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Planning by Dynamic Programming (RL)

Notes of lectures by D. Silver.

Introduction

  • Dynamic: sequential or temporal components to the problem
  • Programming: optimizing a “program”, i.e. a policy
    c.f. linear programming

  • Breaking down the complex problems into subproblems

    • Solve the problems
    • Combine solutions to subproblems

Requirements for dynamic programming

Dynamic programming(DP) is general solution for problems with two properties:

  • Optimal substructure
    • Principle of optimality applies
    • Optimal solution can be decomposed into subproblems
  • Overlapping subproblems
    • Subproblems recur many times
    • Solutions can be cached and reused

MDP satisfy both properties:

  • Bellman equation gives recursive decomposition
  • Value function stores and reuses solutions

Planning by DP

  • DP assumes full knowledge of the MDP
  • It is used for planning in an MDP
  • For prediction:
    1. Input: DMP $<\mathcal{S}, \mathcal{A},\mathcal{P},\mathcal{R},\mathcal{\gamma}>$ and policy $\pi$, or MRP $<\mathcal{S},\mathcal{P}^{\pi},\mathcal{R}^{\pi},\mathcal{\gamma}>$
    2. Output: value function
  • For control:
    1. Input: MDP $<\mathcal{S}, \mathcal{A},\mathcal{P},\mathcal{R},\mathcal{\gamma}>$
    2. Output: optimal value function , and optimal policy

Other applications by DP

  • Scheduling algorithms
  • String algorithms (e.g. sequence alignment)
  • Graph algorithms (e.g. shortest path algorithms)
  • Graphical models (e.g. Viterbi algorithm)
  • Bioinformatics (e.g. lattice model)

Iterative policy evaluation

  • Problem: evaluate a given policy $\pi$
  • Solution: iterative application of Bellman expectation backup

Using synchronous backups:

  • At each iteration $k+1$
  • For all states $s \in \mathcal{S}$
  • Update from , where $s’$ is a successor state of $s$

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Policy iteration

How to improve a policy

Given a policy $\pi$:

  • Evaluate the policy $\pi$
  • Improve the policy by acting greedily w.r.t

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  • Policy evaluation: estimate (iterative policy evaluation)

  • Policy improvement: generate (greedy policy improvement)

Policy improvement

Consider a deterministic policy $a=\pi(s)$

  • Improve the policy by acting greedily

This improves the value from any state $s$ over one step

It therefore improves the value function

  • If improvements stop,
  • Then the Bellman optimality equation has been satisfied

  • Therefore for all $s \in \mathcal{S}$. So $\pi$ is an optimal policy

Value iteration

Value iteration in MDPs

Principle of Optimality

An optimal policy can be subdivided into two components:

  • An optimal first action
  • Followed by an optimal policy from successor state $S’$

Deterministic value iteration

If we know the solution to subproblems , then solution can be found by one-step lookahead:

  • The idea of value iteration is to apply these update iteratively
  • Intuition: start with final rewards and work backwards
  • Still works loopy, stochastic MDPs

Value iteration

  • Problems: find optimal policy $\pi$
  • Solution: iterative application of Bellman optimality backup

  • Using synchronous backups at each iteration $k+1$, for all states , update from

  • Unlike policy iteration , there is no explicit policy

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Synchronous DP algorithms

Problem Bellman equation Algorithm
Prediction Bellman Expectation Equation Iterative Policy Evaluation
Control Bellman Expectation Equation + Greedy Policy Improvement Policy Iteration
Control Bellman Optimality Equation Value Iteration

Asynchronous DP

In-place DP

  • Synchronous value iteration stores two copies of value function for all $s$ in $\mathcal{S}$:

  • In-place value iteration only stores one copy of value function for all $s$ in $\mathcal{S}$:

Prioritized sweeping

Use magnitude of Bellman error to select state

Backup the state with the largest remaining Bellman error

Real-time DP

  • Idea: only states that are relevant to agent
  • Use agent’s experience to guide the selection of states
  • After each time-step
  • Backup the state
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