Mdps and reinforcement learning
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MDPs and Reinforcement Learning. Overview. MDPs Reinforcement learning. Sequential decision problems. In an environment, find a sequence of actions in an uncertain environment that balance risks and rewards Markov Decision Process (MDP):

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  • MDPs

  • Reinforcement learning

Sequential decision problems
Sequential decision problems

  • In an environment, find a sequence of actions in an uncertain environment that balance risks and rewards

  • Markov Decision Process (MDP):

    • In a fully observable environment we know initial state (S0) and state transitions T(Si, Ak, Sj) = probability of reaching Sj from Si when doing Ak

    • States have a reward associated with them R(Si)

  • We can define a policy π that selects an action to perform given a state, i.e., π(Si)

  • Applying a policy leads to a history of actions

  • Goal: find policy maximizing expected utility of history

4x3 grid world1
4x3 Grid World

  • Assume R(s) = -0.04 except where marked

  • Here’s an optimal policy

4x3 grid world2
4x3 Grid World

Different default rewards produce different optimal policies

life=pain, get out quick

Life = struggle, go for +1, accept risk

Life = good, avoid exits

Life = ok, go for +1, minimize risk

Finite and infinite horizons
Finite and infinite horizons

  • Finite Horizon

    • There’s a fixed time N when the game is over

    • U([s1…sn]) = U([s1…sn…sk])

    • Find a policy that takes that into account

  • Infinite Horizon

    • Game goes on forever

  • The best policy for with a finite horizon can change over time: more complicated


  • The utility of a sequence is usually additive

    • U([s0…s1]) = R(s0) + R(s1) + … R(sn)

  • But future rewards might be discounted by a factor γ

    • U([s0…s1]) = R(s0) + γ*R(s1) + γ2*R(s2)…+ γn*R(sn)

  • Using discounted rewards

    • Solves some technical difficulties with very long or infinite sequences and

    • Is psychologically realistic

Value functions
Value Functions

  • The value of a state is the expected return starting from that state; depends on the agent’s policy:

  • The value of taking an action in a stateunder policy p is the expected return starting from that state, taking that action, and thereafter following p :

Bellman equation for a policy p
Bellman Equation for a Policy p

The basic idea:


Or, without the expectation operator: