Heuristic Search & Local Search
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1 Heuristic Search & Local Search CS171 Week 3 Discussion July 7, 2016
2 Consider the following graph, with initial state S and goal G, and the heuristic function h. Fill in the form using greedy best-first search. Indicate the f value in parenthesis after the node label, e.g. A(8). Assume the algorithm does not re-visit each node. Note: Normally there shouldn t be paths pointing out from the goal node. You may choose to ignore it in this problem.
3 Consider the following graph, with initial state S and goal G, and the heuristic function h. Fill in the form using greedy best-first search. Indicate the f value in parenthesis after the node label, e.g. A(8). Assume the algorithm does not re-visit each node. S(8) C(3), B(7) S(8)
4 Greedy best-first search Recall: f(n) = h(n) S(8) C(3), B(7) S(8)
5 Greedy best-first search S(8) C(3), B(7) S C(3) B(7), H(100) S, C
6 Greedy best-first search S(8) C(3), B(7) S C(3) B(7), H(100) S, C B(7) H(100), D(1), E(4) S, C, B
7 Greedy best-first search S(8) C(3), B(7) S C(3) B(7), H(100) S, C B(7) H(100), D(1), E(4) S, C, B D(1) H(100), E(4), G(0), F(6) S, C, B, D
8 Greedy best-first search S(8) C(3), B(7) S C(3) B(7), H(100) S, C B(7) H(100), D(1), E(4) S, C, B D(1) H(100), E(4), G(0), F(6) S, C, B, D G(0)
9 A* search Recall: f(n) = g(n) + h(n) S(8) B(8)
10 A* search Recall: f(n) = g(n) + h(n) S(8) B(8), C(4) S
11 A* search Recall: f(n) = g(n) + h(n) S(8) B(8), C(4) S C(4) B(8), H(102) S, C
12 A* search Recall: f(n) = g(n) + h(n) S(8) B(8), C(4) S C(4) B(8), H(102) S, C B(8) H(102), D(11)
13 A* search Recall: f(n) = g(n) + h(n) S(8) B(8), C(4) S C(4) B(8), H(102) S, C B(8) H(102), D(11), E(6) S, C, B
14 A* search Recall: f(n) = g(n) + h(n) S(8) B(8), C(4) S C(4) B(8), H(102) S, C B(8) H(102), D(11), E(6) S, C, B E(6) H(102), D(11), F(9) S, C, B, E
15 A* search Recall: f(n) = g(n) + h(n) S(8) B(8), C(4) S C(4) B(8), H(102) S, C B(8) H(102), D(11), E(6) S, C, B E(6) H(102), D(11), F(9) S, C, B, E F(9) H(102), D(11) S, C, B, E, F
16 A* search Recall: f(n) = g(n) + h(n) S(8) B(8), C(4) S C(4) B(8), H(102) S, C B(8) H(102), D(11), E(6) S, C, B E(6) H(102), D(11), F(9) S, C, B, E F(9) H(102), D(11) S, C, B, E, F D(11) H(102), G(11) S, C, B, E, F, D
17 A* search Recall: f(n) = g(n) + h(n) S(8) B(8), C(4) S C(4) B(8), H(102) S, C B(8) H(102), D(11), E(6) S, C, B E(6) H(102), D(11), F(9) S, C, B, E F(9) H(102), D(11) S, C, B, E, F D(11) H(102), G(11) S, C, B, E, F, D G(11)
18 Is this heuristic admissible? Explain why or why not.
19 Is this heuristic admissible? Explain why or why not. Recall: An admissible heuristic is one that never overestimates the cost to reach the goal, which is: h(n) h*(n), for all n, where h*(n) is the true cost to reach the goal state from n.
20 Is this heuristic admissible? Explain why or why not. n h(n) h*(n) S 8 11 B 7 10 C 3 D 1 1 E 4 F 6 G 0 0 H 100 Yes, it is admissible.
21 Gradient Descent (Ascent): Review An iterative method to find the max/min* values *: Local max/min; sensitive to starting point Intuition: At each step, walk towards the steepest descent/ascent direction. Used widely in practice Steps: 1. Pick a starting point 2. Repeat a. Calculate the gradient of the function at current point b. Move a step towards the direction of the gradient to reduce cost function 3. Stop when is small enough
22 Local Search: Exercise Use gradient descent/ascent method to find an optimum, which point will be returned? D E G Starting at X to find a minimum returns Starting at X to find a maximum returns B X Y Z Starting at Y to find a minimum returns Starting at Z to find a maximum returns A C F
23 Local Search: Exercise Use gradient descent/ascent method to find an optimum, which point will be returned? Cost function D E G C Starting at X to find a minimum returns Starting at X to find a maximum returns D Starting at Y to find a minimum returns Y B X Y Z Starting at Z to find a maximum returns G In standard gradient descent method, the stopping condition is the gradient of the cost function being small. So if we start at Y, the gradient of the cost function = 0. It will just return Y. A C F You could design more clever methods to deal with the plateau.
24 Example: Gradient Descent in Linear Regression Recall: Linear Regression (See lecture note 30 June - Lecture 1) This example illustrates how gradient descent can be used in machine learning problems involving finding optimum values. In fact, gradient descent is a practical approach to solving many complex machine learning problems. It is ok if you are not familiar with the math.
25 Example: Gradient Descent in Linear Regression At each iteration, update the parameter, where is the step size.
26 Simulated Annealing: Illustration
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