# Path is irrelevant the goal state itself is the

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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.

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15 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.
16 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.

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17 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.
18 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.

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19 Hill climbing Problem:  depending on initial state, may get stuck in local  extremum.
20 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   0 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.   B C A Basin of  Attraction for C D E

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21 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.
22 Simulated annealing: basic idea From current state, pick a  random  successor state;

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