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# Hill climbing algorithm terminates when

Thank you so much for publishing this! I have a final tomorrow in AI and I appreciate you unique point of view. Thanks for Sharing Quality information With us Aido Aido Robot. Nice post. Keep updating Artificial Intelligence Online Training. Hill climbing. Hill climbing is a variant of generate-and-test in which feedback from the procedure is used to help the generator decide which direction to move in the search space.

In a pure generate-and-test procedure, the test function responds with only a yes or no.

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But if the test function is augmented with a heuristic function that provides an estimate of how close a given state is to a goal state, the generate procedure can exploit it.

This is particularly nice because often the computation of a heuristic function can be done at almost no cost at the same time that the test for a solution is being performed. Hill climbing is often used when a good heuristic function is available for evaluating states but when no other useful knowledge is available.

Suppose you are in an unfamiliar city without a map and you want to get downtown. You simply aim for the tall buildings. The heuristic function is just distance between the current location and the location of the tall buildings and the desirable states are those in which this distance is minimized.

Getting downtown is an example of such a problem. For these problems, hill climbing can terminate whenever a goal state is reached. For maximization or minimization problems, such as the traveling salesman problem. In these problems, there is no a priori goal state.

For problems of this sort, it makes sense to terminate hill climbing when there is no reasonable alternative state to move to. The simplest way to implement hill climbing. Evaluate the initial state.

If it is also a goal state, then return it and quit. Otherwise, continue with the initial state as the current state. Loop until a solution is found or until there are no new operators left to applied in the current state:.

If it is a goal state, then return it and quit. If it is not a goal state but it is better than the current state, then make it the current state. If it is not better than the current state, then continue in the loop. The key difference between this algorithm and the one we gave for generate-and-test is the use of an evaluation function as a way to inject task-specific knowledge into the control process.

The puzzle of the four colored blocks.

Artificial Intelligence - Tutorial #5 - Hill Climbing Approach

To solve the problem, we first need to define a heuristic function that describes how close a particular configuration is to being a solution.On Y-axis we have taken the function which can be an objective function or cost function, and state-space on the x-axis. If the function on Y-axis is cost then, the goal of search is to find the global minimum and local minimum.

If the function of Y-axis is Objective function, then the goal of the search is to find the global maximum and local maximum. Local Maximum: Local maximum is a state which is better than its neighbor states, but there is also another state which is higher than it. Global Maximum: Global maximum is the best possible state of state space landscape. It has the highest value of objective function.

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Current state: It is a state in a landscape diagram where an agent is currently present. Flat local maximum: It is a flat space in the landscape where all the neighbor states of current states have the same value.

Simple hill climbing is the simplest way to implement a hill climbing algorithm. It only evaluates the neighbor node state at a time and selects the first one which optimizes current cost and set it as a current state. It only checks it's one successor state, and if it finds better than the current state, then move else be in the same state.

This algorithm has the following features:. The steepest-Ascent algorithm is a variation of simple hill climbing algorithm. This algorithm examines all the neighboring nodes of the current state and selects one neighbor node which is closest to the goal state.

This algorithm consumes more time as it searches for multiple neighbors. Stochastic hill climbing does not examine for all its neighbor before moving. Rather, this search algorithm selects one neighbor node at random and decides whether to choose it as a current state or examine another state.

Local Maximum: A local maximum is a peak state in the landscape which is better than each of its neighboring states, but there is another state also present which is higher than the local maximum. Solution: Backtracking technique can be a solution of the local maximum in state space landscape. Create a list of the promising path so that the algorithm can backtrack the search space and explore other paths as well.

Plateau: A plateau is the flat area of the search space in which all the neighbor states of the current state contains the same value, because of this algorithm does not find any best direction to move. A hill-climbing search might be lost in the plateau area. Solution: The solution for the plateau is to take big steps or very little steps while searching, to solve the problem. Randomly select a state which is far away from the current state so it is possible that the algorithm could find non-plateau region.

Ridges: A ridge is a special form of the local maximum. It has an area which is higher than its surrounding areas, but itself has a slope, and cannot be reached in a single move. Solution: With the use of bidirectional search, or by moving in different directions, we can improve this problem. A hill-climbing algorithm which never makes a move towards a lower value guaranteed to be incomplete because it can get stuck on a local maximum.

And if algorithm applies a random walk, by moving a successor, then it may complete but not efficient. Simulated Annealing is an algorithm which yields both efficiency and completeness. In mechanical term Annealing is a process of hardening a metal or glass to a high temperature then cooling gradually, so this allows the metal to reach a low-energy crystalline state.

The same process is used in simulated annealing in which the algorithm picks a random move, instead of picking the best move. If the random move improves the state, then it follows the same path.Hill climbing search is a local search problem.

It is based on the heuristic search technique where the person who is climbing up on the hill estimates the direction which will lead him to the highest peak. The topographical regions shown in the figure can be defined as:. Simple hill climbing is the simplest technique to climb a hill. The task is to reach the highest peak of the mountain.

If he finds his next step better than the previous one, he continues to move else remain in the same state. This search focus only on his previous and next step. Steepest-ascent hill climbing is different from simple hill climbing search. Unlike simple hill climbing search, It considers all the successive nodes, compares them, and choose the node which is closest to the solution. Steepest hill climbing search is similar to best-first search because it focuses on each node instead of one.

Note: Both simple, as well as steepest-ascent hill climbing search, fails when there is no closer node. Stochastic hill climbing does not focus on all the nodes. It selects one node at random and decides whether it should be expanded or search for a better one. Random-restart algorithm is based on try and try strategy. It iteratively searches the node and selects the best one at each step until the goal is not found.

The success depends most commonly on the shape of the hill. If there are few plateaus, local maxima, and ridges, it becomes easy to reach the destination.

Hill climbing algorithm is a fast and furious approach. It finds the solution state rapidly because it is quite easy to improve a bad state. But, there are following limitations of this search:. Simulated annealing is similar to the hill climbing algorithm. It works on the current situation. It picks a random move instead of picking the best move.

If the move leads to the improvement of the current situation, it is always accepted as a step towards the solution state, else it accepts the move having a probability less than 1. This search technique was first used in to solve VLSI layout problems. It is also applied for factory scheduling and other large optimization tasks.

Local beam search is quite different from random-restart search. It keeps track of k states instead of just one. It selects k randomly generated states, and expand them at each step. If any state is a goal state, the search stops with success. Else it selects the best k successors from the complete list and repeats the same process.

In random-restart search where each search process runs independently, but in local beam search, the necessary information is shared between the parallel search processes. Note: A variant of Local Beam Search is Stochastic Beam Search which selects k successors at random rather than choosing the best k successors.

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The topographical regions shown in the figure can be defined as: Global Maximum: It is the highest point on the hill, which is the goal state. Local Maximum: It is the peak higher than all other peaks but lower than the global maximum.Hill climbing is a technique for certain classes of optimization problems.

The idea is to start with a sub-optimal solution to a problem i. One of the most popular hill-climbing problems is the network flow problem. Although network flow may sound somewhat specific it is important because it has high expressive power: for example, many algorithmic problems encountered in practice can actually be considered special cases of network flow.

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After covering a simple example of the hill-climbing approach for a numerical problem we cover network flow and then present examples of applications of network flow. This new solution can be the starting point for applying the same procedure again.

Thus, in general a better approximation can be constructed by repeatedly applying. As shown in the illustration, this is nothing else but the construction of the zero from the tangent at the initial guessing point.

This would give the extremal positions of the function, its maxima and minima. Starting Newton's method close enough to a maximum this way, we climb the hill.

The net present value function is a function of time, an interest rate, and a series of cash flows. A related function is Internal Rate of Return. Instead of regarding continuous functions, the hill-climbing method can also be applied to discrete networks.

Suppose you have a directed graph possibly with cycles with one vertex labeled as the source and another vertex labeled as the destination or the "sink". The source vertex only has edges coming out of it, with no edges going into it. Similarly, the destination vertex only has edges going into it, with no edges coming out of it. We can assume that the graph is fully connected with no dead-ends; i.

We assign a "capacity" to each edge, and initially we'll consider only integral-valued capacities. The following graph meets our requirements, where "s" is the source and "t" is the destination:. We'd like now to imagine that we have some series of inputs arriving at the source that we want to carry on the edges over to the sink.

The number of units we can send on an edge at a time must be less than or equal to the edge's capacity. You can think of the vertices as cities and the edges as roads between the cities and we want to send as many cars from the source city to the destination city as possible.

The constraint is that we cannot send more cars down a road than its capacity can handle. After using this path, we can compute the residual graph by subtracting 1 from the capacity of each edge:. To be maximal means that there is no other flow assignment that obeys the constraints that would have a higher value. The traffic example can describe what the four flow constraints mean:.

The following algorithm computes the maximal flow for a given graph with non-negative capacities. What the algorithm does can be easy to understand, but it's non-trivial to show that it terminates and provides an optimal solution.

To do: explain, hopefully using pictures, what the algorithm is doing.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Artificial Intelligence Stack Exchange is a question and answer site for people interested in conceptual questions about life and challenges in a world where "cognitive" functions can be mimicked in purely digital environment.

It only takes a minute to sign up. Stochastic Hill Climbing generally performs worse than Steepest Hill Climbingbut what are the cases in which the former performs better? The steepest hill climbing algorithms works well for convex optimization.

However, real world problems are typically of the non-convex optimization type: there are multiple peaks. In such cases, when this algorithm starts at a random solution, the likelihood of it reaching one of the local peaks, instead of the global peak, is high.

Improvements like Simulated Annealing ameliorate this issue by allowing the algorithm to move away from a local peak, and thereby increasing the likelihood that it will find the global peak. Obviously, for a simple problem with only one peak, the steepest hill climbing is always better.

It can also use early stopping if a global peak is found. In comparison, a simulated annealing algorithm would actually jump away from a global peak, return back and jump away again. This would repeat until its cooled down enough or a certain preset number of iterations have completed.

Real world problems deal with noisy and missing data. A stochastic hill climbing approach, while slower, is more robust to these issues, and the optimization routine has a higher likelihood of reaching the global peak in comparison to the steepest hill climbing algorithm.

Epilogue: This is a good question which raises a persistent question when designing a solution or choosing between various algorithms: the performance-computational cost trade-off. As you might have suspected, the answer is always: it depends on your algorithm's priorities.

If it is part of some online learning system that is operating on a batch of data, then there is a strong time constraint, but weak performance constraint next batches of data will correct for erroneous bias introduced by first batch of data.

On the other hand, if it is an offline learning task with the entire available data in hand, then performance is the main constraint, and the stochastic approaches are advisable. Let's begin with some definitions first. Hill-climbing is a search algorithm simply runs a loop and continuously moves in the direction of increasing value-that is, uphill. The loop terminates when it reaches a peak and no neighbour has a higher value.

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Stochastic hill climbinga variant of hill-climbing, chooses a random from among the uphill moves. The probability of selection can vary with the steepness of the uphill move. Two well-known methods are: First-choice hill climbing: generates successors randomly until one is generated that is better than the current state.

Random-restart hill climbing: Works on the philosophy of "If you don't succeed, try, try again". Now to your answer. Stochastic hill climbing can actually perform better in many cases.

Consider the following case. The image shows state-space landscape.GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. If nothing happens, download GitHub Desktop and try again. If nothing happens, download Xcode and try again. If nothing happens, download the GitHub extension for Visual Studio and try again.

Here's the implementation of Hill Climbing Algorithm to find the minimum value of Ackley's function. Initial position was randomly chosen, then the next positions were using the following equations. At every point, the value of the function is calculated and checked it with the minimum value which is updated whenever it finds a lesser value.

## Algorithms/Hill Climbing

The iteration is terminated when a value minimum than the minimum is not found till now in continuous steps. This process is repeated times and minimum value is found in every iteration.

We can clearly see that, Hill Climbing Algorithm gets struck at a local minimum rather than reaching Global Optimum every time. In this part, Evolutionary algorithm is used to find the minimum value of Ackley's function starting from a random point.

As the algorithm used was evolutionary, it has the following parts, reproduction, mutation, evalution and selection. Here a total of 20 random points are selected at first, then for every point, three other points are selected which are different from the former and then a mutation is done using.

A random number is generated and compared with Cross Over Probability to check if it is eligible for crossover.

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If it is, then the donor vector replaces the original vector and then the Ackley's function is calculated for the new vector. At every step, we compare the minimum value of the iteration with the minimum found till now, and the program is terminated when the minimum doesn't change till steps.

Using this algorithm we can observe that, we reach the global optimal value So it is clear that it doesn't get struck at local optima's now and then, as it is continuously learning from the previous values as we are generating values from the parents. Skip to content. Dismiss Join GitHub today GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together.

Sign up. Python Branch: master. Find file. Sign in Sign up. Go back. Launching Xcode If nothing happens, download Xcode and try again. Latest commit Fetching latest commitâ€¦. Differential Evolution Algorithm In this part, Evolutionary algorithm is used to find the minimum value of Ackley's function starting from a random point. You signed in with another tab or window. Reload to refresh your session.

You signed out in another tab or window.Hill Climbing is a heuristic search used for mathematical optimization problems in the field of Artificial Intelligence. Given a large set of inputs and a good heuristic function, it tries to find a sufficiently good solution to the problem. This solution may not be the global optimal maximum. Generate possible solutions. Test to see if this is the expected solution. If the solution has been found quit else go to step 1.

Hence we call Hill climbing as a variant of generate and test algorithm as it takes the feedback from the test procedure. Then this feedback is utilized by the generator in deciding the next move in search space. Uses the Greedy approach : At any point in state space, the search moves in that direction only which optimizes the cost of function with the hope of finding the optimal solution at the end.

Types of Hill Climbing. Step 1 : Evaluate the initial state. If it is a goal state then stop and return success. Otherwise, make initial state as current state. Step 2 : Loop until the solution state is found or there are no new operators present which can be applied to the current state. If the current state is a goal state, then stop and return success.

If it is better than the current state, then make it current state and proceed further. If it is not better than the current state, then continue in the loop until a solution is found. Step 3 : Exit. If it is goal state then exit else make the current state as initial state Step 2 : Repeat these steps until a solution is found or current state does not change.

State space diagram is a graphical representation of the set of states our search algorithm can reach vs the value of our objective function the function which we wish to maximize. X-axis : denotes the state space ie states or configuration our algorithm may reach.

Y-axis : denotes the values of objective function corresponding to a particular state. The best solution will be that state space where objective function has maximum value global maximum.

To overcome plateaus : Make a big jump. Randomly select a state far away from the current state. Chances are that we will land at a non-plateau region Ridge : Any point on a ridge can look like peak because movement in all possible directions is downward. Hence the algorithm stops when it reaches this state.