In the previous lectures, the path to the goal was the entire point of the search (e.g., finding the shortest sequence of boat trips or puzzle slides). In local search, the path is completely irrelevant; the only thing that matters is the final state or configuration itself.

Because they don't keep track of the paths they traverse, local search algorithms have two massive advantages: they use very little memory (usually a constant amount), and they can find reasonable solutions in exceptionally large or infinite (continuous) state spaces where systematic algorithms like A* would completely crash.

Here is a detailed breakdown of the four algorithms covered:

1. Hill-Climbing Search (Greedy Local Search)

Imagine the state space as a landscape of hills and valleys, where elevation represents the objective function (the "fitness" or quality of the state). Hill-climbing operates by looking at its immediate neighbors and simply moving to the highest one. It does not think ahead about where to go next.

• The Flaw: While it makes rapid progress initially because it is easy to improve a bad state, it frequently gets stuck.

• Local Maxima: It reaches a peak that is higher than its immediate neighbors, but lower than the true global maximum. Because all immediate steps go "down," the algorithm halts, permanently trapped.

• Plateaux & Ridges: It can also get completely lost wandering on flat areas (plateaux) or struggle to navigate narrow sequences of local maxima (ridges).

2. Simulated Annealing

To fix the "getting stuck" problem of Hill-Climbing, Simulated Annealing introduces controlled randomness. It is inspired by metallurgy, where metals are heated to high temperatures and gradually cooled to allow the material to reach a low-energy, stable crystalline state.

• The Mechanism: Instead of always picking the best move, it picks a random move. If the move improves the situation, it is accepted. If the move is worse, it might still be accepted based on a specific probability: $e^{\mathrm{\Delta }E/T}$.

• The Role of Temperature ($T$): The search begins with a high temperature. When $T$ is high, the probability of accepting a bad move is close to 1, allowing the algorithm to freely explore and bounce out of local maxima. As the algorithm progresses, the temperature slowly drops based on a cooling schedule. When $T$ becomes low, it strictly exploits good moves, behaving just like hill-climbing to settle on the absolute highest peak.

3. Local Beam Search

Instead of keeping just one node in memory, Local Beam Search keeps track of $k$ randomly generated states. At each step, it generates all successors for all $k$ states, pools them together into one complete list, and selects the absolute best $k$ successors to continue the search.

4. Genetic Algorithms (GAs)

A Genetic Algorithm is a variant of stochastic beam search inspired by natural selection. Instead of just modifying single states, it generates successors by combining two "parent" states.

• Population: The set of candidate solutions.

• Fitness Function: Evaluates how "good" a solution is (e.g., measuring the speed of a rabbit).

• Selection & Crossover: Parents with superior fitness are selected to combine their "genes" to produce offspring, with the hope that the combination yields even higher fitness.

• Mutation: Occasional random changes are applied to the offspring to maintain diversity in the population and prevent the algorithm from stagnating.