📗 For some problems, every state is a solution, only some states are better than other states specified by a cost function (sometimes score or reward): Wikipedia.
📗 The search strategy will go from state to state, but the path between states is not important.
📗 Local search assumes similar (nearby) states have similar costs, and search through the state space by iteratively improving the costs to find an optimal state.
📗 The successor states are called neighbors (or move set).
📗 Hill climbing is the discrete version of gradient descent: Wikipedia.
➩ It starts at a random state.
➩ Move to the best neighbor (successor) state.
➩ Stop when all neighbors are no better than the current state (local minimum).
📗 Random restarts can be used to pick multiple random initial states and find the best local minimum (similar to neural network training).
📗 If there are too many neighbors, first choice hill climbing randomly generates neighbors until a better neighbor is found.
TopHat Discussion
ID:
📗 [1 points] Given the initial state (red points) \(i\), the neighbors are \(i - 1\) and \(i + 1\), and is used for imization of the score, which solution will be found? Click on the red point to restart, and click anywhere else to move to the next iteration.
Temperature: \(T\) = , \(dT\) = .
📗 Answer: .
TopHat Quiz
(Past Exam Question) ID:
📗 [3 points] Given the scores in the following table, if hill-climbing (valley-finding) is used, how many states will lead to the global imum? Note: the neighbors of state \(i\) are states \(i - 1\) and \(i + 1\) (if they exist).
📗 Simulated annealing uses a process similar to heating solids (heating and slow cooling to toughen and reduce brittleness): Wikipedia.
➩ Each time, a random neighbor is generated.
➩ If the neighbor has a lower cost, move to the neighbor.
➩ If the neighbor has a higher cost, move to the neighbor with a small probability: \(p = e^{- \dfrac{\left| f\left(s'\right) - f\left(s\right) \right|}{T\left(t\right)}}\), where \(f\) is the cost and \(T\left(t\right)\) is the temperature and decreasing in \(t\).
➩ Stop until bored.
📗 Simulated annealing is a version of Metropolis-Hastings algorithm: Wikipedia.
Example
📗 The traveling salesman problem is often solved by simulated annealing: Link.
📗 The temperature function should be decreasing over time. They can change arithmetically or geometrically.
➩ Arithmetic sequence: for example, \(T\left(t + 1\right) = \displaystyle\max\left\{T\left(t\right) - 1, 1\right\}\).
➩ Geometric sequence: for example, \(T\left(t + 1\right) = 0.9 T\left(t\right)\).
📗 When the temperature is high: almost always accept any state.
📗 When the temperature is low: first choice hill climbing.
TopHat Discussion
ID:
📗 [1 points] Given the initial state (red points) \(i\), the neighbors are \(i - 1\) and \(i + 1\), and is used for imization of the score, which solution will be found? Click on the red point to restart, and click anywhere else to move to the next iteration.
Temperature: \(T\) = , \(dT\) = .
📗 Answer: .
testhcd,hls,sadq
📗 Notes and code adapted from the course taught by Professors Jerry Zhu, Yingyu Liang, and Charles Dyer.
📗 Content from note blocks marked "optional" and content from Wikipedia and other demo links are helpful for understanding the materials, but will not be explicitly tested on the exams.
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