1. 1 DCP 1172 Introduction to Artificial Intelligence Lecture notes for Chap. 4 [AIMA] Chang-Sheng Chen

2. DCP 1172, Ch.4 2 Last time: Problem-Solving Problem solving:  Goal formulation  Problem formulation (states, operators)  Search for solution Problem formulation:  Initial state  Operators  Goal test  Path cost Problem types:  single state:fully observable and deterministic environment  multiple state: partially observable and deterministic environment  contingency: partially observable and nondeterministic environment  exploration:unknown state-space

3. DCP 1172, Ch.4 3 Last time: Finding a solution Function General-Search(problem, strategy) returns a solution, or failure initialize the search tree using the initial state problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solution else expand the node and add resulting nodes to the search tree end  Solution: is a sequence of operators that bring you from current state to the goal state  Basic idea: offline, systematic exploration of simulated state-space by generating successors of explored states (expanding) Strategy: The search strategy is determined by the order in which the nodes are expanded.

4. DCP 1172, Ch.4 4 A Clean Robust Algorithm Function UniformCost-Search(problem, Queuing-Fn) returns a solution, or failure open  make-queue(make-node(initial-state[problem])) closed  [empty] loop do if open is empty then return failure currnode  Remove-Front(open) if Goal-Test[problem] applied to State(currnode) then return currnode children  Expand(currnode, Operators[problem]) while children not empty [… see next slide …] end closed  Insert(closed, currnode) open  Sort-By-PathCost(open) end

5. DCP 1172, Ch.4 5 A Clean Robust Algorithm [… see previous slide …] children  Expand(currnode, Operators[problem]) while children not empty child  Remove-Front(children) if no node in open or closed has child’s state open  Queuing-Fn(open, child) else if there exists node in open that has child’s state if PathCost(child) < PathCost(node) open  Delete-Node(open, node) open  Queuing-Fn(open, child) else if there exists node in closed that has child’s state if PathCost(child) < PathCost(node) closed  Delete-Node(closed, node) open  Queuing-Fn(open, child) end [… see previous slide …]

6. DCP 1172, Ch.4 6 Last time: search strategies Uninformed: Use only information available in the problem formulation Breadth-first Uniform-cost Depth-first Depth-limited Iterative deepening Informed: Use heuristics to guide the search Best first A*

7. DCP 1172, Ch.4 7 Evaluation of search strategies Search algorithms are commonly evaluated according to the following four criteria: Completeness: does it always find a solution if one exists? Time complexity: how long does it take as a function of number of nodes? Space complexity: how much memory does it require? Optimality: does it guarantee the least-cost solution? Time and space complexity are measured in terms of: b – max branching factor of the search tree d – depth of the least-cost solution m – max depth of the search tree (may be infinity)

8. DCP 1172, Ch.4 8 Last time: uninformed search strategies Uninformed search: Use only information available in the problem formulation Breadth-first Uniform-cost Depth-first Depth-limited Iterative deepening

9. DCP 1172, Ch.4 9 This time: informed search Informed search: Use heuristics to guide the search Best first A* Heuristics Hill-climbing Simulated annealing

10. DCP 1172, Ch.4 10 Best-first search Idea: use an evaluation function for each node; estimate of “desirability”  expand most desirable unexpanded node. Implementation: QueueingFn = insert successors in decreasing order of desirability Special cases: greedy search A* search

11. DCP 1172, Ch.4 11 Romania with step costs in km 374 329 253

12. DCP 1172, Ch.4 12 Greedy search Estimation function: h(n) = estimate of cost from node n to goal (heuristic) For example: h SLD (n) = straight-line distance from n to Bucharest Greedy search expands first the node that appears to be closest to the goal, according to h(n).

13. DCP 1172, Ch.4 13

14. DCP 1172, Ch.4 14

15. DCP 1172, Ch.4 15

16. DCP 1172, Ch.4 16

17. DCP 1172, Ch.4 17 Properties of Greedy Search Complete?No – can get stuck in loops e.g., Iasi > Neamt > Iasi > Neamt > …  Complete in finite space with repeated-state checking. Time?O(b^m) but a good heuristic can give dramatic improvement Space?O(b^m) – keeps all nodes in memory Optimal?No.

18. DCP 1172, Ch.4 18 A* search Idea: avoid expanding paths that are already expensive  evaluation function: f(n) = g(n) + h(n)with: g(n) – cost so far to reach n h(n) – estimated cost to goal from n f(n) – estimated total cost of path through n to goal A* search uses an admissible heuristic, that is, h(n)  h*(n) where h*(n) is the true cost from n.  For example: h SLD (n) never overestimates actual road distance. Theorem: A* search is optimal

19. DCP 1172, Ch.4 19

20. DCP 1172, Ch.4 20

21. DCP 1172, Ch.4 21

22. DCP 1172, Ch.4 22

23. DCP 1172, Ch.4 23

24. DCP 1172, Ch.4 24

25. DCP 1172, Ch.4 25 1 Optimality of A* (standard proof) Suppose some suboptimal goal G 2 has been generated and is in the queue. Let n be an unexpanded node on a shortest path to an optimal goal G 1.

26. DCP 1172, Ch.4 26 Optimality of A* (more useful proof)

27. DCP 1172, Ch.4 27 f-contours How do the contours look like when h(n) =0?

28. DCP 1172, Ch.4 28 Properties of A* Complete?Yes, unless infinitely many nodes with f  f(G) Time?Exponential in [(relative error in h) x (length of solution)] Space?Keeps all nodes in memory Optimal?Yes – cannot expand f i+1 until f i is finished

29. DCP 1172, Ch.4 29 Proof of lemma: pathmax

30. DCP 1172, Ch.4 30 Admissible heuristics

31. DCP 1172, Ch.4 31 Relaxed Problem Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem. If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h 1 (n) gives the shortest solution. If the rules are relaxed so that a tile can move to any adjacent square, then h 2 (n) gives the shortest solution.

32. DCP 1172, Ch.4 32 Recall: breadth-first search, step by step

33. DCP 1172, Ch.4 33 Implementation of search algorithms Function General-Search(problem, Queuing-Fn) returns a solution, or failure nodes  make-queue(make-node(initial-state[problem])) loop do if nodes is empty then return failure node  Remove-Front(nodes) if Goal-Test[problem] applied to State(node) succeeds then return node nodes  Queuing-Fn(nodes, Expand(node, Operators[problem])) end Queuing-Fn(queue, elements) is a queuing function that inserts a set of elements into the queue and determines the order of node expansion. Varieties of the queuing function produce varieties of the search algorithm.

34. DCP 1172, Ch.4 34 Recall: breath-first search, step by step

35. DCP 1172, Ch.4 35 Breadth-first search Node queue:initialization #statedepthpath costparent # 1Arad00--

36. DCP 1172, Ch.4 36 Breadth-first search Node queue:add successors to queue end; empty queue from top (i.e., FIFO) #statedepthpath costparent # 1Arad00-- 2Zerind111 3Sibiu111 4Timisoara111

37. DCP 1172, Ch.4 37 Breadth-first search Node queue:add successors to queue end; empty queue from top #statedepthpath costparent # 1Arad00-- 2Zerind111 3Sibiu111 4Timisoara111 5Arad222 6Oradea222 (get smart: e.g., avoid repeated states like node #5)

38. DCP 1172, Ch.4 38 Depth-first search

39. DCP 1172, Ch.4 39 Depth-first search Node queue:initialization #statedepthpath costparent # 1Arad00--

40. DCP 1172, Ch.4 40 Depth-first search Node queue:add successors to queue front; empty queue from top (i.e., FILO, or stack ) #statedepthpath costparent 2Zerind111 3Sibiu111 4Timisoara111 1Arad00--

41. DCP 1172, Ch.4 41 Depth-first search Node queue:add successors to queue front; empty queue from top #statedepthpath costparent # 5Arad222 6Oradea222 2Zerind111 3Sibiu111 4Timisoara111 1Arad00--

42. DCP 1172, Ch.4 42 Last time: search strategies Uninformed: Use only information available in the problem formulation Breadth-first Uniform-cost Depth-first Depth-limited Iterative deepening Informed: Use heuristics to guide the search Best first:  Greedy search -- queue first nodes that maximize heuristic “desirability” based on estimated path cost from current node to goal; A* search – queue first nodes that minimize sum of path cost so far and estimated path cost to goal.

43. DCP 1172, Ch.4 43 This time – Local Search Iterative improvement Hill climbing Simulated annealing

44. DCP 1172, Ch.4 44 Iterative improvement In many optimization problems, path is irrelevant; the goal state itself is the solution. Then, state space = space of “complete” configurations. Algorithm goal: - find optimal configuration (e.g., TSP), or, - find configuration satisfying constraints (e.g., n-queens) In such cases, can use iterative improvement algorithms: keep a single “current” state, and try to improve it.

45. DCP 1172, Ch.4 45 Iterative improvement example: vacuum world Simplified world: 2 locations, each may or not contain dirt, each may or not contain vacuuming agent. Goal of agent: clean up the dirt. If path does not matter, do not need to keep track of it.

46. DCP 1172, Ch.4 46 Iterative improvement example: n-queens Goal: Put n chess-game queens on an n x n board, with no two queens on the same row, column, or diagonal. Here, goal state is initially unknown but is specified by constraints that it must satisfy.

47. DCP 1172, Ch.4 47 Hill climbing (or gradient ascent/descent) Iteratively maximize “value” of current state, by replacing it by successor state that has highest value, as long as possible. ( 健忘症 )

48. DCP 1172, Ch.4 48 Question: What is the difference between this problem and our problem (finding global minima)?

49. DCP 1172, Ch.4 49 Hill climbing Note:  minimizing a “value” function v(n) is equivalent to maximizing –v(n), thus both notions are used interchangeably. Notion of “extremization”: find extrema (minima or maxima) of a value function.

50. DCP 1172, Ch.4 50 Hill climbing Problem: depending on initial state, may get stuck in local extremum.

51. DCP 1172, Ch.4 51 Minimizing energy Let’s now change the formulation of the problem a bit, so that we can employ new formalism: - let’s compare our state space to that of a physical system that is subject to natural interactions, - and let’s compare our value function to the overall potential energy E of the system. On every updating, we have  E  0

52. DCP 1172, Ch.4 52 Minimizing energy Hence the dynamics of the system tend to move E toward a minimum. We stress that there may be different such states — they are local minima. Global minimization is not guaranteed.

53. DCP 1172, Ch.4 53 Local Minima Problem Question: How do you avoid this local minima? starting point descend direction local minima global minima barrier to local search

54. DCP 1172, Ch.4 54 Consequences of the Occasional Ascents Help escaping the local optima. desired effect Might pass global optima after reaching it adverse effect (easy to avoid by keeping track of best-ever state)

55. DCP 1172, Ch.4 55 Boltzmann machines h The Boltzmann Machine of Hinton, Sejnowski, and Ackley (1984) uses simulated annealing to escape local minima. To motivate their solution, consider how one might get a ball- bearing traveling along the curve to "probably end up" in the deepest minimum. The idea is to shake the box "about h hard" — then the ball is more likely to go from D to C than from C to D. So, on average, the ball should end up in C's valley.

56. DCP 1172, Ch.4 56 Simulated annealing: basic idea From current state, pick a random successor state; If it has better value than current state, then “accept the transition,” that is, use successor state as current state; Otherwise, do not give up, but instead flip a coin and accept the transition with a given probability (that is lower as the successor is worse). So we accept to sometimes “un-optimize” the value function a little with a non-zero probability.

57. DCP 1172, Ch.4 57 Boltzmann’s statistical theory of gases In the statistical theory of gases, the gas is described not by a deterministic dynamics, but rather by the probability that it will be in different states. The 19th century physicist Ludwig Boltzmann developed a theory that included a probability distribution of temperature (i.e., every small region of the gas had the same kinetic energy). Hinton, Sejnowski and Ackley’s idea was that this distribution might also be used to describe neural interactions, where low temperature T is replaced by a small noise term T (the neural analog of random thermal motion of molecules). While their results primarily concern optimization using neural networks, the idea is more general.

58. DCP 1172, Ch.4 58 Boltzmann distribution At thermal equilibrium at temperature T, the Boltzmann distribution gives the relative probability that the system will occupy state A vs. state B as: where E(A) and E(B) are the energies associated with states A and B.

59. DCP 1172, Ch.4 59 Simulated annealing Kirkpatrick et al. 1983: Simulated annealing is a general method for making likely the escape from local minima by allowing jumps to higher energy states. The analogy here is with the process of annealing used by a craftsman in forging a sword from an alloy. He heats the metal, then slowly cools it as he hammers the blade into shape. If he cools the blade too quickly the metal will form patches of different composition; If the metal is cooled slowly while it is shaped, the constituent metals will form a uniform alloy.

60. DCP 1172, Ch.4 60 Real annealing: Sword He heats the metal, then slowly cools it as he hammers the blade into shape. If he cools the blade too quickly the metal will form patches of different composition; If the metal is cooled slowly while it is shaped, the constituent metals will form a uniform alloy.

61. DCP 1172, Ch.4 61 Simulated annealing in practice -set T -optimize for given T -lower T -repeat MDSA: Molecular Dynamics Simulated Annealing

62. DCP 1172, Ch.4 62 Simulated annealing in practice -set T -optimize for given T -lower T(see Geman & Geman, 1984) -repeat Geman & Geman (1984): if T is lowered sufficiently slowly (with respect to the number of iterations used to optimize at a given T), simulated annealing is guaranteed to find the global minimum. Caveat: this algorithm has no end (Geman & Geman’s T decrease schedule is in the 1/log of the number of iterations, so, T will never reach zero), so it may take an infinite amount of time for it to find the global minimum.

63. DCP 1172, Ch.4 63 Simulated annealing algorithm Idea: Escape local extrema by allowing “bad moves,” but gradually decrease their size and frequency. Note: goal here is to maximize E. -

64. DCP 1172, Ch.4 64 Simulated annealing algorithm Idea: Escape local extrema by allowing “bad moves,” but gradually decrease their size and frequency. Algorithm when goal is to minimize E. < - -

65. DCP 1172, Ch.4 65 Note on simulated annealing: limit cases Boltzmann distribution: accept “bad move” with  E<0 (goal is to maximize E) with probability P(  E) = exp(  E/T) If T is large:  E < 0  E/T < 0 and very small exp(  E/T) close to 1 accept bad move with high probability If T is near 0:  E < 0  E/T < 0 and very large exp(  E/T) close to 0 accept bad move with low probability

66. DCP 1172, Ch.4 66 Note on simulated annealing: limit cases Boltzmann distribution: accept “bad move” with  E<0 (goal is to maximize E) with probability P(  E) = exp(  E/T) If T is large:  E < 0  E/T < 0 and very small exp(  E/T) close to 1 accept bad move with high probability If T is near 0:  E < 0  E/T < 0 and very large exp(  E/T) close to 0 accept bad move with low probability Random walk Deterministic down-hill

67. DCP 1172, Ch.4 67 Evolutionary Computation Several different methods of evolutionary computation are now known. They all simulate natural evolution, generally by: creating a populations of individuals, evaluating their fitness, generating a new population through genetic operations, and repeating a number of times.

68. DCP 1172, Ch.4 68 Overview of Genetic Algorithm Difference with traditional search techniques: Coding of the design variables as opposed to the design variables themselves, allowing both discrete and continuous variables Works with population of designs as opposed to single design, thus reducing the risk of getting stuck at local minima Only requires the objective function value, not the derivatives. This aspect makes GAs domain-independent The fitness function defines how well each solution solves the problem objective GA is a probabilistic search method, not deterministic, making the search highly exploitative.

69. DCP 1172, Ch.4 69 Nature Genetics vs. Genetic Algorithm

70. DCP 1172, Ch.4 70 Model of Genetic Algorithm

71. DCP 1172, Ch.4 71 Evolutionary mechanism of the Genetic Algorithm Definition of GA: Genetic algorithms are a class of stochastic search algorithms based on biological evolution.

72. DCP 1172, Ch.4 72 Overview of GA stochastic, directed and highly parallel search technique based on principles of population genetics Darwin's principle of survival of the fittest: evolution is performed by genetically breeding the population of individuals over a number of generations crossover combines good information from the parents mutation prevents premature convergence

73. DCP 1172, Ch.4 73 Genetic Algorithm Flow-chart

74. DCP 1172, Ch.4 74 GA - Gene Encoding Legal Representation after Variation/Recombination 無 論 用 什 麼 方 法 來 做 gene encoding ， 在 其 執 行 reproduction, crossover, mutation 等 運 算 後， 其 出 現 的 基 因 表 示 方 式 必 須 是 合 法 的 表 示 方 式。crossovermutation 也 就 是 說， 好 的 基 因 encoding 方 法 不 但 能 夠 表 示 所 有 合 法 的 基 因 組 合， 同 時， 不 會 因 為 執 行 遺 傳 演 算 法 的 基 本 運 算， 而 演 化 出 不 合 法 的 個 體。 Gene Encoding ( Representation scheme) 11010101 0.31.22.31.903.20.44.0 Discrete Floating

75. DCP 1172, Ch.4 75 Genetic Algorithm - Operators Mutation Crossover 11001011+11011111 = 11001111 11001001 => 10001001 Target function = prediction accuracy + feature subset size

76. DCP 1172, Ch.4 76 Evaluation Function of GA

77. DCP 1172, Ch.4 77 Solving the 8-queen problem using GA

78. DCP 1172, Ch.4 78 Solving the 8-queen problem using GA (cont)

79. DCP 1172, Ch.4 79 GA Parameters - Population Size Population SizePopulation 選 擇 一 適 當 的 population size 可 以 使 得 遺 傳 演 算 法 兼 顧 效 率 和 效 用。 在 遺 傳 演 算 法 中 population 的 大 小 通 常 是 固 定 的， 在 較 大 的 population 中 每 一 代 training 的 時 間 較 長， 但 是 較 大 的 population 的 training 品 質 較 好， 較 小 的 population 則 每 一 代 training 的 時 間 較 快， 但 其 training 品 質 可 能 較 差 。

80. DCP 1172, Ch.4 80 GA Parameters – Mutation Rates Mutation Rate mutation 是 為 了 得 到 更 多 的 資 訊， 但 是 通 常 來 說 mutation rate 的 值 是 很 小 的， 因 為 太 高 的 mutation rate 反 而 會 使 chromosomes 中 的 有 用 資 訊 因 為 mutation 而 遺 失。 而 mutation rate 是可以動態的， 例 如 當 連 續 一 段 時 間 chromosomes 都 沒 什 麼 進 步 時， 便 可 以 將 mutation rate 調 大，以 期 有 更 多 變 化。

81. DCP 1172, Ch.4 81 GA Parameters – Crossover Rate Crossover Rate Crossover 主 要 是 為 了 讓 chromosomes 互 相 交 換 有 用 的 資 訊， 使 得 chromosomes 獲 得 更 高 的 fitness ， 以 期 望 能 在 下 一 代 有 更 好 的 chromosomes ， 而 獲 得 更 好 的 performance 。 但 是 有 時 為 了 讓 某 些 chromosomes 的 基 因 可 以 完 全 保 留 給 到 下 一 代， 所 以 便 有 了 crossover rate 。 其 大 小 依 照 各 問 題 而 定， 適 當 的 crossover rate 對 於 訓 練 品 質 是 非 常 重 要 的。

82. DCP 1172, Ch.4 82 Summary Best-first search => general search, where the minimum-cost nodes (according to some measure) are expanded first. Greedy search = best-first with the estimated cost to reach the goal as a heuristic measure. - Generally faster than uninformed search - not optimal - not complete. A* search = best-first with measure = path cost so far + estimated path cost to goal.  combines advantages of uniform-cost and greedy searches  complete, optimal and optimally efficient  space complexity still exponential

83. DCP 1172, Ch.4 83 Summary Time complexity of heuristic algorithms depend on quality of heuristic function.  Good heuristics can sometimes be constructed by examining the problem definition or by generalizing from experience with the problem class. Iterative improvement algorithms keep only a single state in memory.  Can get stuck in local extrema;  simulated annealing provides a way to escape local extrema, and is complete and optimal given a slow enough cooling schedule.

84. DCP 1172, Ch.4 84 Case Study - E-mail Filtering by Search E-mail basics Overview of Anti-SPAM filtering Pattern Matching Filtering by automatic learning E-mail filtering using Heuristic Search Simple Static Pattern matching White List Black List Dynamic Pattern Matching Grey List Automatic E-mail Filtering GA Bayesian Network

85. DCP 1172, Ch.4 85 Internet Bouncing server Incoming SMTP Gateway Farm Mail Spool server Outgoing SMTP Gateway Farm Transparent Firewall Model of the E-mail System

86. DCP 1172, Ch.4 86 Sample SPAM Message-20041018

87. DCP 1172, Ch.4 87 Sample SPAM Mail-20041018a (1)

88. DCP 1172, Ch.4 88 Generic Mail Filtering Generic Mail Filtering Functions F( n) = g (n) + h (n) G(n): exact value known H(n): Heuristic / estimate value Generic Mail Filtering Anti-SPAM Search Engine Reject Mail Spool Accept Pass Fail Client

89. DCP 1172, Ch.4 89 Generic Mailing Operation (1) MTA Anti-SPAM Search Engine-1 Anti-SPAM Search Engine-K Anti-SPAM Learning Engine-N Reject Mail Spool Accept Discard Account database Bounce Client Mail Delivery Account verification Milter-like API

90. DCP 1172, Ch.4 90 Generic Mail Filtering (cont) Generic Mail Filtering White List Black List Automatic SPAM Learning Reject Mail Spool Accept Grey List (1) (2) (3) (4) Pass Fail temporarily Client Update

91. DCP 1172, Ch.4 91 Sample SPAM Mail-20041018a (2)

92. DCP 1172, Ch.4 92 Sample SPAM Mail-20041018a (3)

93. DCP 1172, Ch.4 93 SPAM Mail -20041018c(0)

94. DCP 1172, Ch.4 94 SPAM Message-20041018c(1)

95. DCP 1172, Ch.4 95 SPAM Mail-20041018c(2)

96. DCP 1172, Ch.4 96 SPAM Mail Filtering Tool - Netscape Communicator