Test 1: gradient descent issues

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

1. 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.
2. 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.
3. Stochastic hill climbing is guaranteed to arrive at a global optimum.
4. In a continuous space, gradient descent with a fixed step size is guaranteed to converge to a local or global optimum.
5. Gradient descent finds the global optimum from any starting point if and only if the function is convex.