Analyzing Local Search Algorithms

Analyzing Local Search Algorithms

For each of the following assertions, support your answer with examples or counterexamples:

For any local-search problem, hill-climbing will return a global optimum if the algorithm is run starting at any state that is a neighbor of a neighbor of a globally optimal state.
In a nondeterministic, partially observable problem, improving an agent’s transition model (i.e., eliminating spurious outcomes that never actually occur) will never increase the size of the agent’s belief states.
Stochastic hill climbing is guaranteed to arrive at a global optimum.
In a continuous space, gradient descent with a fixed step size is guaranteed to converge to a local or global optimum.
Gradient descent finds the global optimum from any starting point if and only if the function is convex.

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Analysis: This is false.
- Hill-climbing moves towards states that improve the objective function. However, it can get stuck in local optima, plateaus, or ridges.
Counterexample:
- Consider a search space where a globally optimal state is surrounded by states of lower values (a “valley”). If the algorithm starts in a state outside this valley, it will likely converge to a local optimum instead of the global one.
- Example: Imagine a terrain where the globally optimal state (highest peak) is surrounded by steep descents leading to other local peaks. Starting in a neighbor of the valley may not lead to the global peak.

Analysis: This is true.
- Improving the transition model eliminates impossible outcomes, which reduces the uncertainty in the system. As a result, the belief state may shrink or remain the same but will not increase.
Example:
- Suppose an agent in a grid world is uncertain about its position. If the transition model erroneously includes transitions to inaccessible areas, the belief state is larger. Removing suc…