy, takes in a state and returns an action.

The goal of an MDP is to find a good policy.

Markov Decision Processes - Definition and terminologies

In 6.390,

  • \(\mathcal{S}\) and \(\mathcal{A}\) are small discrete sets, unless otherwise specified.
  • \(s^{\prime}\) and \(a^{\prime}\) are short-hand for the next-timestep state and action.
  • \(\mathrm{R}(s, a)\) is deterministic and bounded.
  • \(\pi(s)\) is deterministic.
  • \(a_t = \pi(s_t)\)
  • \(r_t = \mathrm{R}(s_t,a_t)\)

Policy \(\pi(s)\)

Transition \(\mathrm{T}\left(s, a, s^{\prime}\right)\)

Reward \(\mathrm{R}(s, a)\)

time

a trajectory (also called an experience or rollout) of horizon \(h\)

 \(\quad \tau=\left(s_0, a_0, r_0, s_1, a_1, r_1, \ldots s_{h-1}, a_{h-1}, r_{h-1}\right)\) 

\underbrace{\hspace{4.2cm}}

initial state

all depends on \(\pi\)

\dots
s_2
s_3
s_4
s_5
a_2
a_3
a_4
r_2
r_3
r_4
s_0
s_1
a_0
a_1
r_0
r_1
s_{h-2}
a_{h-2}
r_{h-2}
s_{h-1}
a_{h-1}
r_{h-1}
  • \(\mathrm{T}\left(s, a, s^{\prime}\right)\)

Outline

  • Markov Decision Processes Definition

  • Policy Evaluation

    • State value functions: \(\mathrm{V}^{\pi}\)

    • Bellman recursions and Bellman equations

  • Policy Optimization

    • Optimal policies \(\pi^*\)

    • Optimal action value functions: \(\mathrm{Q}^*\)

    • Value iteration

Starting in state \(s\), how good is it to follow a given policy \(\pi\) for \(h\) time steps?

One idea:

But if we start at \(s_0=6\) and follow the "always-up" policy:

??
-10
\uparrow
6

👈

80\%
20\%

states and one special transition:

rewards:

trajectory:

\mathrm{R}(s_0, \pi(s_0))
\gamma \mathrm{R}(s_1, \pi(s_1))
\gamma^3\mathrm{R}(s_3, \pi(s_3))
\gamma^2 \mathrm{R}(s_2, \pi(s_2))
\dots
+
+
+
+
\gamma^{h-1}\mathrm{R}(s_{h-1}, \pi(s_{h-1}))
+

Reward \(\mathrm{R}(s, a)\)

r_2
r_3
r_4
r_0
r_1
r_{h-2}
r_{h-1}

Policy \(\pi(s)\)

Transition \(\mathrm{T}\left(s, a, s^{\prime}\right)\)

\dots
s_2
s_3
s_4
s_5
a_2
a_3
a_4
s_0
s_1
a_0
a_1
s_{h-2}
a_{h-2}
s_{h-1}
a_{h-1}

time

\mathbb{E}\Big[
\Big]

Value functions:

  • \(\mathrm{V}_h^\pi(s):\) expected sum of discounted rewards starting in state \(s\) and follow \(\pi\) for \(h\) steps
  • Value is long-term; reward is immediate (one-time)
  • Convention: \(\mathrm{V}_0^\pi(s)=0\) for all states
  • The value expectation in 6.390 is only w.r.t. the transition probabilities \(\mathrm{T}\left(s, a, s^{\prime}\right)\)
\mathrm{R}(s_0, \pi(s_0))
\gamma \mathrm{R}(s_1, \pi(s_1))
\gamma^3\mathrm{R}(s_3, \pi(s_3))
\gamma^2 \mathrm{R}(s_2, \pi(s_2))
\dots
+
+
+
+
\gamma^{h-1}\mathrm{R}(s_{h-1}, \pi(s_{h-1}))
+
=\mathbb{E}\left[\sum_{t=0}^{h-1} \gamma^t \mathrm{R}\left(s_t, \pi\left(s_t\right)\right) \mid s_0=s, \pi\right]
\mathrm{V}_h^\pi(s):=

(eq. 1️⃣)

\( h\) terms

evaluate \(\mathrm{V}_h^\pi(s)\) under the "always-up" policy

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
\underbrace{\hspace{4.6cm}}
\mathrm{V}_h^\uparrow(s)=\mathbb{E}\left[\sum_{t=0}^{h-1} \gamma^t \mathrm{R}\left(s_t, \uparrow \right) \mid s_0=s\right]
= \mathbb{E}\Big[\mathrm{R}(s_0, \uparrow)+\gamma \mathrm{R}(s_1, \uparrow) + \dots + \gamma^{h-1}\mathrm{R}(s_{h-1}, \uparrow)\Big]
\mathrm{V}^{\uparrow}_0(s) = 0
0
0
0
0
0
0
0

horizon \(h\) = 0: no step left

0
0
0
0
0
0
1
-10

horizon \(h\) = 1: receive the rewards

\mathrm{V}^{\uparrow}_1(s) = \mathrm{R}(s, \uparrow)
0
0
0
0
0
\mathrm{R}(1, \uparrow) + \gamma \mathrm{R}(1, \uparrow)
0
\mathrm{R}(2, \uparrow) + \gamma \mathrm{R}(2, \uparrow)
1.9
\mathrm{R}(3, \uparrow) + \gamma \mathrm{R}(3, \uparrow)
0
\mathrm{R}(4, \uparrow) + \gamma \mathrm{R}(1, \uparrow)
= 1 + 0.9 *(1) \Rightarrow 1.9
\mathrm{R}(5, \uparrow) + \gamma \mathrm{R}(2, \uparrow)

horizon \(h = 2\)

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
\mathrm{V}^{\uparrow}_2(s): \mathbb{E}\Big[\mathrm{R}(s_0, \uparrow)+\gamma \mathrm{R}(s_1, \uparrow)\Big]

\( 2\) terms

\underbrace{\hspace{2cm}}
-9
0
0
-9.28
\mathrm{R}(6, \uparrow) + \gamma [.2 \mathrm{R}(2, \uparrow) + .8 \mathrm{R}(3, \uparrow)]
\mathrm{R}(8, \uparrow) + \gamma \mathrm{R}(5, \uparrow)
\mathrm{R}(7, \uparrow) + \gamma \mathrm{R}(4, \uparrow)
\mathrm{R}(9, \uparrow) + \gamma \mathrm{R}(6, \uparrow)
= -10 + 0.9*(0.2*0+0.8*1)
\Rightarrow -9.28
= 0 + 0.9*(-10)\Rightarrow -9

action \(\uparrow\)

\mathrm{R}(6, \uparrow)
20\%
80\%

action \(\uparrow\)

\mathrm{R}(2, \uparrow)
\gamma

action \(\uparrow\)

\mathrm{R}(3, \uparrow)
\gamma

horizon \(h = 2\)

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
0
0
0
1.9
0

\( 2\) terms

\underbrace{\hspace{2cm}}
\mathrm{V}^{\uparrow}_2(s): \mathbb{E}\Big[\mathrm{R}(s_0, \uparrow)+\gamma \mathrm{R}(s_1, \uparrow)\Big]
V^{\uparrow}_3(6)

action \(\uparrow\)

action \(\uparrow\)

action \(\uparrow\)

action \(\uparrow\)

action \(\uparrow\)

20\%
80\%
80\%
\mathrm{R}(6, \uparrow)
\gamma \mathrm{R}(2, \uparrow)
20\%
+
+
\gamma \mathrm{R}(2, \uparrow)
\gamma^2 \mathrm{R}(2, \uparrow)
\gamma \mathrm{R}(3, \uparrow)
\mathrm{R}(6, \uparrow)
\gamma^2\mathrm{R}(3, \uparrow)
\gamma^2 \mathrm{R}(2, \uparrow)
\gamma \mathrm{R}(3, \uparrow)
20\%
+
80\%
\gamma^2\mathrm{R}(3, \uparrow)
+
=

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
\mathrm{R}(6, \uparrow)
\gamma \mathrm{R}(2, \uparrow)
20\%
+
+
\gamma^2 \mathrm{R}(2, \uparrow)
\gamma \mathrm{R}(3, \uparrow)
80\%
+
\gamma^2\mathrm{R}(3, \uparrow)
=
+
[
]
]
[
=
+
\left[\mathrm{R}(3, \uparrow) + \gamma \mathrm{R}(3, \uparrow)\right]
\gamma
+
80\%
\mathrm{R}(6, \uparrow)
\left[\mathrm{R}(2, \uparrow) + \gamma \mathrm{R}(2, \uparrow)\right]
\gamma
20\%
=
80\%
\gamma
20\%
+
V^{\uparrow}_2(2)
\gamma
+
\mathrm{R}(6, \uparrow)
\mathrm{V}^{\uparrow}_2(3)

horizon \(h = 3\)

\( 3\) terms

\underbrace{\hspace{3.5cm}}
\mathrm{V}^{\uparrow}_3(s): \mathbb{E}\Big[\mathrm{R}(s_0, \uparrow)+\gamma \mathrm{R}(s_1, \uparrow)+\gamma^2 \mathrm{R}(s_2, \uparrow)\Big]
\mathrm{V}_h^\pi(s)=\mathrm{R}\left(s, \pi(s)\right)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}_{h-1}^\pi\left(s^{\prime}\right)

the immediate reward for taking the policy-prescribed action \(\pi(s)\) in state \(s\).

horizon-\(h\) value in state \(s\): the expected sum of discounted rewards, starting in state \(s\) and following policy \(\pi\) for \(h\) steps.

\mathrm{V}^{\uparrow}_3(6)
=
80\%
\gamma
20\%
+
\mathrm{V}^{\uparrow}_2(2)
\gamma
+
\mathrm{R}(6, \uparrow)
\mathrm{V}^{\uparrow}_2(3)

\((h-1)\) horizon future value at a next state \(s^{\prime}\)

sum of future values weighted by the probability of reaching that next state \(s^{\prime}\) 

discounted by \(\gamma\) 

(eq. 2️⃣)

\mathrm{V}_h^\pi(s)=\mathrm{R}\left(s, \pi(s)\right)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}_{h-1}^\pi\left(s^{\prime}\right)

Bellman Recursion (eq. 2️⃣)

= 0 + 0.9\times[-10]
\mathrm{V}^{\uparrow}_{2}(9) = \mathrm{R}(9, \uparrow) + \gamma [\mathrm{V}^{\uparrow}_{1}(6)]
= -9

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
\mathrm{V}^{\uparrow}_1(s) = \mathrm{R}(s,\uparrow)
\mathrm{V}^{\uparrow}_2(s)
\mathrm{V}_h^\pi(s)=\mathrm{R}\left(s, \pi(s)\right)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}_{h-1}^\pi\left(s^{\prime}\right)
= -10 +.9[.2\times0+0.8\times1.9]
\mathrm{V}^{\uparrow}_{3}(6) = \mathrm{R}(6, \uparrow) + \gamma [.2 \times \mathrm{V}^{\uparrow}_{2}(2) + .8\times \mathrm{V}^{\uparrow}_{2}(3)]
= -8.632

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
\mathrm{V}^{\uparrow}_1(s) = \mathrm{R}(s,\uparrow)
\mathrm{V}^{\uparrow}_2(s)
\mathrm{V}^{\uparrow}_3(s)

Bellman Recursion (eq. 2️⃣)

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
\mathrm{V}^{\uparrow}_4(s)
\mathrm{V}^{\uparrow}_5(s)
\mathrm{V}^{\uparrow}_6(s)
= -10 +.9[.2\times0+0.8\times4.10]
\mathrm{V}^{\uparrow}_{6}(6) = \mathrm{R}(6, \uparrow) + \gamma [.2 \times \mathrm{V}^{\uparrow}_{5}(2) + .8\times \mathrm{V}^{\uparrow}_{5}(3)]
= -7.048
\mathrm{V}_h^\pi(s)=\mathrm{R}\left(s, \pi(s)\right)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}_{h-1}^\pi\left(s^{\prime}\right)
\dots

Bellman Recursion (eq. 2️⃣)

\mathrm{V}^{\uparrow}_{60}(s)
\mathrm{V}^{\uparrow}_{61}(s)

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
\mathrm{V}_h^\pi(s)=\mathrm{R}\left(s, \pi(s)\right)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}_{h-1}^\pi\left(s^{\prime}\right)
\mathrm{V}^{\uparrow}_{59}(s)
= -10 +.9[.2\times0+0.8\times9.98]
\mathrm{V}^{\uparrow}_{61}(6) = \mathrm{R}(6, \uparrow) + \gamma [.2 \times \mathrm{V}^{\uparrow}_{60}(2) + .8\times \mathrm{V}^{\uparrow}_{60}(3)]
= -2.8144
\dots

Bellman Recursion (eq. 2️⃣)

\mathrm{V}^{\uparrow}_{60}(s)
\mathrm{V}^{\uparrow}_{61}(s)

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%
\dots
\mathrm{V}^{\uparrow}_{\infty}(s)
  • As we extend the horizon, value differences shrink
  • because longer-term rewards are heavily discounted 
  • so, as \(h \to \infty,\) the value functions stop changing
  • convergence can be seen, e.g., via \(\mathrm{V}^{\uparrow}_{\infty}(3)=1+.9+.9^2+.9^3 + \dots =10\)

Value functions converge as \(h \to \infty\)

Typically, \(\gamma < 1\) to ensure \(\mathrm{V}_{\infty}\) is finite.

states and

one special transition:

rewards

  • \(\pi(s) = ``\uparrow",\  \forall s\)
  • \(\gamma = 0.9\)
80\%
20\%

Recursion (finite \(h\)) 2️⃣

As horizon \(h \to \infty,\) the Bellman recursion becomes the Bellman equation

Equation \((h\to \infty)\)  3️⃣

\mathrm{V}^\pi_{\infty}(s)= \mathrm{R}(s, \pi(s))+ \gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}^\pi_{\infty}\left(s^{\prime}\right)
\mathrm{V}_h^\pi(s)=\mathrm{R}\left(s, \pi(s)\right)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}_{h-1}^\pi\left(s^{\prime}\right)
\mathrm{V}^{\uparrow}_{\infty}(s)
\mathrm{V}^{\uparrow}_{\infty}(3) = \mathrm{R}(3, \uparrow) + \gamma [\mathrm{V}^{\uparrow}_{\infty}(3)]
= 1 + .9 \times 10 \Rightarrow 10
= -10 +.9[.2\times0+0.8\times10]\Rightarrow -2.8
\mathrm{V}^{\uparrow}_{\infty}(6) = \mathrm{R}(6, \uparrow) + \gamma [.2 \times \mathrm{V}^{\uparrow}_{\infty}(2) + .8\times \mathrm{V}^{\uparrow}_{\infty}(3)]

A system of \(|\mathcal{S}|\) self-consistent linear equations, one for each state

finite-horizon Bellman recursions 

infinite-horizon Bellman equations 

\mathrm{V}^\pi_{\infty}(s)= \mathrm{R}(s, \pi(s))+ \gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}^\pi_{\infty}\left(s^{\prime}\right)
\mathrm{V}_{h}^\pi(s)= \mathrm{R}(s, \pi(s))+ \gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}_{h-1}^\pi\left(s^{\prime}\right)
\pi(s)
\mathrm{V}^{\pi}_{h}(s)

Policy Evaluation

Quick summary

Use the definition and sum up expected rewards:

Or, leverage the recursive structure:

\mathrm{V}_h^\pi(s):=\mathbb{E}\left[\sum_{t=0}^{h-1} \gamma^t \mathrm{R}\left(s_t, \pi\left(s_t\right)\right) \mid s_0=s, \pi\right]

1️⃣

2️⃣

3️⃣

Outline

  • Markov Decision Processes Definition

  • Policy Evaluation

    • State value functions: \(\mathrm{V}^{\pi}\)

    • Bellman recursions and Bellman equations

  • Policy Optimization

    • Optimal policies \(\pi^*\)

    • Optimal action value functions: \(\mathrm{Q}^*\)

    • Value iteration

  • Intuitively, an optimal policy \(\pi^*\) is a policy that                                                                               yields the highest possible value \(\mathrm{V}_h^{*}({s})\) from every state
  • An MDP in 6.390 has a unique optimal value \(\mathrm{V}_h^{*}({s})\)
  • Optimal policy \(\pi^*\) might not be unique

Optimal policy \(\pi^*\)

e.g. in the "Luigi game", any policy is an optimal policy

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%

Rewards:

  • Formally: an optimal policy \(\pi^*\) is such that: \[\mathrm{V}_h^{\pi^*}(s) = \max_{\pi} \mathrm{V}_h^{\pi}(s) = \mathrm{V}_h^*(s), \forall s \in \mathcal{S}\]
  • How to search for an optimal policy \(\pi^*\)?
  • Even if we tediously enumerate over all \(\pi\), do policy evaluation, compare values to get \(\mathrm{V}^{*}_h(s)\)...it's not yet clear how to choose actions.

\(\mathrm{V}^*(s)\) is defined over states, not actions.
It tells us where we'd like to be — not what we should do to get there.

Optimal policy \(\pi^*\)

\(\mathrm{V}_h^*(s) = \max_{a} \big[\mathrm{R}(s, a) + \gamma \sum_{s'} \mathrm{T}(s, a, s') \mathrm{V}_{h-1}^*(s') \big]\)

Policy \(\pi(s)\)

Transition \(\mathrm{T}\left(s, a, s^{\prime}\right)\)

Reward \(\mathrm{R}(s, a)\)

time

\dots
s_2
s_3
s_4
s_5
a_2
a_3
a_4
r_2
r_3
r_4
s_0
s_1
a_0
a_1
r_0
r_1
s_{h-2}
a_{h-2}
r_{h-2}
s_{h-1}
a_{h-1}
r_{h-1}
\underbrace{\hspace{6.8cm}}
\underbrace{\hspace{7.8cm}}

if we've acted optimally for \(h\) steps: \(\mathrm{V}_h^*(s)\)

we must have acted optimally from the first step onward \(\mathrm{V}_{h-1}^*(s')\)

(new, eq. 4️⃣, for an optimal policy)

(recall, eq. 2️⃣, for any policy)

\mathrm{V}_h^\pi(s)=\mathrm{R}\left(s, \pi(s)\right)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, \pi(s), s^{\prime}\right) \mathrm{V}_{h-1}^\pi\left(s^{\prime}\right)

with the first step action

that led to the optimal future

Define the optimal state-action value functions \(\mathrm{Q}^*_h(s, a):\)

the expected sum of discounted rewards, obtained by

  • starting in state \(s\)
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*\) satisfies:

\[\mathrm{Q}^*_h (s, a)=\mathrm{R}(s, a)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, a, s^{\prime} \right) \max _{a^{\prime}} \mathrm{Q}^*_{h-1}\left(s^{\prime}, a^{\prime}\right)\]

(eq. 5️⃣)

\underbrace{\hspace{4.3cm}}

\(\mathrm{Q}^*_{h}(s, a)\)

(eq. 4️⃣)

\(\mathrm{V}_h^*(s) = \max_{a} \big[\mathrm{R}(s, a) + \gamma \sum_{s'} \mathrm{T}(s, a, s') \mathrm{V}_{h-1}^*(s') \big]\)

  • starting in state \(s\),
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*_h(s, a)\): the value for

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%

Rewards:

\mathrm{Q}^*_0(s, a)
\mathrm{Q}^*_1(s, a)
=\mathrm{R}(s, a)

Consider \(\mathrm{Q}^*_2(3, \downarrow)\)

  • receive \(\mathrm{R}(3,\downarrow)\)

\( = 1 + .9\times -10\)

  • next state \(s'\) = 6, act optimally for the remaining one timestep
    • receive \(\max _{a^{\prime}} \mathrm{Q}_{1}^*\left(6, a^{\prime}\right)\)

\( = -8\)

\(\mathrm{Q}_2^*(3, \downarrow) = \mathrm{R}(3,\downarrow)  + \gamma \max _{a^{\prime}} \mathrm{Q}_{1}^*\left(6, a^{\prime}\right)\)

\mathrm{Q}^*_2(s, a)
  • starting in state \(s\),
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*_h(s, a)\): the value for

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%

Rewards:

\mathrm{Q}^*_1(s, a)

Let's consider \(\mathrm{Q}_2^*(3, \leftarrow)\)

  • receive \(\mathrm{R}(3,\leftarrow)\)

\( = 1 + .9\times0 \)

  • next state \(s'\) = 2, act optimally for the remaining one timestep
    • receive \(\max _{a^{\prime}} \mathrm{Q}_{1}^*\left(2, a^{\prime}\right)\)

\( = 1\)

\(\mathrm{Q}_2^*(3, \leftarrow) = \mathrm{R}(3,\leftarrow)  + \gamma \max _{a^{\prime}} \mathrm{Q}_{1}^*\left(2, a^{\prime}\right)\)

\mathrm{Q}^*_2(s, a)
  • starting in state \(s\),
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*_h(s, a)\): the value for

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%

Rewards:

\mathrm{Q}^*_1(s, a)

Let's consider \(\mathrm{Q}^*_2(3, \uparrow)\)

  • receive \(\mathrm{R}(3,\uparrow)\)

\( = 1 + .9 \times 1\)

  • next state \(s'\) = 3, act optimally for the remaining one timestep
    • receive \(\max _{a^{\prime}} \mathrm{Q}^*_{1}\left(3, a^{\prime}\right)\)

\( = 1.9\)

\(\mathrm{Q}^*_2(3, \uparrow) = \mathrm{R}(3,\uparrow)  + \gamma \max _{a^{\prime}} \mathrm{Q}^*_{1}\left(3, a^{\prime}\right)\)

\mathrm{Q}^*_2(s, a)
  • starting in state \(s\),
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*_h(s, a)\): the value for

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%

Rewards:

\mathrm{Q}^*_1(s, a)
\mathrm{Q}^*_2(s, a)

Let's consider \(\mathrm{Q}^*_2(3, \rightarrow)\)

  • receive \(\mathrm{R}(3,\rightarrow)\)

\( = 1 + .9\times1\)

  • next state \(s'\) = 3, act optimally for the remaining one timestep
    • receive \(\max _{a^{\prime}} \mathrm{Q}^*_{1}\left(3, a^{\prime}\right)\)

\( = 1.9\)

\(\mathrm{Q}^*_2(3, \rightarrow) = \mathrm{R}(3,\rightarrow)  + \gamma \max _{a^{\prime}} \mathrm{Q}^*_{1}\left(3, a^{\prime}\right)\)

  • starting in state \(s\),
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*_h(s, a)\): the value for

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%

Rewards:

\mathrm{Q}^*_1(s, a)
  • act optimally at the next state \(s^{\prime}=6\)
    receive \(\max _{a^{\prime}} \mathrm{Q}_{1}^*\left(6, a^{\prime}\right)\)
  • receive \(\mathrm{R}(6,\rightarrow)\)

Let's consider \(\mathrm{Q}_2^*(6, \rightarrow)  \)

\(\mathrm{Q}_2^*(6, \rightarrow)=\mathrm{R}(6,\rightarrow)  + \gamma[\max _{a^{\prime}} \mathrm{Q}_{1}^*\left(6, a^{\prime}\right)] \)

\mathrm{Q}^*_2(s, a)

\( = -10 + .9 \times -10 \Rightarrow -19\)

  • starting in state \(s\),
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*_h(s, a)\): the value for

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%

Rewards:

\mathrm{Q}^*_1(s, a)
  • act optimally at the next state \(s^{\prime}\) 
  • 20% chance, \(s'\) = 2, act optimally, get \(\max _{a^{\prime}} \mathrm{Q}_{1}^*\left(2, a^{\prime}\right)\)
  • 80% chance, \(s'\) = 3, act optimally, get \(\max _{a^{\prime}} \mathrm{Q}_{1}^*\left(3, a^{\prime}\right)\)

\(= -10 + .9 [.2 \times 0+ .8 \times 1] \Rightarrow -9.28\)

  • receive \(\mathrm{R}(6,\uparrow)\)

Let's consider \(\mathrm{Q}_2^*(6, \uparrow)  \)

\(\mathrm{Q}_2^*(6, \uparrow)=\mathrm{R}(6,\uparrow)  + \gamma[.2 \max _{a^{\prime}} \mathrm{Q}_{1}^*\left(2, a^{\prime}\right)+ .8\max _{a^{\prime}} \mathrm{Q}_{1}^*\left(3, a^{\prime}\right)] \)

\mathrm{Q}^*_1(s, a)
\mathrm{Q}^*_2(s, a)
  • starting in state \(s\),
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*_h(s, a)\): the value for

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%

Rewards:

  • starting in state \(s\),
  • take action \(a\), for one step
  • act optimally thereafter for the remaining \((h-1)\) steps

\(\mathrm{Q}^*_h(s, a)\): the value for

\(\gamma = 0.9\)

States and one special transition:

80\%
20\%
\mathrm{Q}^*_3(s, a)

Rewards:

\mathrm{Q}^*_2(s, a)

\(= -10 + .9 [.2 \times 0.9 + .8 \times 1.9] \Rightarrow -8.47\)

\(\mathrm{Q}_3^*(6, \uparrow)=\mathrm{R}(6,\uparrow)  + \gamma[.2 \max _{a^{\prime}} \mathrm{Q}_{2}^*\left(2, a^{\prime}\right)+ .8\max _{a^{\prime}} \mathrm{Q}_{2}^*\left(3, a^{\prime}\right)] \)

Let's consider \(\mathrm{Q}_3^*(6, \uparrow)  \)

  • receive \(\mathrm{R}(6,\uparrow)\)
  • 20% chance, \(s'\) = 2, act optimally, get \(\max _{a^{\prime}} \mathrm{Q}_{2}^*\left(2, a^{\prime}\right)\)
  • act optimally at the next state \(s^{\prime}\) 
  • 80% chance, \(s'\) = 3, act optimally, get \(\max _{a^{\prime}} \mathrm{Q}_{2}^*\left(3, a^{\prime}\right)\)
  1. for \(s \in \mathcal{S}, a \in \mathcal{A}\) :
  2.       \(\mathrm{Q}_{\text {old }}(\mathrm{s}, \mathrm{a})=0\)
  3. while True:
  4.       for \(s \in \mathcal{S}, a \in \mathcal{A}\) :
  5.             \(\mathrm{Q}_{\text {new }}(s, a) \leftarrow \mathrm{R}(s, a)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, a, s^{\prime}\right) \max _{a^{\prime}} \mathrm{Q}_{\text {old }}\left(s^{\prime}, a^{\prime}\right)\)
  6.       if \(\max _{s, a}\left|Q_{\text {old }}(s, a)-Q_{\text {new }}(s, a)\right|<\epsilon:\)
  7.             return \(\mathrm{Q}_{\text {new }}\)
  8.       \(\mathrm{Q}_{\text {old }} \leftarrow \mathrm{Q}_{\text {new }}\)
\mathrm{Q}^*_h (s, a)=\mathrm{R}(s, a)+\gamma \sum_{s^{\prime}} \mathrm{T}\left(s, a, s^{\prime} \right) \max _{a^{\prime}} \mathrm{Q}^*_{h-1}\left(s^{\prime}, a^{\prime}\right)
Value Iteration

if run this block \(h\) times and break, then the returns are exactly \(\mathrm{Q}^*_h\)

\{

\(\mathrm{Q}^*_{\infty}(s, a)\)

Value iteration: what we just did, iteratively invoke (eq. 5️⃣):

\{
\mathrm{Q}_1^*(s, a)
\mathrm{Q}_2^*(s, a)
\pi_h^*(s)=\arg \max _a \mathrm{Q}^*_h(s, a)

Optimal policy easily extracted: 6️⃣

\mathrm{Q}_3^*(s, a)
\mathrm{Q}_{\infty}^*(s, a)
\dots

e.g. the best actions to take in state 5

  • For finite \(h\), optimal policy \(\pi^*_h\) depends on how many time steps are left
  • When \(h \rightarrow \infty\), time no longer matters, i.e., there exists a stationary \(\pi^*\)

Summary

  • A Markov decision process \((\mathcal{S}, \mathcal{A}, T, R, \gamma)\) is the mathematical framework for sequential decision-making and the foundation of reinforcement learning.

  • To evaluate a given policy \(\pi\), we compute state value functions \(\mathrm{V}^{\pi}(s)\) via the Bellman recursion (finite horizon) or the Bellman equation (infinite horizon).

  • To find an optimal policy, we compute \(\mathrm{Q}^*(s,a)\) via the value iteration algorithm, then act greedily: \(\pi^*(s) = \arg\max_a \mathrm{Q}^*(s,a)\).