Reinforcement Learning

Transcription

1 Reinforcement Learning LU 2 - Markov Decision Problems and Dynamic Programming Dr. Joschka Bödecker AG Maschinelles Lernen und Natürlichsprachliche Systeme Albert-Ludwigs-Universität Freiburg jboedeck@informatik.uni-freiburg.de Acknowledgement Slides courtesy of Martin Riedmiller and Martin Lauer Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (1)

2 LU 2: Markov Decision Problems and DP Goals: Definition of Markov Decision Problems (MDPs) Introduction to Dynamic Programming (DP) Outline short review definition of MDPs DP: principle of optimality the DP algorithm (backward DP) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (2)

3 Review Process, can be influenced by actions Agent: Sensory input, output of action Feedback RL: Training information through evaluation only Delayed Reinforcement Learning: Decision, decision, decision,... evaluation Multi-stage decision process Optimization Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (3)

4 The Agent Concept Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (4)

5 Multi-stage decision problems Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (5)

6 Three components System, process Rewards, costs Policy, strategy Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (6)

7 Requirements for the model Goal: Describing the system s behaviour (also a system: Process, world, environment) requirements for a model: situations activities current situation can be influenced adjustments possible at discrete points in time noise, interference, random goal specification: definition of costs / rewards Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (7)

8 System description Discrete decision points t T = {0, 1,..., N} or (stages) T = {0, 1,... } Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (8)

9 System description Discrete decision points t T = {0, 1,..., N} or (stages) T = {0, 1,... } System state (situation) s t S here: S finite Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (8)

10 System description Discrete decision points t T = {0, 1,..., N} or (stages) T = {0, 1,... } System state (situation) s t S here: S finite Actions u t U here: U finite Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (8)

11 System description Discrete decision points t T = {0, 1,..., N} or (stages) T = {0, 1,... } System state (situation) s t S here: S finite Actions u t U here: U finite Transition function s t+1 = f (s t, u t) reaction of the system Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (8)

12 Goal formulation: Introducing costs At every decision (= in every stage) direct costs arise Direct costs Refinement: dependant on state and action c : S R c : S U R Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (9)

13 Goal formulation: Introducing costs At every decision (= in every stage) direct costs arise Direct costs Refinement: dependant on state and action c : S R c : S U R Reward, cost, punishment? Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (9)

14 Summary: Deterministic systems discrete decision points t T = {0, 1,..., N} or stages T = {0, 1,... } system state (situation) actions s t S u t U transition function s t+1 = f (s t, u t) direct costs c : S U R 5-tuple (T, S, U, f, c) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (10)

15 Example: Shortest path problems Find the shortest path from start node to finish node. Every edge has a specific cost that can be interpreted as length. Optimization goal over multiple stages Evaluation of whole sequence (reminder: decision, decision,... evaluation) Look at accumulated total costs: t T c(st, ut) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (11)

16 Stochastic systems Again: requirements for a model: situations activities current situation can be influenced adjustments possible at discrete points in time noise, interference,random goal specification: definition of costs / rewards Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (12)

17 Markov Decision Processes Deterministic system: 5-Tuple (T, S, U, f, c) Stochastic system: The deterministic transition function f is replaced by a conditional probability distribution. In the following, we re looking at a finite state set S = (1, 2,..., N). Let i, j S be states: Notation: Markov Decision Process (MDP): 5-Tuple (T, S, U, p ij (u), c(s, u)) P(s t+1 = j s t = i, u t = u) = p ij (u) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (13)

18 Markov property It holds that: P(s t+1 = j s t, u t) = P(s t+1 = j s t, s t 1,..., u t, u t 1,...) The probability distribution of the following state s t+1 is uniquely defined given the knowledge of the current state s t and the action u t. It especially does not depend on the previous history of the system. Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (14)

19 Remarks (1) Deterministic system is a special case of an MDP: { 1, st+1 = f (s t, u t) P(s t+1 s t, u t) = 0, otherwise Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (15)

20 Remarks (2) Equivalent description with deterministic transition function f : Approach: additional argument - random variable w t (noise): s t+1 = f (s t, u t, w t) with w t random variable with given probability distribution P(w t s t, u t) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (16)

21 Remarks (2) Equivalent description with deterministic transition function f : Approach: additional argument - random variable w t (noise): s t+1 = f (s t, u t, w t) with w t random variable with given probability distribution P(w t s t, u t) Transformation into previous form: Let W (i, u, j) = {w j = f (i, u, w)} be the set of all values of w, for which the system transitions from state i on input of u into state j. Then it holds: p ij (u) = P(w W (i, u, j)) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (16)

22 Summary: MDPs discrete decision points t T = {0, 1,..., N} or stages T = {0, 1,... } system state (situation) actions transition probabilites p ij (u) s t S u t U P(s t+1 = j s t = i, u t = u) = p ij (u) alternatively: Transition function s t+1 = f (s t, u t, w t) with w t random variable direct costs 5-tuple (T, S, U, p ij (u), c(s, u)) c : S U R Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (17)

23 Summary: MDPs Model: State, action, following state Deterministic and stochastic transition function Information about history summarized in state Very general description: OR, control engineering, games,... Generalizations (not covered here) Transition function not stationary p ij,t (u) Costs not stationary c t(i, u) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (18)

24 Example stock keeping Assume you are the owner of a toys shop at an exhibition. Exhibition lasts N days. state: number of toys in your shop action: ordered number of toys to be delivered on the next day disturbance : number of toys sold s t u t w t system equation: s t+1 = s t + u t w t Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (19)

25 Example stock keeping Assume you are the owner of a toys shop at an exhibition. Exhibition lasts N days. state: number of toys in your shop action: ordered number of toys to be delivered on the next day disturbance : number of toys sold s t u t w t system equation: costs for toys in stock acquisition costs for each toy which was ordered minus gain for sold toys s t+1 = s t + u t w t c(s, u) = c 1(s) + c 2(u) gain Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (19)

26 Example stock keeping Assume you are the owner of a toys shop at an exhibition. Exhibition lasts N days. state: number of toys in your shop action: ordered number of toys to be delivered on the next day disturbance : number of toys sold s t u t w t system equation: costs for toys in stock acquisition costs for each toy which was ordered minus gain for sold toys s t+1 = s t + u t w t c(s, u) = c 1(s) + c 2(u) gain there are also terminal costs g(s), if there are still toys in stock after the N days. Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (19)

27 Policy and selection function Policy: The selection function π t : S U, π t(s) = u chooses at time t an action u U as function of the current state s S. Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (20)

28 Policy and selection function Policy: The selection function π t : S U, π t(s) = u chooses at time t an action u U as function of the current state s S. Selection function chooses an action in dependence of the situation (see graphic agent ) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (20)

29 Policy and selection function Policy: The selection function π t : S U, π t(s) = u chooses at time t an action u U as function of the current state s S. Selection function chooses an action in dependence of the situation (see graphic agent ) Refinement: π t : S U, π t(s) = u, with u U(s) situation dependent action set (example: chess) A policy ˆπ consists of N selection functions (N being the number of decision points) ˆπ = (π 0, π 1,..., π t,...) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (20)

30 Non-stationary policies The selection function π t can be dependent on the time of the decision. Meaning: The same situation at different points in time can lead to different decisions of the agent. ˆπ = (π 0, π 1,..., π t,...) If the selection functions differ for single time points, we call it a non-stationary policy. Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (21)

31 Non-stationary policies The selection function π t can be dependent on the time of the decision. Meaning: The same situation at different points in time can lead to different decisions of the agent. ˆπ = (π 0, π 1,..., π t,...) If the selection functions differ for single time points, we call it a non-stationary policy. Example soccer: Situation s: Midfield player has the ball. Reasonable action in the first minute: π 1(s) = return pass Reasonable action in the last minute: π 90(s) = shoot on goal General rationale: The limited optimization time frame ( finite horizon, see below) usually requires a non-stationary policy! Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (21)

32 Stationary policies We will look mostly at stationary policies. Then it holds that π 0 = π 1 =... π t... =: π and ˆπ = (π, π,..., π,...) With stationary policies, the terms policy and selection function become interchangeable. We will call the selection function π - as generally done in literature - our policy. Bertsekas uses the term µ for the selection function. Therefore there arise minor differences from the notation used there. Remark: In the following only deterministic selection functions will be used Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (22)

33 Goal of the policy Reach the optimization goal over multiple stages (sequence of decisions) Solving a dynamic optimization problem Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (23)

34 Cumulated costs (costs-to-go) Interesting: Cumulated costs for a given state s with given policy π: J π (s) = t T c(s t, π(s t)), s 0 = s Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (24)

35 Cumulated costs (costs-to-go) Interesting: Cumulated costs for a given state s with given policy π: J π (s) = t T c(s t, π(s t)), s 0 = s Wanted: Optimal policy π so that for all s it holds that: J π (s) = min c(s t, π(s t)), π ˆπ t T under the constraint that s t+1 = f (s t, u t) s 0 = s Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (24)

36 Cumulated costs in MDPs Expected cumulated costs for a given state s using a given policy π: J π (s) = E w c(s t, π(s t)), s 0 = s t T Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (25)

37 Cumulated costs in MDPs Expected cumulated costs for a given state s using a given policy π: J π (s) = E w c(s t, π(s t)), s 0 = s t T Wanted: Optimal policy π so that for all s it holds that: J π (s) = min π Π Ew t T c(s t, π(s t)), s 0 = s under the constraint that s t+1 = f (s t, u t, w t), or with given probability distribution P(s t+1 = j s t = i, u t = u) = p ij (u) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (25)

38 Problem types Definition horizon: The horizon N of a problem denotes the number of decision stages to be traversed. Finite horizon: Problems with given termination time Infinite horizon: Approximation for very long processes or processes with an unknown end (e.g. control system) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (26)

39 Finite horizon N-stage decision problem Each state has terminal costs g(i) that are due if the system ends in i after N stages. Costs of a policy π N 1 JN π (s) = E[g(s N ) + c(s t, π t(s t)) s 0 = s] Generally: Non-stationary policy t=0 Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (27)

40 Infinite horizon Costs of a policy π J π (s) = lim E[ N c(s t, π t(s t)) s 0 = s] N Problem: Finite costs? t=0 Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (28)

41 Infinite horizon Costs of a policy π Problem: Finite costs? J π (s) = lim E[ N c(s t, π t(s t)) s 0 = s] N Solution: Discount α < 1 J π (s) = lim E[ N α t c(s t, π t(s t)) s 0 = s] N t=0 t=0 Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (28)

42 Solution of dynamic optimization problems Central question: How do we find the policy that leads (on average) to minimal costs? Solution method: Dynamic Programming (Bellman, 1957) Backward Dynamic Programming (finite horizon) Value Iteration (LU 3 ff., infinite horizon) Policy Iteration (LU 3 ff., infinite horizon) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (29)

43 Backward Dynamic Programming - idea Problem: Stochastic multistage decision problems with finite horizon Idea: Calculate the costs starting from the last stage to the first stage. Example: Find the cheapest path in a graph Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (30)

44 Backward Dynamic Programming - problem specification (1) finite horizon N MDP N discrete decision points t T = {0, 1,..., N} State set finite s t S = {1, 2,..., n} Action set finite u t U = {u 1,..., u m} Transition prob. p ij (u) P(s t+1 = j s t = i, u t = u) = p ij (u) direct costs c : S U R in the last stage N every stage causes terminal costs g(s N ) := c N (s N ) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (31)

45 Backward Dynamic Programming - objective Wanted: π with J π = min π J π with JN π (i) = E[g(s N ) + N 1 t=0 c(st, πt(st)) s0 = i] the costs belonging to π are called the optimal cumulated costs J := J π. Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (32)

46 Backward Dynamic Programming - objective Wanted: π with J π = min π J π with JN π (i) = E[g(s N ) + N 1 t=0 c(st, πt(st)) s0 = i] the costs belonging to π are called the optimal cumulated costs J := J π. Approach: 1. Calcuation of optimal cumulated costs ( cost-to-go ) J k ( ) for all states (J k ( ) is a n dimensional vector). k is the number of remaining steps. 2. from J k follows the optimal policy for the k step problem. (k steps until process terminates). Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (32)

47 Backward Dynamic Programming - motivation Thesis - Bellman s Principle of Optimality: If I have k more steps to go, the optimal costs for a state i are given with the minimal expected value of the sum of the direct transition costs + optimal cumulated costs of the next state, if there are k 1 more steps to be done from there. The minimization here goes over all possible actions Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (33)

48 Bellman s Principle of Optimality Formal: For the optimal cumulated costs J k (i) of the k-stage decision problem, it holds that: Jk (i) = min u U(i) Ew {c(i, u) + k J k 1(f (i, u, w k )} n = min u U(i) {p ij (u)(c(i, u) + Jk 1(j))} i = 1... n (1) j=1 Hence we can calculate the optimal cumulated costs of the N stage optimization problem recursively starting with k = 0. Backward-DP algorithm Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (34)

49 Bellman s Principle of Optimality - proof (1) Policy ˆπ (k) for k stages: ˆπ (k) = (π k, π k 1, π k 2,...) = (π k, ˆπ (k 1) ) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (35)

50 Bellman s Principle of Optimality - proof (1) Policy ˆπ (k) for k stages: ˆπ (k) = (π k, π k 1, π k 2,...) = (π k, ˆπ (k 1) ) Let S (k) (i) = (s N k = i, s (N k)+1,..., s N ) be a possible state sequence starting in state i with k transitions. Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (35)

51 Bellman s Principle of Optimality - proof (1) Policy ˆπ (k) for k stages: ˆπ (k) = (π k, π k 1, π k 2,...) = (π k, ˆπ (k 1) ) Let S (k) (i) = (s N k = i, s (N k)+1,..., s N ) be a possible state sequence starting in state i with k transitions. J k (i) = min J ˆπ (k) ˆπ (k) k (i) (2) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (35)

52 Bellman s Principle of Optimality - proof (1) Policy ˆπ (k) for k stages: ˆπ (k) = (π k, π k 1, π k 2,...) = (π k, ˆπ (k 1) ) Let S (k) (i) = (s N k = i, s (N k)+1,..., s N ) be a possible state sequence starting in state i with k transitions. J k (i) = min J ˆπ (k) ˆπ (k) k = min ˆπ (k) { S (k) (i) (i) (P(S (k) (i) ˆπ (k) )( k c(s N l, π l (s N l )) + g(s N )))} l=1 (2) (3) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (35)

53 Bellman s Principle of Optimality - proof (1) Policy ˆπ (k) for k stages: ˆπ (k) = (π k, π k 1, π k 2,...) = (π k, ˆπ (k 1) ) Let S (k) (i) = (s N k = i, s (N k)+1,..., s N ) be a possible state sequence starting in state i with k transitions. J k (i) = min J ˆπ (k) ˆπ (k) k = min ˆπ (k) { S (k) (i) (i) (P(S (k) (i) ˆπ (k) )( = min ˆπ (k) {c(i, π k (i)) + S (k) (i) k c(s N l, π l (s N l )) + g(s N )))} l=1 (P(S (k) (i) ˆπ (k) ) k 1 ( c(s N l, π l (s N l )) + g(s N )))} l=1 (2) (3) (4) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (35)

54 Bellman s Principle of Optimality - proof (2) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (36)

55 Bellman s Principle of Optimality - proof (2) = min ˆπ (k) {c(i, π k (i)) + S (k) (i) (P(S (k) (i) ˆπ (k) ) k 1 ( c(s N l, π l (s N l )) + g(s N )))} l=1 (5) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (36)

56 Bellman s Principle of Optimality - proof (2) = min ˆπ (k) {c(i, π k (i)) + S (k) (i) (P(S (k) (i) ˆπ (k) ) k 1 ( c(s N l, π l (s N l )) + g(s N )))} l=1 = min{c(i, π k (i)) + P(s (N k)+1 = j s N k = i, π k ) ˆπ (k) j S (5) S (k 1) (j) k 1 (P(S (k 1) (j) ˆπ (k 1) ) ( c(s N l, π l (s N l )) + g(s N )))} l=1 (6) (7) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (36)

57 Bellman s Principle of Optimality - proof (3) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (37)

58 Bellman s Principle of Optimality - proof (3) = min{c(i, π k (i)) + P(s (N k)+1 = j s N k = i, π k ) ˆπ (k) j S S (k 1) (j) k 1 (P(S (k 1) (j) ˆπ (k 1) ) ( c(s N l, π l (s N l )) + g(s N )))} l=1 (8) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (37)

59 Bellman s Principle of Optimality - proof (3) = min{c(i, π k (i)) + P(s (N k)+1 = j s N k = i, π k ) ˆπ (k) j S S (k 1) (j) k 1 (P(S (k 1) (j) ˆπ (k 1) ) ( c(s N l, π l (s N l )) + g(s N )))} = min { c(i, u) + P(s (N k)+1 = j s N k = i, u) u U(i) min { ˆπ (k 1) S (k 1) (j) j S l=1 k 1 (P(S (k 1) (j) ˆπ (k 1) ) ( c(s N l, π l (s N l )) + g(s N )))} } l=1 (8) (9) (10) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (37)

60 Bellman s Principle of Optimality - proof (4) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (38)

61 Bellman s Principle of Optimality - proof (4) = min { c(i, u) + P(s (N k)+1 = j s N k = i, u) u U(i) min { ˆπ (k 1) S (k 1) (j) j S k 1 (P(S (k 1) (j) ˆπ (k 1) ) ( c(s N l, π l (s N l )) + g(s N )))} } l=1 (11) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (38)

62 Bellman s Principle of Optimality - proof (4) = min { c(i, u) + P(s (N k)+1 = j s N k = i, u) u U(i) min { ˆπ (k 1) S (k 1) (j) j S = min { c(i, u) + u U(i) k 1 (P(S (k 1) (j) ˆπ (k 1) ) ( c(s N l, π l (s N l )) + g(s N )))} } j S l=1 P(s (N k)+1 = j s N k = i, u) min {J ˆπ (k 1) ˆπ (k 1) (11) k 1 (j)} } (12) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (38)

63 Bellman s Principle of Optimality - proof (4) = min { c(i, u) + P(s (N k)+1 = j s N k = i, u) u U(i) min { ˆπ (k 1) S (k 1) (j) j S = min { c(i, u) + u U(i) k 1 (P(S (k 1) (j) ˆπ (k 1) ) ( c(s N l, π l (s N l )) + g(s N )))} } j S l=1 P(s (N k)+1 = j s N k = i, u) min {J ˆπ (k 1) ˆπ (k 1) = min {c(i, u) + P(s (N k)+1 = j s N k = i, u) Jk 1(j)} u U(i) j S (11) k 1 (j)} } (12) (13) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (38)

64 Bellman s Principle of Optimality - proof (4) = min { c(i, u) + P(s (N k)+1 = j s N k = i, u) u U(i) min { ˆπ (k 1) S (k 1) (j) j S = min { c(i, u) + u U(i) k 1 (P(S (k 1) (j) ˆπ (k 1) ) ( c(s N l, π l (s N l )) + g(s N )))} } j S l=1 P(s (N k)+1 = j s N k = i, u) min {J ˆπ (k 1) ˆπ (k 1) = min {c(i, u) + P(s (N k)+1 = j s N k = i, u) Jk 1(j)} u U(i) j S (11) k 1 (j)} } (12) = min {c(i, u) + p ij (u) Jk 1(j)} u U(i) j S (13) (14) (15) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (38)

65 Backward Dynamic Programming - algorithm k = 0: For k = 1 To N, i S J 0 (i) = g(i) or Jk (i) = min E wk {c(i, u) + Jk 1(f (i, u, w k ))} u U(i) J k (i) = min u U(i) n p ij (u)(c(i, u) + Jk 1(j)) j=1 Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (39)

66 Choosing an action Requirement: Jk (i) is known for all k N. Approach: We simply calculate for all possible actions the expected costs and choose the best action (with minimal expected cumulated costs). π k (i) arg min u U(i) E wk {c(i, u) + J k 1(f (i, u, w k )) the chosen optimal action minimizes the sum of the expected transition costs plus the expected cumulated costs of the remaining problem. Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (40)

67 Choosing an action Requirement: Jk (i) is known for all k N. Approach: We simply calculate for all possible actions the expected costs and choose the best action (with minimal expected cumulated costs). π k (i) arg min u U(i) E wk {c(i, u) + J k 1(f (i, u, w k )) the chosen optimal action minimizes the sum of the expected transition costs plus the expected cumulated costs of the remaining problem. Remark: J k defines an optimal policy The policy is not unique, but J k is Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (40)

68 Remarks Complexity for deterministic systems O(N n m) Complexity for stochastic systems O(N n 2 m) Exact solution rarely computable, numeric solution; but: very complex! (N = number of stages, n = number of states, m = number of actions) Prof. Dr. M. Riedmiller, Dr. M. Lauer, Dr. J. Boedecker Machine Learning Lab, University of Freiburg Reinforcement Learning (41)