Markov Decision Process - I


CSE 440: Introduction to Artificial Intelligence

Vishnu Boddeti


Content Credits: CMU AI, http://ai.berkeley.edu

Today

  • Markov Decision Processes (MDPs)

Non-Determinstic Search

Example: Grid World

  • A maze like problem
    • The agent lives in a grid
    • Walls block the agent's path
  • Noisy movement: actions do not always go as planned
    • 80% of the time, the action North takes the agent North (if there is no wall there)
    • 10% of the time, North takes the agent West; 10% East
    • If there is a wall in the direction the agent would have been taken, the agent stays put
  • The agent receives rewards each time step
    • Small "living" reward each step (can be negative)
    • Big rewards come at the end (good or bad)
  • Goal: maximize sum of rewards

Grid World Actions

Deterministic Grid World
Stochastic Grid World

Markov Decision Processes

  • An MDP is defined by:
    • A set of states $s \in S$
    • A set of actions $a \in A$
    • A transition function $T(s, a, s')$
      • Probability that $a$ from $s$ leads to $s'$, i.e., $P(s'| s, a)$
      • Also called the model or the dynamics
    • A reward function $R(s, a, s')$
      • Sometimes just $R(s)$ or $R(s')$
    • A start state, $s_0$
    • Maybe a terminal state

  • MDPs are non-deterministic search problems
    • One way to solve them is with expectimax search
    • We will have a new tool soon

What is Markov about MDPs

  • "Markov" generally means that given the present state, the future and the past are independent
  • For Markov decision processes, "Markov" means action outcomes depend only on the current state
    • $$\begin{equation} \begin{aligned} & p(S_{t+1}=s'|S_t=s_t, A_t=a_t, S_{t-1}=s_{t-1},A_{t-1},\dots,S_0=s_0) \nonumber \\ & = p(S_{t+1}=s'|S_t=s_t, A_t=a_t) \nonumber \end{aligned} \end{equation}$$
  • This is just like search, where the successor function could only depend on the current state (not the history)

Policies

  • In deterministic single-agent search problems, we wanted an optimal plan, or sequence of actions, from start to a goal

  • For MDPs, we want an optimal policy $\pi^*:S\rightarrow A$
    • A policy $\pi$ gives an action for each state
    • An optimal policy is one that maximizes expected utility if followed
    • An explicit policy defines a reflex agent

  • Expectimax did not compute entire policies
    • It computed the action for a single state only

Optimal Policies

$R(s) = -0.01$
$R(s) = -0.4$

Example: Car Racing

  • A robot car wants to travel far, quickly
  • Three states: Cool, Warm, Overheated
  • Two actions: Slow, Fast
  • Going faster gets double reward

Racing Search Tree

MDP Search Tree

  • Each MDP state projects an expectimax-like search tree

Utility of Sequences

  • What preferences should an agent have over reward sequences?


  • More or less? [1,2,2] or [2,3,4]


  • Now or later? [0,0,1] or [1,0,0]

Discounting

  • It is reasonable to maximize the sum of rewards
  • It is also reasonable to prefer rewards now to rewards later
  • One solution: values of rewards decay exponentially
Worth Now
Worth Next Step
Worth in Two Steps

Discounting

  • How to discount?
    • Each time we descend a level, we multiply in the discount once
  • Why discount?
    • Sooner rewards probably do have higher utility than later rewards
    • Also helps our algorithms converge
  • Example: discount of 0.5
    • $U([1,2,3]) = 1*1 + 0.5*2 + 0.25*3$
    • $U([1,2,3]) < U([3,2,1])$

Stationary Preferences

  • Theorem: if we assume stationary preferences:
    • $$\begin{eqnarray} [a_1, a_2, \dots] &\succ& [b_1, b_2, \dots] \\ & \Updownarrow &\\ [r, a_1, a_2, \dots] &\succ& [r, b_1, b_2, \dots]\\ \end{eqnarray}$$
  • Then: there are only two ways to define utilities
    • Additive utility: \[U([r_0, r_1, r_2, \dots]) = r_0 + r_1 + r_2 + \dots\]
    • Discounted utility: \[U([r_0, r_1, r_2, \dots]) = r_0 + \gamma r_1 + \gamma^2 r_2 + \dots\]

Infinite Utilities

  • Problem: What if the game lasts forever? Do we get infinite rewards?
  • Solutions:
    • Finite horizon: (similar to depth-limited search)
      • Terminate episodes after a fixed T steps (e.g. life)
      • Gives non-stationary policies ($\pi$ depends on time left)
    • Discounting: use $0 < \gamma < 1$
      • \[U([r_0,\dots,r_{\infty}]) = \sum_{t=0}^{\infty}\gamma^tr_t \leq \frac{R_{max}}{1-\gamma}\]
      • Smaller $\gamma$ means smaller "horizon" – shorter term focus
    • Absorbing state: guarantee that for every policy, a terminal state will eventually be reached (like "overheated" for racing)

Recap: Defining MDPs

  • Markov decision processes:
    • Set of states $S$
    • Start state $s_0$
    • Set of actions $A$
    • Transitions $P(s'|s,a)$ (or $T(s,a,s')$)
    • Rewards R(s,a,s') (and discount $\gamma$)

  • MDP quantities so far:
    • Policy = Choice of action for each state
    • Utility = sum of (discounted) rewards

Q & A

Image
XKCD